A New Homotopy Transformation Method for Solving the Fuzzy Fractional Black–Scholes European Option Pricing Equations under the Concept of Granular Differentiability

Abstract

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper.

1. Introduction

Option pricing is a major subject in financial investing and a significant component of contemporary financial theory research. Not only has the development of option pricing theory facilitated financial derivative innovation, but also the flow and growth of financial markets. How to reasonably price options is a serious problem facing investors during the forming and developing of international derivative financial markets. The Black–Scholes model [1] is one of the most influential mathematical models in the financial sector. In 1973, Fisher Black and Myron Scholes established an option pricing model by using non-arbitrage pricing theory. Soon after, the Chicago Board Options Exchange applied the Black–Scholes model into practical problems by programing it with computer. Their study results laid new foundation for option pricing theory and also gave significant instructions on the application and management of financial derivative instruments, for which they received th Nobel Memorial Prize in Economic Sciences in 1997. By using computers and advanced communication technology, it is possible to express complex option pricing equation in a function.

The Black–Scholes equation is a partial differential equation depicting options’ price changes. However, this model needs to be improved due to the huge differences between its ideal hypothesis and the actual financial markets [2,3,4]. In the perspective of mathematics, a fractional Black–Scholes equation is more accurate than an integer order equation to reflect the middle process of price changes. In finance, some researchers believe that asset prices have long-term relativity [5], which indicates that the price of an item at a specific point is related to both the present price and the price from a long time ago. Decision makers can make decisions based on past market experiences [6]. A fractional Black–Scholes equation can depict the long-term relativity in behavioral finance [7,8]. In recent decades, fractional derivatives have drawn the attention of mathematicians and physicists [9]. Fractional derivatives have over 20 different kinds of definitions, including the most commonly used ones, such as Riemann–Liouville’s, Caputo’s, Jumare’s, Conformable’s, etc. The theories of fractional calculus have been widely studied and applied in many fields, including economics, sociology, computer science, biology, material science, etc. [10]. Many scholars applied the fractional calculus model to the modeling of natural phenomenon. Fractional calculus has its own characteristics, such as memorizing and inheriting. Therefore, fractional calculus equations are highly suitable for describing complex systems with heredity and memorization features. Examples include the anthrax disease model in animals [11], the mathematical model for COVID-19 transmission [12,13], the fractional-order epidemic model for childhood diseases [14] and hearing loss in children caused by the mumps virus model [15]. However, it is extremely difficult to find the accurate result of fractional differential equations (FDE). Thus, people turned to approximate analytical solutions for solving FDE. Therefore, some approximation methods have been applied to solve FDE.

The uncertain factors in the actual financial market are inevitable, while fuzzy analysis can draw the uncertain phenomenon into an algorithmic system. In reality, massive problems provide uncertainties from different angles. Integrating the fuzzy phenomenon into FDE can result in fuzzy FDE. This kind of mixed equation can describe option pricing models under the uncertain circumstances of financial markets. The following are the study results of solving fuzzy derivative equations: Puri and Ralescu introduced Hukuhara differentiability (H-differentiability) of fuzzy functions in 1983; Bede and Stefanini came up with general Hukuhara differentiability (gH-differentiability) of fuzzy functions in 2013. Moreover, FDE would lead to two different geometric-meaning types of solutions under this differentiability. Mazandarani and others [16,17,18] introduced how to gain fuzzy-number derivatives by using granular differentiability (gr-differentiability) and horizontal membership function and how to solve fuzzy differential equations in 2018. In the research of fuzzy differential equations, gr-differentiability has the following advantages [18]:

• An FDE has only one solution so as to avoid the multiplicity of solutions drawback.
• It avoids the doubling property problem for solving each FDE, which means solving just one individual differential equation.
• This method does not lead to the unnatural behavior in modeling (UBM) phenomenon.

The Homotopy Perturbation Method (HPM) [19] is a new type of asymptotic numerical algorithm developed with the foundation of artificial parameter perturbation method in recent years. HPM, which combines homotopy theory and the perturbation method, is proposed by He [20]. Because homotopy perturbation is utilized to solve nonlinear differential equations and integral equations, it is considered to be very accurate and not overly difficult to calculate. Currently, HPM has been widely applied in solving all kinds of fractional partial differential equations and fractional integral equations [21,22,23,24,25,26]. The main character of fractional derivatives is their non-limitation. Relatively, the numerical simulation process of fractional equations has a massive amount of calculations. However, the computational efficiency of the option pricing model has an impact on the actual application effect, and a longer calculation time will lead to the deviation of option pricing results [27]. We added fractional Elzaki transformation, an advanced Laplace and Sumudu transform, into the HPM algorithm, then combined HPM and Elzaki transform together [28,29,30]. This approach makes the calculation more uncomplicated and more efficient.

Actual option pricing is affected by the following factors, such as the risk preference of investors, the market environment, and regional economic policies [31]. Due to the uncertainty of these factors, we describe these factors by using fuzzy information, while some use random information. The fuzzy options could better consider the above uncertain factors at a comprehensive level to improve option pricing accuracy. Thus, studying fuzzy option pricing is meaningful.

The remainder of this article is laid out as follows: In Section 2, some fundamental definitions involved in fractional derivatives and fuzzy differentiation are introduced. In Section 3, the ETHPM steps are presented. In Section 4, some examples are provided to illustrate the application results. In Section 5, the conclusion of our research is presented.

2. Preliminaries

This section presents some necessary definitions and theorems that will be used later. Throughout this paper, the set of all real numbers is denoted by $\mathbb{R}$, the set of complex numbers is denoted by $\mathbb{C}$, and the set of all the fuzzy numbers on $\mathbb{R}$ by $\mathbb{E}$.

Definition 1

([32]). A real function $f\left(x\right),$ $x>0,$ is said to be in the space $C_{\mu },$ $\mu \in \mathbb{R}$ if there exists a real number $p>\mu$, such that $f\left(x\right)=x^{p}h\left(x\right)$, where $h\left(x\right)\in [0,\infty )$ and it is said to be in space $C_{\mu }^m$ if $f^{\left(m\right)}\in C_{\mu },$ $m\in N$.

Definition 2

([32]). The Riemann–Liouville fractional integral operator of order $\alpha >0$, of a function $f\in C_m$, is defined as

$$\begin{array}{c}\hfill J^{\alpha }f\left(t\right)=\left\{\begin{array}{cc}\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t\frac{f\left(s\right)}{{(t-s)}^{1-\alpha }}ds=\frac{1}{\Gamma \left(\alpha \right)}{t}^{\alpha -1}\ast f\left(t\right)\hfill & \alpha >0,t>0,\hfill \\ f\left(t\right)\hfill & \alpha =0,\hfill \end{array}\right.\end{array}$$

(1)

where $t^{\alpha -1}\ast f\left(t\right)$ is the convolution product of $t^{\alpha -1}$ and $f\left(t\right)$.

For the Riemann–Liouville fractional integral, we have

$$\begin{array}{c}\hfill \begin{array}{c}1.J^{\alpha }{t}^{\beta }=\frac{\Gamma (\beta +1)}{\Gamma (\beta +\alpha +1)}{t}^{\alpha +\beta },\beta >-1,\hfill \\ 2.J^{\alpha }(\lambda f\left(t\right)+\mu g\left(t\right))=\lambda J^{\alpha }f\left(t\right)+\mu J^{\alpha }g\left(t\right),\hfill \end{array}\end{array}$$

where $\lambda$ and $\mu$ are real constants.

This paper utilizes the Caputo derivative condition, which is defined as follows.

Definition 3

([33,34]). Let $f\left(t\right):[0,+\infty )\to \mathbb{R}$ be a function, and n be the upper positive integer of $\alpha (\alpha >0)$. The Caputo fractional derivative is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}& {\int }_0^t\frac{f^{\left(n\right)}\left(s\right)}{{(t-s)}^{\alpha +1-n}}ds,\hfill \\ & n-1<\alpha \le n,n\in N.\hfill \end{array}\end{array}$$

(2)

For the Caputo derivative, we have

$$\begin{array}{c}\hfill \begin{array}{c}1.D^{\alpha }{J}^{\alpha }f\left(t\right)=f\left(t\right),\hfill \\ 2.J^{\alpha }{D}^{\alpha }f\left(t\right)=f\left(t\right)-\sum _{i=0}^{n-1}{y}^{\left(i\right)}\left(0\right)\frac{t^i}{i!},\hfill \\ 3.D^{\alpha }{t}^{\beta }=\left\{\begin{array}{cc}\frac{\mathsf{\Gamma }(\beta +1)}{\mathsf{\Gamma }(\beta +1-\alpha )}{t}^{\beta -\alpha }\hfill & \beta \ge \alpha \hfill \\ 0\hfill & \beta <\alpha \hfill \end{array}\right.,\hfill \\ 4.D^{\alpha }c=0,\hfill \\ 5.D^{\alpha }(\lambda f\left(t\right)+\mu g\left(t\right))=\lambda D^{\alpha }f\left(t\right)+\mu D^{\alpha }g\left(t\right),\hfill \end{array}\end{array}$$

where $\lambda$, $\mu$ and c are real constants.

The derivative of conformable fractional derivative is defined as follows.

Definition 4

([35]). Given a function $f:[0,\infty )\to \mathbb{R}$. Then the conformable derivative of f order $\alpha \in (0,1]$ is defined by

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\left(f\right)\left(t\right)=\underset{\epsilon \to 0}{lim}\frac{f\left(t+\epsilon t^{1-\alpha }\right)-f\left(t\right)}{\epsilon },\end{array}$$

(3)

for all $t>0$.

Definition 5

([35,36]). Let f be n-times differentiable at t. Then the conformable derivative of f order α is defined as

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\left(f\left(t\right)\right)=\underset{\epsilon \to 0}{lim}\frac{f^{(\lceil \alpha \rceil -1)}\left(t+\epsilon t^{(\lceil \alpha \rceil -\alpha )}\right)-f^{(\lceil \alpha \rceil -1)}\left(t\right)}{\epsilon },\end{array}$$

(4)

for all $t>0,\alpha \in (n,n+1]$. Here $\lceil \alpha \rceil$ is the smallest integer greater than or equal to α.

Theorem 1

([35]). Let f be n-times differentiable at t. Then

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\left(f\left(t\right)\right)=t^{\lceil \alpha \rceil -\alpha }{f}^{\lceil \alpha \rceil }\left(t\right),\end{array}$$

(5)

for all $t>0,\alpha \in (n,n+1]$.

Definition 6

([37]). The single parameter and the two parameters variants of the Mittag–Leffler function are denoted by $E_{\alpha }\left(t\right)$ and $E_{\alpha ,\beta }\left(t\right)$, respectively, which are relevant for their connection with fractional calculus, and are defined as

$$\begin{array}{c}\hfill E_{\alpha }\left(t\right)=\sum _{j=0}^{\infty }\frac{t^j}{\mathsf{\Gamma }(\alpha j+1)},\alpha >0,t\in \mathbb{C},\end{array}$$

(6)

$$\begin{array}{c}\hfill E_{\alpha ,\beta }\left(t\right)=\sum _{j=0}^{\infty }\frac{t^j}{\mathsf{\Gamma }(\alpha j+\beta )},\alpha ,\beta >0,t\in \mathbb{C}.\end{array}$$

(7)

Based on the HMF, Mazandarani proposed the concepts of relative distance measure (RDM) and granular differentiability.

Definition 7

([16,17]). For a fuzzy number $\tilde{u}:[a,b]\to [0,1]$ with parametric form ${\left[u\right]}^{\mu }=\left[{\underline{u}}^{\mu },{\overline{u}}^{\mu }\right],$ ${\underline{u}}^{\mu }$ is a bounded non-decreasing left continuous function in (0,1], and it is right continuous at $\mu =0$, ${\overline{u}}^{\mu }$ is a bounded non-increasing left continuous function in (0,1], and it is right continuous at $\mu =0$, ${\underline{u}}^{\mu }\le {\overline{u}}^{\mu }$. The HMF $u^{\mathrm{gr}}:[0,1]\times [0,1]\to [a,b]$ with $x=u^{gr}\left(\mu ,{\alpha }_u\right)={\underline{u}}^{\mu }+\left({\overline{u}}^{\mu }-{\underline{u}}^{\mu }\right){\alpha }_u$ stands for the granule of information included in $x\in [a,b],\mu \in [0,1]$ is the membership degree of x in $\tilde{u}\left(x\right)$, ${\alpha }_u\in [0,1]$ is called RDM.

Note 1 [16,17]. We can also denote the HMF of $\tilde{u}\in \mathbb{E}$ by $\mathcal{H}\left(\tilde{u}\right)\triangleq u^{\mathrm{gr}}\left(\mu ,{\alpha }_u\right)$, In particular, if $\tilde{u}=(a,b,c),$ $a\le b\le c$ is a triangular fuzzy number, then the HMF of $\tilde{u}$ is $\mathcal{H}\left(\tilde{u}\right)=a+(b-a)\mu +(1-\mu )(c-a){\alpha }_u$.

Note 2 [16]. The $\mu$ -level sets of $\tilde{u}\in \mathbb{E}$ which are the span of the information granule can be obtained by using

$$\begin{array}{c}\hfill {\mathcal{H}}^{-1}\left(u^{gr}\left(\mu ,{\alpha }_u\right)\right)={\left[\tilde{u}\right]}^{\mu }=\left[\underset{\beta \ge \mu }{inf}\underset{{\alpha }_u}{min}{u}^{gr}\left(\beta ,{\alpha }_u\right),\underset{\beta \ge \mu }{sup}\underset{{\alpha }_u}{max}{u}^{gr}\left(\beta ,{\alpha }_u\right)\right].\end{array}$$

(8)

In Figure 1, the picture shows the triangular fuzzy number $\tilde{u}=(1,2,9)$.

Definition 8

([16]). The fuzzy-valued function $\tilde{f}:[a,b]\to \mathbb{E}$ is said to be granular differentiable (gr-differentiable for short) at a point $t_0\in [a,b]$ if there exists an element $\frac{d^{gr}\tilde{f}\left(t_0\right)}{dt}\in \mathbb{E}$ such that the following limit

$$\begin{array}{c}\hfill \underset{\mathsf{\Delta }t\to 0}{lim}\frac{\tilde{f}\left(t_0+\mathsf{\Delta }t\right){\ominus }^{\mathrm{gr}}\tilde{f}\left(t_0\right)}{\mathsf{\Delta }t}=\frac{d^{gr}\tilde{f}\left(t_0\right)}{dt},\end{array}$$

(9)

exists for $\mathsf{\Delta }t$ sufficiently near 0. $\frac{d^{gr}\tilde{f}\left(t_0\right)}{dt}$ is called gr-derivative of fuzzy-valued function $\tilde{f}$ at the point $t_0$. If the gr-derivative exists for all points $t\in [a,b]\subseteq \mathbb{R}$, we say that $\tilde{f}$ is gr-differentiable on $[a,b]\subseteq \mathbb{R}$. We use $C^1(U,\mathbb{E})$ to denote the space of all continuously gr-differentiable fuzzy-valued functions on $U\subseteq \mathbb{R}$.

Theorem 2

([16]). The fuzzy function $\tilde{f}:[a,b]\subseteq \mathbb{R}\to \mathbb{E}$ is gr-differentiable at the point $t\in [a,b]$ if and only if its HMF is differentiable with respect to t at that point. Moreover,

$$\begin{array}{c}\hfill \mathcal{H}\left(\frac{d\tilde{f}\left(t\right)}{dt}\right)=\frac{\partial f^{gr}\left(t,\mu ,{\alpha }_f\right)}{\partial t}.\end{array}$$

(10)

The definition and properties of the Elzaki transform are as follows.

Definition 9

([29]). Elzaki transform, a new transform, defining for function of exponential order we consider functions in the set A, defined by

$$\begin{array}{c}\hfill A=f\left(t\right):\exists M,k_1,k_2>0,\left|f\left(t\right)\right|<M{e}^{\frac{\left|t\right|}{k_j}},ift\in {(-1)}^j\times [0,\infty ).\end{array}$$

(11)

For a given function in the set, the constant G must be finite number, $k_1$, $k_2$ may be finite or infinite. The Elzaki transform which is defined by the integral equation.

$$\begin{array}{c}\hfill E\left[f\left(t\right)\right]=T\left(v\right)=v{\int }_0^{\infty }f\left(t\right)e^{\frac{-t}{v}}dt,t\ge 0,k_1\le v\le k_2.\end{array}$$

(12)

The following results can be obtained from the definition and simple calculations.

• $E\left[t^n\right]=n!v^{n+2},$
• $E\left[f^{\prime }\left(t\right)\right]=\frac{T\left(v\right)}{v}-vf\left(0\right),$
• $E\left[f^{″}\left(t\right)\right]=\frac{T\left(v\right)}{v^2}-f\left(0\right)-v{f}^{\prime }\left(0\right),$
• $E\left[f^{\left(n\right)}\left(t\right)\right]=\frac{T\left(v\right)}{v^n}-{\sum }_{k=0}^{n-1}{v}^{2-n+k}{f}^{\left(k\right)}\left(0\right),$
• $E\left[t^{\alpha }\right]={\int }_0^{\infty }{e}^{-vt}{t}^{\alpha }dt=v^{\alpha +1}\mathsf{\Gamma }(\alpha +1),\mathbb{R}\left(\alpha \right)>0.$

Theorem 3

([30]). If $T\left(v\right)$ is Elzaki transform of $f\left(t\right)$, one can consider the following Elzaki transform of the Riemann–Liouville derivative

$$\begin{array}{c}\hfill E\left[D^{\alpha }f\left(t\right)\right]=v^{-\alpha }[T\left(v\right)-\sum _{k=1}^n{v}^{\alpha -k+2}\left[D^{\alpha -k}f\left(0\right)\right]],-1<n-1\le \alpha <n.\end{array}$$

(13)

Definition 10

([30]). The Elzaki transform of the Caputo fractional derivative by using Theorem 3 is defined as follows

$$\begin{array}{c}\hfill E\left[D^{\alpha }f\left(t\right)\right]=v^{-\alpha }E\left[f\left(t\right)\right]-\sum _{k=0}^{m-1}{v}^{2-\alpha +k}{f}^{\left(k\right)}\left(0\right),\end{array}$$

(14)

where $m-1<\alpha <m$.

3. Elzaki Transform Homotopy Perturbation Method (ETHPM)

Consider the following fractional differential equations to explain the core steps of this method

$$\begin{array}{c}\hfill D_t^{\alpha }\tilde{u}(x,t)=\mathcal{L}\left(\tilde{u}(x,t)\right)+\mathcal{N}\left(\tilde{u}(x,t)\right)+\tilde{f}(x,t),\end{array}$$

(15)

with initial condition

$$\begin{array}{c}\hfill D_t^k\tilde{u}(x,0)={\tilde{g}}_k,k=0,\dots ,n-1,\end{array}$$

(16)

where $D_t^{\alpha }$ is the fractional derivative, $\mathcal{L}$ is a linear operator, $\mathcal{N}$ is a nonlinear differential operator, and $\tilde{f}$ is a known fuzzy function.

Step 1. By using the HMF for Equation (15), we get

$$\begin{array}{c}\hfill D_t^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]=\mathcal{L}\left[\mathcal{H}\left(\tilde{u}(x,t)\right)\right]+\mathcal{N}\left[\mathcal{H}\left(\tilde{u}(x,t)\right)\right]+\mathcal{H}\left[\tilde{f}(x,t)\right],\end{array}$$

(17)

with the initial condition

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,0)\right]=u^{gr}(\mu ,{\alpha }_c,x,0).\end{array}$$

(18)

Step 2. After using the Elzaki transform of the Equation (17), we obtain

$$\begin{array}{c}\hfill E\left[D_t^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=E\left[\mathcal{L}\left[\mathcal{H}\left(\tilde{u}(x,t)\right)\right]+\mathcal{N}\left[\mathcal{H}\left(\tilde{u}(x,t)\right)\right]+\mathcal{H}\left[\tilde{f}(x,t)\right]\right].\end{array}$$

(19)

Applying Equation (14), we obtain

$$\begin{array}{c}\hfill E\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=v^{\alpha }E\left[\mathcal{L}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]+\mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]\right]+\mathcal{H}\left[\tilde{g}(x,t)\right].\end{array}$$

(20)

Step 3. By applying the inverse Elzaki transform for the Equation (20), we get

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\mathcal{H}\left[\tilde{G}(x,t)\right]+E^{-1}\left[v^{\alpha }E[\mathcal{L}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]+\mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]]\right],\end{array}$$

(21)

where $\mathcal{H}\left[\tilde{G}(x,t)\right]$ is the result of the integration of the initial condition and the known functions $\mathcal{H}\left[\tilde{f}(x,t)\right]$.

Step 4. Now, we construct the following homotopy equation

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right].\end{array}$$

(22)

In order to make the calculation of the equation simple, we use the He polynomial

$$\begin{array}{c}\hfill \mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=\sum _{n=0}^{\infty }{p}^n{H}_n\left[\mathcal{H}\left(\tilde{u}\right)\right],\end{array}$$

(23)

where $H_n\left[\mathcal{H}\left(\tilde{u}\right)\right]$ stands for He’s polynomial in nature,

$$\begin{array}{c}\hfill H_n\mathcal{H}\left[\left({\tilde{u}}_0,{\tilde{u}}_1,{\tilde{u}}_2\dots {\tilde{u}}_n\right)\right]=\frac{1}{n!}\frac{{\partial }^n}{\partial p^n}{\left[\mathcal{N}\left(\sum _{i=0}^{\infty }{p}^i\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]\right)\right]}_{p=0},n=0,1,2,\cdots .\end{array}$$

(24)

Equation (21) can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =\mathcal{H}\left[\tilde{G}(x,t)\right]+p\left[E^{-1}\left[v^{\alpha }E\left[\mathcal{L}\left(\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\right)\right]\right.\right.\hfill \\ & \left.\left.+v^{\alpha }E\left[\mathcal{N}\left(\sum _{n=0}^{\infty }{p}^n{H}_n\left[\mathcal{H}\left(\tilde{u}\right)\right]\right)\right]\right]\right].\hfill \end{array}\end{array}$$

(25)

Comparing the coefficient of like powers of p, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}{p}^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=\mathcal{H}\left[\tilde{G}(x,t)\right],\\ \\ p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]=E^{-1}\left[v^{\alpha }E\left[\mathcal{L}\left[\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]\right]+H_0\left(\mathcal{H}\left(\tilde{u}\right)\right)\right]\right],\\ \\ p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]=E^{-1}\left[v^{\alpha }E\left[\mathcal{L}\left[\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\right]+H_1\left(\mathcal{H}\left(\tilde{u}\right)\right)\right]\right],\\ \\ p^3:\mathcal{H}\left[{\tilde{u}}_3(x,t)\right]=E^{-1}\left[v^{\alpha }E\left[\mathcal{L}\left[\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\right]+H_2\left(\mathcal{H}\left(\tilde{u}\right)\right)\right]\right],\\ ⋮\\ p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=E^{-1}\left[v^{\alpha }E\left[\mathcal{L}\left[\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]\right]+H_{n-1}\left(\mathcal{H}\left(\tilde{u}\right)\right)\right]\right].\end{array}\end{array}$$

(26)

The result is as follows

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\underset{p\to 1}{lim}\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]+\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]+\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]+\cdots .\end{array}$$

(27)

Remark 1.

The value of $|\widetilde{Res}(x,t)|$ describes the difference between the precise solution and approximate solution. Normally, the precise value of $|\widetilde{Res}(x,t)|$ is zero for any $\alpha \in (0,1]$. The definition of $|\widetilde{Res}(x,t)|$ is as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill max|\widetilde{Res}(x,t)|=& \underset{\mu \in [0,1],{\alpha }_c\in [0,1]}{max}|D_t^{\alpha }\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]-\mathcal{L}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]\hfill \\ & -\mathcal{N}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]-f^{gr}(\mu ,{\alpha }_c,x,t)|.\hfill \end{array}\end{array}$$

(28)

4. Illustrative Examples

We use the Elzaki transform homotopy perturbation method [38] (ETHPM), homotopy perturbation method [39] (HPM), residual power series method [40] (RPSM) and the conformable fractional Adomian decomposition method [41] (CFADM) respectively to calculate the following three examples. Literature [38,39,40,41] mainly studied the problems of the deterministic fractional Black–Scholes equations. We use these methods to recalculate the fuzzy fractional Black–Scholes equation under the concept of granular differentiability.

Example 1.

Consider the fractional Black–Scholes equation

$$\begin{array}{c}\hfill \frac{{\partial }^{\alpha }\tilde{u}}{\partial t^{\alpha }}=\frac{{\partial }^2\tilde{u}}{\partial x^2}+(k-1)\frac{\partial \tilde{u}}{\partial x}-k\tilde{u},\end{array}$$

(29)

with the initial condition

$$\begin{array}{c}\hfill \tilde{u}(x,0)=\tilde{c}max\left(e^x-1,0\right),\end{array}$$

(30)

$$\begin{array}{c}\hfill \tilde{c}=\left(0,1,2\right).\end{array}$$

(31)

By using the HMF, we have

$$\begin{array}{c}\hfill \frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}=\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-k\mathcal{H}\left[\tilde{u}(x,t)\right],\end{array}$$

(32)

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,0)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left(e^x-1,0\right),\end{array}$$

(33)

for each $\mu ,{\alpha }_c\in [0,1]$ and

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=u^{gr}(\mu ,{\alpha }_c,x,t).\end{array}$$

(34)

Firstly, the ETHPM is used to solving the Example 1.

After using the Elzaki transform for the Equation (30), we get

$$\begin{array}{c}\hfill E[\frac{{\partial }^{\alpha }\mathcal{H}[\tilde{u}(x,t)]}{\partial t^{\alpha }}]=E[\frac{{\partial }^2\mathcal{H}[\tilde{u}(x,t)]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}[\tilde{u}(x,t)]}{\partial x}-k\mathcal{H}[\tilde{u}(x,t)]].\end{array}$$

(35)

By using the Elzaki transform’s differential property, we obtain

$$\begin{array}{cc}\hfill v^{-\alpha }[E\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]-v^2\mathcal{H}\left[\tilde{u}(x,0)\right]]& =E[\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & +(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-k\mathcal{H}\left[\tilde{u}(x,t)\right]],\hfill \end{array}$$

(36)

and,

$$\begin{array}{cc}\hfill E\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]& =v^{\alpha }\left[E\right[\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}\hfill \\ \hfill & -k\mathcal{H}\left[\tilde{u}(x,t)\right]\left]\right]+v^2\mathcal{H}\left[\tilde{u}(x,0)\right].\hfill \end{array}$$

(37)

By applying the inverse Elzaki transform for the Equation (35), we get

$$\begin{array}{cc}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]& =E^{-1}[v^{\alpha }\left[E\right[\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}\hfill \\ \hfill & -k\mathcal{H}\left[\tilde{u}(x,t)\right]\left]\right]]+\mathcal{H}\left[\tilde{u}(x,0)\right],\hfill \end{array}$$

(38)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+p{E}^{-1}\left[v^{\alpha }E[\sum _{n=0}^{\infty }{p}^n\mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]]\right],\end{array}$$

(39)

where

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]=\mathcal{H}[\frac{{\partial }^2{\tilde{u}}_n(x,t)}{\partial x^2}+(k-1)\frac{\partial {\tilde{u}}_n(x,t)}{\partial x}-k{\tilde{u}}_n(x,t)].\end{array}$$

(40)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left({\mathrm{e}}^x-1,0\right),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{(-k{t}^{\alpha })}{\mathsf{\Gamma }(\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{(-k{t}^{\alpha })}{\mathsf{\Gamma }(\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{{(-k{t}^{\alpha })}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{{(-k{t}^{\alpha })}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{{(-k{t}^{\alpha })}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{{(-k{t}^{\alpha })}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)].\hfill \end{array}\end{array}$$

(41)

As a result, the exact solution $\mathcal{H}\left[\tilde{u}(x,t)\right]$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u^{gr}(\mu ,{\alpha }_c,x,t)& =\mathcal{H}\left[\tilde{u}(x,t)\right]\hfill \\ & =\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })],\hfill \end{array}\end{array}$$

(42)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{H}}^{-1}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]& =[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })].\hfill \end{array}\end{array}$$

(43)

The trigonometric fuzzy number form of the exact solution is as follow

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })]\hfill \\ & =(0,1,2)[max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })].\hfill \end{array}\end{array}$$

(44)

Secondly, the HPM is used to solving the Example 1.

$$\begin{array}{c}\hfill D_t^{\alpha }\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\right]=p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right],\end{array}$$

(45)

where

$$\begin{array}{c}\hfill \mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]=\mathcal{H}[\frac{{\partial }^2{\tilde{u}}_n(x,t)}{\partial x^2}+(k-1)\frac{\partial {\tilde{u}}_n(x,t)}{\partial x}-k{\tilde{u}}_n(x,t)],\end{array}$$

(46)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+J^{\alpha }\left[p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right]\right].\end{array}$$

(47)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left({\mathrm{e}}^x-1,0\right),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{(-k{t}^{\alpha })}{\mathsf{\Gamma }(\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{(-k{t}^{\alpha })}{\mathsf{\Gamma }(\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{{(-k{t}^{\alpha })}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{{(-k{t}^{\alpha })}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-\frac{{(-k{t}^{\alpha })}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left({\mathrm{e}}^x,0\right)\hfill \\ & +\frac{{(-k{t}^{\alpha })}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left({\mathrm{e}}^x-1,0\right)].\hfill \end{array}\end{array}$$

(48)

So, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })]\hfill \\ & =(0,1,2)[max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })].\hfill \end{array}\end{array}$$

(49)

Thirdly, the RPSM is used to solving the Example 1.

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{\alpha n}}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(50)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]=\sum _{n=0}^i\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{n\alpha }}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(51)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{f}}_0\left(x\right)\right]=\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=\left[\mu +2(1-\mu ){\alpha }_c\right]max\left({\mathrm{e}}^x-1,0\right).\end{array}$$

(52)

The ith residual function as follows

$$\begin{array}{c}\hfill Re{s}_i^{gr}\left(x,t,\mu ,{\alpha }_c\right)=\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial t^{\alpha }}-\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x^2}-(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x}+k\mathcal{H}\left[{\tilde{u}}_i(x,t)\right],\end{array}$$

(53)

$$\begin{array}{c}\hfill D_t^{(i-1)\alpha }Re{s}_i^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0.\end{array}$$

(54)

Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Re{s}_1^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial t^{\alpha }}-\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}-(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}+k\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\hfill \\ & =f_1^{gr}(x,\mu ,{\alpha }_c)-D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]-D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & -(k-1)\left(D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)\hfill \\ & +k\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right).\hfill \end{array}\end{array}$$

(55)

Thereby, from $Re{s}_1^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we obtain

$$\begin{array}{c}\hfill f_1^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c][kmax\left({\mathrm{e}}^x,0\right)-kmax\left({\mathrm{e}}^x-1,0\right)],\end{array}$$

(56)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)+[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)]\frac{(-k{t}^{\alpha })}{\Gamma (\alpha +1)}\right].\end{array}$$

(57)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Re{s}_2^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial t^{\alpha }}-\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}-(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}+k\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\hfill \\ & =f_1^{gr}(x,\mu ,{\alpha }_c)+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & -D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]-D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}-D_x^2\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\hfill \\ & -(k-1)\left[D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+D_x\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right]\hfill \\ & +k\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right).\hfill \end{array}\end{array}$$

(58)

And, from $D_x^{\alpha }Re{s}_2^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we get

$$\begin{array}{c}\hfill f_2^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c][-k^{2}max\left({\mathrm{e}}^x,0\right)+k^{2}max\left({\mathrm{e}}^x-1,0\right)],\end{array}$$

(59)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)+[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)][\frac{(-k{t}^{\alpha })}{\Gamma (\alpha +1)}+\frac{{(-k{t}^{\alpha })}^2}{\Gamma (2\alpha +1)}]\right].\end{array}$$

(60)

Continuing this way, we can get the exact solution

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })]\hfill \\ & =(0,1,2)[max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })].\hfill \end{array}\end{array}$$

(61)

Finally, the CFADM is used to solving the Example 1.

Assume that $L_{\alpha }={}^{CF}{D}_{\ast }^{\alpha }=\frac{{\partial }^{\alpha }}{\partial t^{\alpha }}$ is a linear operator

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]=\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-k\mathcal{H}\left[\tilde{u}(x,t)\right],\end{array}$$

(62)

$$\begin{array}{c}\hfill t^{1-\alpha }{\partial }_t\mathcal{H}\left[\tilde{u}(x,t)\right]=\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-k\mathcal{H}\left[\tilde{u}(x,t)\right].\end{array}$$

(63)

From $L_{\alpha }^{-1}={\int }_0^t\frac{1}{{\zeta }^{1-\alpha }}(.)d\zeta$, we get

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+L_{\alpha }^{-1}\left[\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-k\mathcal{H},\left[\tilde{u}(x,t)\right]\right],\end{array}$$

(64)

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{u}}_n(x,t)\right].\end{array}$$

(65)

In the Adomian decomposition method, we assume that the nonlinear operator may be decomposed into an infinite polynomial series, then

$$\begin{array}{c}\hfill \mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=\sum _{n=0}^{\infty }{A}_n,\end{array}$$

(66)

where $A_n\left[\mathcal{H}\left(\tilde{u}\right)\right]$ are Adomian polynomials, which are defined as

$$\begin{array}{c}\hfill A_n\mathcal{H}\left[\left({\tilde{u}}_0,{\tilde{u}}_1,{\tilde{u}}_2\dots {\tilde{u}}_n\right)\right]=\frac{1}{n!}\frac{d^n}{d{\lambda }^n}{\left[\mathcal{N}\left(\sum _{i=0}^n{\lambda }^i\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]\right)\right]}_{\lambda =0},n=0,1,2,\cdots .\end{array}$$

(67)

So, from $A_n\left[\mathcal{H}\left(\tilde{u}\right)\right]$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]& =[\mu +2(1-\mu ){\alpha }_c]max\left({\mathrm{e}}^x-1,0\right),\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =L_{\alpha }^{-1}\left[\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x}-k\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)\right]\frac{-k{t}^{\alpha }}{\alpha },\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =L_{\alpha }^{-1}\left[\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}-k\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)\right]\frac{{(-k{t}^{\alpha })}^2}{2!{\alpha }^2},\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_3(x,t)\right]& =L_{\alpha }^{-1}\left[\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}-k\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)\right]\frac{{(-k{t}^{\alpha })}^3}{3!{\alpha }^3},\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =L_{\alpha }^{-1}\left[\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x^2}+(k-1)\frac{\partial \mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x}-k\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left({\mathrm{e}}^x-1,0\right)-max\left({\mathrm{e}}^x,0\right)\right]\frac{{(-k{t}^{\alpha })}^n}{n!{\alpha }^n},\hfill \\ \hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-e^{-\frac{kt}{\alpha }}))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)e^{-\frac{kt}{\alpha }})]\hfill \\ & =(0,1,2)[max\left({\mathrm{e}}^x,0\right)(1-e^{-\frac{kt}{\alpha }}))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)e^{-\frac{kt}{\alpha }})].\hfill \end{array}\end{array}$$

(68)

In

Table 1. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.5,k=2$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.72536	1.69448	$[\mu +2(1-\mu ){\alpha }_c]0.72536$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]0.72536$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]1.02362$	1.34924
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$0.96892	0.42818	$[\mu +2(1-\mu ){\alpha }_c]0.96892$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]0.96892$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.36431$	3.81622
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.24634	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.24634$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.24634$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.76501$	7.01085
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.57115	2.75655	$[\mu +2(1-\mu ){\alpha }_c]1.57115$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]1.57115$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]2.24970$	10.79391
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.94486	4.76615	$[\mu +2(1-\mu ){\alpha }_c]1.94486$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]1.94486$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]2.82107$	15.08494
0.4	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.99578	1.69448	$[\mu +2(1-\mu ){\alpha }_c]0.99578$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]0.99578$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]1.29405$	1.34924
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$1.23935	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.23935$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.23935$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.63473$	3.81622
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.51676	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.51676$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.51676$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]2.03543$	7.01085
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.84158	2.75655	$[\mu +2(1-\mu ){\alpha }_c]1.84158$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]1.84158$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]2.52012$	10.79391
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$2.21529	4.76615	$[\mu +2(1-\mu ){\alpha }_c]2.21529$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]2.21529$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]3.09149$	15.08494
0.6	0.1	$[\mu +2(1-\mu ){\alpha }_c]$1.32608	1.69448	$[\mu +2(1-\mu ){\alpha }_c]1.32608$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]1.32608$	1.69448	$[\mu +2(1-\mu ){\alpha }_c]1.62434$	1.34924
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$1.56964	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.56964$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.56964$	0.42818	$[\mu +2(1-\mu ){\alpha }_c]1.96503$	3.81622
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.84706	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.84706$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]1.84706$	1.03090	$[\mu +2(1-\mu ){\alpha }_c]2.36572$	7.01085
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$2.17187	2.75655	$[\mu +2(1-\mu ){\alpha }_c]2.17187$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]2.17187$	2.75655	$[\mu +2(1-\mu ){\alpha }_c]2.85042$	10.79391
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$2.54558	4.76615	$[\mu +2(1-\mu ){\alpha }_c]2.54558$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]2.54558$	4.76615	$[\mu +2(1-\mu ){\alpha }_c]3.42178$	15.08494

,

Table 2. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.75,k=2$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.53087	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.53087$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.53087$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.60095$	0.07109
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$0.68703	1.54889	$[\mu +2(1-\mu ){\alpha }_c]0.68703$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]0.68703$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]0.78545$	0.33817
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$0.81811	0.78071	$[\mu +2(1-\mu ){\alpha }_c]0.81811$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]0.81811$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]0.9286$	0.84205
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$0.95402	0.01695	$[\mu +2(1-\mu ){\alpha }_c]0.95402$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]0.95402$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.06532$	1.60861
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.11121	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.11121$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.11121$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.21435$	2.65765
0.4	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.80130	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.80130$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.80130$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]0.87137$	0.07109
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$0.95745	1.54889	$[\mu +2(1-\mu ){\alpha }_c]0.95745$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]0.95745$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]1.05587$	0.33817
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.08854	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.08854$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.08854$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.19906$	0.84205
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.22445	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.22445$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.22445$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.33575$	1.60861
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.38163	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.38163$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.38163$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.48477$	2.65765
0.6	0.1	$[\mu +2(1-\mu ){\alpha }_c]$1.13159	2.43534	$[\mu +2(1-\mu ){\alpha }_c]1.13159$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]1.13159$	2.43534	$[\mu +2(1-\mu ){\alpha }_c]1.20166$	0.07109
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$1.28774	1.54889	$[\mu +2(1-\mu ){\alpha }_c]1.28774$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]1.28774$	1.54889	$[\mu +2(1-\mu ){\alpha }_c]1.38616$	0.33817
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.41883	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.41883$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.41883$	0.78071	$[\mu +2(1-\mu ){\alpha }_c]1.52935$	0.84205
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.55474	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.55474$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.55474$	0.01695	$[\mu +2(1-\mu ){\alpha }_c]1.66604$	1.60861
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.71193	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.71193$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.71193$	0.93152	$[\mu +2(1-\mu ){\alpha }_c]1.81506$	2.65765

and

Table 3. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =1,k=2$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.40274	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.40274$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.40274$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.40274$	0.00534
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$0.55207	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.55207$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.55207$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.55207$	0.04267
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$0.67740	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.67740$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.67740$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.67740$	0.14400
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$0.78674	0.34133	$[\mu +2(1-\mu ){\alpha }_c]0.78674$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]0.78674$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]0.78674$	0.34133
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$0.88807	0.66667	$[\mu +2(1-\mu ){\alpha }_c]0.88807$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]0.88807$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]0.88807$	0.66667
0.4	0.1	$[\mu +2(1-\mu ){\alpha }_c]$0.67316	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.67316$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.67316$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]0.67316$	0.00534
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$0.82249	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.82249$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.82249$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]0.82249$	0.04267
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$0.94782	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.94782$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.94782$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]0.94782$	0.14400
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.05716	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.05716$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.05716$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.05716$	0.34133
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.15849	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.15849$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.15849$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.15849$	0.66667
0.6	0.1	$[\mu +2(1-\mu ){\alpha }_c]$1.00345	0.00534	$[\mu +2(1-\mu ){\alpha }_c]1.00345$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]1.00345$	0.00534	$[\mu +2(1-\mu ){\alpha }_c]1.00345$	0.00534
	0.2	$[\mu +2(1-\mu ){\alpha }_c]$1.15279	0.04267	$[\mu +2(1-\mu ){\alpha }_c]1.15279$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]1.15279$	0.04267	$[\mu +2(1-\mu ){\alpha }_c]1.15279$	0.04267
	0.3	$[\mu +2(1-\mu ){\alpha }_c]$1.27812	0.14400	$[\mu +2(1-\mu ){\alpha }_c]1.27812$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]1.27812$	0.14400	$[\mu +2(1-\mu ){\alpha }_c]1.27812$	0.14400
	0.4	$[\mu +2(1-\mu ){\alpha }_c]$1.38745	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.38745$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.38745$	0.34133	$[\mu +2(1-\mu ){\alpha }_c]1.38745$	0.34133
	0.5	$[\mu +2(1-\mu ){\alpha }_c]$1.48879	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.48879$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.48879$	0.66667	$[\mu +2(1-\mu ){\alpha }_c]1.48879$	0.66667

, we show the $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ of ETHPM, HPM, RPSM, and CFADM between different values of x and t when the fractions $\alpha =0.5,0.75,1$, parameter $k=2$, respectively. Although the approximate solutions for ETHPM, HPM, and RPSM are the same, HPM requires fractional integration and RPSM requires fractional differentiation, which are computationally more complex. In contrast, the ETHPM doesn’t require fractional integration or differentiation and the ETHPM has a smaller $max|\widetilde{Res}(x,t)|$ than the CFADM.

Remark 2.

In Example 1, the value of $max|\widetilde{Res}(x,t)|$ doesn’t vary with x because when $x>0$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]& =\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[{\mathrm{e}}^x-1-(\frac{(-k{t}^{\alpha })}{\Gamma (\alpha +1)}+\frac{{(-k{t}^{\alpha })}^2}{\Gamma (2\alpha +1)}+...+\frac{{(-k{t}^{\alpha })}^n}{\Gamma (n\alpha +1)})\right],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill max|\widetilde{Res}(x,t)|& =\underset{\mu \in [0,1],{\alpha }_c\in [0,1]}{max}|\frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}-\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}-(k-1)\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+k\mathcal{H}\left[\tilde{u}(x,t)\right]|\hfill \\ & =\underset{\mu \in [0,1],{\alpha }_c\in [0,1]}{max}|[\mu +2(1-\mu ){\alpha }_c][\frac{1}{\mathsf{\Gamma }(1-\alpha )}\sum _{n=1}^{\infty }\left[-\frac{{(-k)}^n}{\mathsf{\Gamma }(n\alpha +1)}{\int }_0^t\frac{s^{n\alpha }}{{(t-s)}^{\alpha }}\mathrm{d}s\right]\hfill \\ & -{\mathrm{e}}^x-(k-1){\mathrm{e}}^x+k[{\mathrm{e}}^x-1-\sum _{n=1}^{\infty }(\frac{{(-k{t}^{\alpha })}^n}{\mathsf{\Gamma }(n\alpha +1)})]\left]\right|\hfill \\ & =\underset{\mu \in [0,1],{\alpha }_c\in [0,1]}{max}|[\mu +2(1-\mu ){\alpha }_c][\frac{1}{\mathsf{\Gamma }(1-\alpha )}\sum _{n=1}^{\infty }\left[-\frac{{(-k)}^n}{\mathsf{\Gamma }(n\alpha +1)}{\int }_0^t\frac{s^{n\alpha }}{{(t-s)}^{\alpha }}\mathrm{d}s\right]\hfill \\ & -k[1+\sum _{n=1}^{\infty }(\frac{{(-k{t}^{\alpha })}^n}{\Gamma (n\alpha +1)})]\left]\right|,\hfill \end{array}\end{array}$$

where the x terms are cancelled out.

Remark 3.

The advantages of the ETHPM are as follows

1. 

Compared with the HPM

HPM requires fractional integration operations. The numerical integration operated by fractional integration needs linearization and discretization. Thus, this operation causes errors, and its higher complexity requires a giant storage cell in a computer. The results of ETHPM and HPM are the same, but after the Elzaki transform, fractional integration is not necessary. ETHPM is more suitable to be achieved by computer programming.

2. 

Compared with the RPSM

RPSM needs to operate the residuals through fractional differentiation. However, computers have insufficient processing power to operate the fractional differentiation due to its massive calculation requirement and high complexity. ETHPM doesn’t need a fractional differentiation operation, which reduces the complexity and the amount of computation.

3. 

Compared with the CFADM

It is most intuitive that the value of $max|\widetilde{Res}(x,t)|$ of ETHPM is smaller, while CFADM also requires fractional integration operation.

Remark 4.

In Table 1, Table 2 and Table 3, $\mathcal{H}\left[\tilde{u}(x,t)\right]$ is converted into trigonometric fuzzy number form as follows

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]& =u^{gr}(\mu ,{\alpha }_c,x,t),\hfill \\ \hfill {\mathcal{H}}^{-1}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]& =\left[\underset{\beta \ge \mu }{inf}\underset{{\alpha }_c}{min}{u}^{gr}\left(\beta ,{\alpha }_c,x,t\right),\underset{\beta \ge \mu }{sup}\underset{{\alpha }_c}{max}{u}^{gr}\left(\beta ,{\alpha }_c,x,t\right)\right]\hfill \\ & =[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })],\hfill \end{array}\end{array}$$

(69)

when $k=2$, by using the μ-level sets representation theorem, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(0.2,0.1)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max\left({\mathrm{e}}^x,0\right)(1-E_{\alpha }(-k{t}^{\alpha }))\hfill \\ & +max\left({\mathrm{e}}^x-1,0\right)E_{\alpha }(-k{t}^{\alpha })]\hfill \\ & =(0,1,2)0.40274.\hfill \end{array}\end{array}$$

(70)

In Figure 2, the picture shows the fuzzy approximate solution $\tilde{u}(x,t)$ of the Black–Scholes equation at the fractional parameter $\alpha =1$ and the auxiliary parameters $k=2$, $\mu =0.5$.

Example 2.

Consider the generalized Black–Scholes equation

$$\begin{array}{c}\hfill \frac{{\partial }^{\alpha }\tilde{u}}{\partial t^{\alpha }}+0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\tilde{u}}{\partial x^2}+0.06x\frac{\partial \tilde{u}}{\partial x}-0.06\tilde{u}=0,\end{array}$$

(71)

with the initial condition

$$\begin{array}{c}\hfill \tilde{u}(x,0)=\tilde{c}max\left(x-25e^{-0.06},0\right),\end{array}$$

(72)

$$\begin{array}{c}\hfill \tilde{c}=\left(0,1,2\right).\end{array}$$

(73)

From Definitions 7 and 8, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}& =-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right],\hfill \end{array}\end{array}$$

(74)

and,

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,0)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left(x-25e^{-0.06},0\right),\end{array}$$

(75)

for each $\mu ,{\alpha }_c\in [0,1]$ and

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=u^{gr}(\mu ,{\alpha }_c,x,t).\end{array}$$

(76)

Firstly, the ETHPM is used to solving the Example 2.

After using the Elzaki transform for the Equation (74), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E\left[\frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}\right]& =E[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right]].\hfill \end{array}\end{array}$$

(77)

Using the properties of the Elzaki transform, we get

$$\begin{array}{cc}\hfill v^{-\alpha }\left[E\right[\mathcal{H}\left[\tilde{u}(x,t)\right]& -v^2\mathcal{H}\left[\tilde{u}(x,0)\right]\left]\right]\hfill \\ \hfill & =E[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & -0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right]],\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill E\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]& =v^{\alpha }\left[E\right[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & -0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right]\left]\right]+v^2\mathcal{H}\left[\tilde{u}(x,0)\right].\hfill \end{array}$$

(79)

By applying the inverse Elzaki transform for the Equation (79), we get

$$\begin{array}{cc}\hfill \mathcal{H}[\tilde{u}(x,t)]& =E^{-1}[v^{\alpha }[E[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}[\tilde{u}(x,t)]}{\partial x^2}\hfill \\ \hfill & -0.06x\frac{\partial \mathcal{H}[\tilde{u}(x,t)]}{\partial x}+0.06\mathcal{H}[\tilde{u}(x,t)]]]]+\mathcal{H}[\tilde{u}(x,0)],\hfill \end{array}$$

(80)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+p{E}^{-1}\left[v^{\alpha }E[\sum _{n=0}^{\infty }{p}^n\mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]]\right],\end{array}$$

(81)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]& =\mathcal{H}[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2{\tilde{u}}_n(x,t)}{\partial x^2}\hfill \\ & -0.06x\frac{\partial {\tilde{u}}_n(x,t)}{\partial x}+0.06{\tilde{u}}_n(x,t)].\hfill \end{array}\end{array}$$

(82)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left(x-25e^{-0.06},0\right),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{0.06t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +\frac{0.06t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}max\left(x-25e^{-0.06},0\right)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{{\left(0.06t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}\hfill \\ & +\frac{{\left(0.06t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left(x-25e^{-0.06},0\right)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{{\left(0.06t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}\hfill \\ & +\frac{{\left(0.06t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left(x-25e^{-0.06},0\right)].\hfill \end{array}\end{array}$$

(83)

So that the solution $\mathcal{H}\left[\tilde{u}(x,t)\right]$ of the problem is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u^{gr}(\mu ,{\alpha }_c,x,t)& =\mathcal{H}\left[\tilde{u}(x,t)\right]\hfill \\ & =\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)],\hfill \end{array}\end{array}$$

(84)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{H}}^{-1}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]& =[\mu ,2-\mu ][x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)].\hfill \end{array}\end{array}$$

(85)

By using the $\mu$-level sets representation theorem, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)]\hfill \\ & =(0,1,2)[x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)].\hfill \end{array}\end{array}$$

(86)

Secondly, the HPM is used to solving the Example 2.

$$\begin{array}{c}\hfill D_t^{\alpha }\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\right]=p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right],\end{array}$$

(87)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]& =\mathcal{H}[-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right],\hfill \end{array}\end{array}$$

(88)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+J^{\alpha }\left[p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right]\right].\end{array}$$

(89)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max\left(x-25e^{-0.06},0\right),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{0.06t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +\frac{0.06t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}max\left(x-25e^{-0.06},0\right)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{{\left(0.06t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}\hfill \\ & +\frac{{\left(0.06t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}max\left(x-25e^{-0.06},0\right)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-x\frac{{\left(0.06t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}\hfill \\ & +\frac{{\left(0.06t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}max\left(x-25e^{-0.06},0\right)].\hfill \end{array}\end{array}$$

(90)

So, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)]\hfill \\ & =(0,1,2)[x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)].\hfill \end{array}\end{array}$$

(91)

Thirdly, the RPSM is used to solving the Example 2.

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{\alpha n}}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(92)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]=\sum _{n=0}^i\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{n\alpha }}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(93)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{f}}_0\left(x\right)\right]=\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=\left[\mu +2(1-\mu ){\alpha }_c\right]max\left(x-25{\mathrm{e}}^{-0.06},0\right).\end{array}$$

(94)

The ith residual function can be written as

$$Re{s}_i^{gr}\left(x,t,\mu ,{\alpha }_c\right)=\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial t^{\alpha }}+0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_i(x,t)\right],$$

(95)

$$\begin{array}{c}\hfill D_t^{(i-1)\alpha }Re{s}_i^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0.\end{array}$$

(96)

Then, we obtain

$$\begin{array}{cc}\hfill Re{s}_1^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial t^{\alpha }}+0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\hfill \\ \hfill & =f_1^{gr}(x,\mu ,{\alpha }_c)+0.08{(2+sinx)}^2{x}^2\left[D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right]\hfill \\ \hfill & +0.06x\left(D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)\hfill \\ \hfill & -0.06\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right).\hfill \end{array}$$

(97)

By calculating $Re{s}_1^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we obtain

$$\begin{array}{c}\hfill f_1^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c]\left[0.06max\left(x-25{\mathrm{e}}^{-0.06},0\right)-0.06x\right],\end{array}$$

(98)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)+[0.06max\left(x-25{\mathrm{e}}^{-0.06},0\right)-0.06x]\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right].\end{array}$$

(99)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Re{s}_2^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial t^{\alpha }}+0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\hfill \\ & =f_1^{gr}(x,\mu ,{\alpha }_c)+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +0.08{(2+sinx)}^2{x}^2\left[D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+D_x^2\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right]\hfill \\ & +0.06x\left[D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+D_x\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right]\hfill \\ & -0.06\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right).\hfill \end{array}\end{array}$$

(100)

By calculating $D_x^{\alpha }Re{s}_2^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we obtain

$$\begin{array}{c}\hfill f_2^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c]0.{06}^2\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x\right],\end{array}$$

(101)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)+[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x][\frac{\left(0.06t^{\alpha }\right)}{\Gamma (\alpha +1)}+\frac{{\left(0.06t^{\alpha }\right)}^2}{\Gamma (2\alpha +1)}]\right].\end{array}$$

(102)

Continuing this way, one may find the values of $f_3^{gr}(x,\mu ,{\alpha }_c),f_4^{gr}(x,\mu ,{\alpha }_c),\dots$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)]\hfill \\ & =(0,1,2)[x(1-E_{\alpha }\left(0.06t^{\alpha }\right))\hfill \\ & +max\left(x-25e^{-0.06},0\right)E_{\alpha }\left(0.06t^{\alpha }\right)].\hfill \end{array}\end{array}$$

(103)

Finally, the CFADM is used to solving the Example 2.

Let $L_{\alpha }={}^{CF}{D}_{\ast }^{\alpha }=\frac{{\partial }^{\alpha }}{\partial t^{\alpha }}$ be a linear operator

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]=-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}-0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right],\end{array}$$

(104)

$$\begin{array}{c}\hfill t^{1-\alpha }{\partial }_t\mathcal{H}\left[\tilde{u}(x,t)\right]=-0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}-0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+0.06\mathcal{H}\left[\tilde{u}(x,t)\right].\end{array}$$

(105)

By the inverse of operator $L_{\alpha }$ which is $L_{\alpha }^{-1}={\int }_0^t\frac{1}{{\zeta }^{1-\alpha }}(.)d\zeta$, we get

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]-L_{\alpha }^{-1}\left[0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[\tilde{u}(x,t)\right]\right],\end{array}$$

(106)

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{u}}_n(x,t)\right].\end{array}$$

(107)

The nonlinear operator can also be decomposed into an infinite polynomial series using the Adomian decomposition method

$$\begin{array}{c}\hfill \mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=\sum _{n=0}^{\infty }{A}_n,\end{array}$$

(108)

where $A_n\left[\mathcal{H}\left(\tilde{u}\right)\right]$ are Adomian polynomials, which are defined as

$$\begin{array}{c}\hfill A_n\mathcal{H}\left[\left({\tilde{u}}_0,{\tilde{u}}_1,{\tilde{u}}_2\dots {\tilde{u}}_n\right)\right]=\frac{1}{n!}\frac{d^n}{d{\lambda }^n}{\left[\mathcal{N}\left(\sum _{i=0}^n{\lambda }^i\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]\right)\right]}_{\lambda =0},n=0,1,2,\dots .\end{array}$$

(109)

So, by using the Adomian decomposition method in conformable sense, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]& =[\mu +2(1-\mu ){\alpha }_c]max\left(x-25{\mathrm{e}}^{-0.06},0\right),\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =-L_{\alpha }^{-1}\left[0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x\right]\frac{0.06t^{\alpha }}{\alpha },\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =-L_{\alpha }^{-1}\left[0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x\right]\frac{{\left(0.06t^{\alpha }\right)}^2}{2!{\alpha }^2},\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_3(x,t)\right]& =-L_{\alpha }^{-1}\left[0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x\right]\frac{{\left(0.06t^{\alpha }\right)}^3}{3!{\alpha }^3},\hfill \\ \hfill ⋮\\ \hfill \mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =-L_{\alpha }^{-1}\left[0.08{(2+sinx)}^2{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x^2}+0.06x\frac{\partial \mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x}-0.06\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c]\left[max\left(x-25{\mathrm{e}}^{-0.06},0\right)-x\right]\frac{{\left(0.06t^{\alpha }\right)}^n}{n!{\alpha }^n}.\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ]\left[x(1-e^{\frac{0.06t}{\alpha }})\right)+max\left(x-25{\mathrm{e}}^{-0.06},0\right)e^{\frac{0.06t}{\alpha }}\left)\right]\hfill \\ & =(0,1,2)\left[x(1-e^{\frac{0.06t}{\alpha }})\right)+max\left(x-25{\mathrm{e}}^{-0.06},0\right)e^{\frac{0.06t}{\alpha }}\left)\right].\hfill \end{array}\end{array}$$

(110)

In

Table 4. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.5$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00435	0.00243	$-[\mu +2(1-\mu ){\alpha }_c]$0.00435	0.00243	$-[\mu +2(1-\mu ){\alpha }_c]$0.00435	0.00243	$-[\mu +2(1-\mu ){\alpha }_c]0.00774$	0.02493
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00620	0.00490	$-[\mu +2(1-\mu ){\alpha }_c]$0.00620	0.00490	$-[\mu +2(1-\mu ){\alpha }_c]$0.00620	0.00490	$-[\mu +2(1-\mu ){\alpha }_c]0.01103$	0.02532
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00764	0.00738	$-[\mu +2(1-\mu ){\alpha }_c]$0.00764	0.00738	$-[\mu +2(1-\mu ){\alpha }_c]$0.00764	0.00738	$-[\mu +2(1-\mu ){\alpha }_c]0.01359$	0.02563
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00886	0.00988	$-[\mu +2(1-\mu ){\alpha }_c]$0.00886	0.00988	$-[\mu +2(1-\mu ){\alpha }_c]$0.00886	0.00988	$-[\mu +2(1-\mu ){\alpha }_c]0.01577$	0.02589
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.00995	0.01240	$-[\mu +2(1-\mu ){\alpha }_c]$0.00995	0.01240	$-[\mu +2(1-\mu ){\alpha }_c]$0.00995	0.01240	$-[\mu +2(1-\mu ){\alpha }_c]0.01771$	0.02612
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00871	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]$0.00871	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]$0.00871	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]0.01547$	0.04986
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01240	0.00980	$-[\mu +2(1-\mu ){\alpha }_c]$0.01240	0.00980	$-[\mu +2(1-\mu ){\alpha }_c]$0.01240	0.00980	$-[\mu +2(1-\mu ){\alpha }_c]0.02205$	0.05065
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01528	0.01476	$-[\mu +2(1-\mu ){\alpha }_c]$0.01528	0.01476	$-[\mu +2(1-\mu ){\alpha }_c]$0.01528	0.01476	$-[\mu +2(1-\mu ){\alpha }_c]0.02717$	0.05126
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01772	0.01976	$-[\mu +2(1-\mu ){\alpha }_c]$0.01772	0.01976	$-[\mu +2(1-\mu ){\alpha }_c]$0.01772	0.01976	$-[\mu +2(1-\mu ){\alpha }_c]0.03154$	0.05178
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01989	0.02478	$-[\mu +2(1-\mu ){\alpha }_c]$0.01989	0.02478	$-[\mu +2(1-\mu ){\alpha }_c]$0.01989	0.02478	$-[\mu +2(1-\mu ){\alpha }_c]0.03542$	0.05225
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.01306	0.00730	$-[\mu +2(1-\mu ){\alpha }_c]$0.01306	0.00730	$-[\mu +2(1-\mu ){\alpha }_c]$0.01306	0.00730	$-[\mu +2(1-\mu ){\alpha }_c]0.02321$	0.07478
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01861	0.01470	$-[\mu +2(1-\mu ){\alpha }_c]$0.01861	0.01470	$-[\mu +2(1-\mu ){\alpha }_c]$0.01861	0.01470	$-[\mu +2(1-\mu ){\alpha }_c]0.03308$	0.07597
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.02291	0.02214	$-[\mu +2(1-\mu ){\alpha }_c]$0.02291	0.02214	$-[\mu +2(1-\mu ){\alpha }_c]$0.02291	0.02214	$-[\mu +2(1-\mu ){\alpha }_c]0.04076$	0.07689
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.02658	0.02964	$-[\mu +2(1-\mu ){\alpha }_c]$0.02658	0.02964	$-[\mu +2(1-\mu ){\alpha }_c]$0.02658	0.02964	$-[\mu +2(1-\mu ){\alpha }_c]0.04731$	0.07767
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.02984	0.03718	$-[\mu +2(1-\mu ){\alpha }_c]$0.02984	0.03718	$-[\mu +2(1-\mu ){\alpha }_c]$0.02984	0.03718	$-[\mu +2(1-\mu ){\alpha }_c]0.05313$	0.07837

,

Table 5. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.75$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00234	0.00242	$-[\mu +2(1-\mu ){\alpha }_c]$0.00234	0.00242	$-[\mu +2(1-\mu ){\alpha }_c]$0.00234	0.00242	$-[\mu +2(1-\mu ){\alpha }_c]0.00287$	0.02434
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00395	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]$0.00395	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]$0.00395	0.00486	$-[\mu +2(1-\mu ){\alpha }_c]0.00484$	0.02458
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00538	0.00732	$-[\mu +2(1-\mu ){\alpha }_c]$0.00538	0.00732	$-[\mu +2(1-\mu ){\alpha }_c]$0.00538	0.00732	$-[\mu +2(1-\mu ){\alpha }_c]0.00659$	0.02479
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00671	0.00978	$-[\mu +2(1-\mu ){\alpha }_c]$0.00671	0.00978	$-[\mu +2(1-\mu ){\alpha }_c]$0.00671	0.00978	$-[\mu +2(1-\mu ){\alpha }_c]0.00821$	0.02499
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.00796	0.01226	$-[\mu +2(1-\mu ){\alpha }_c]$0.00796	0.01226	$-[\mu +2(1-\mu ){\alpha }_c]$0.00796	0.01226	$-[\mu +2(1-\mu ){\alpha }_c]0.00974$	0.02517
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00468	0.00484	$-[\mu +2(1-\mu ){\alpha }_c]$0.00468	0.00484	$-[\mu +2(1-\mu ){\alpha }_c]$0.00468	0.00484	$-[\mu +2(1-\mu ){\alpha }_c]0.00573$	0.04869
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00791	0.00970	$-[\mu +2(1-\mu ){\alpha }_c]$0.00791	0.00970	$-[\mu +2(1-\mu ){\alpha }_c]$0.00791	0.00970	$-[\mu +2(1-\mu ){\alpha }_c]0.00964$	0.04916
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01077	0.01462	$-[\mu +2(1-\mu ){\alpha }_c]$0.01077	0.01462	$-[\mu +2(1-\mu ){\alpha }_c]$0.01077	0.01462	$-[\mu +2(1-\mu ){\alpha }_c]0.01318$	0.04958
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01341	0.01956	$-[\mu +2(1-\mu ){\alpha }_c]$0.01341	0.01956	$-[\mu +2(1-\mu ){\alpha }_c]$0.01341	0.01956	$-[\mu +2(1-\mu ){\alpha }_c]0.01642$	0.04997
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01592	0.02454	$-[\mu +2(1-\mu ){\alpha }_c]$0.01592	0.02454	$-[\mu +2(1-\mu ){\alpha }_c]$0.01592	0.02454	$-[\mu +2(1-\mu ){\alpha }_c]0.01949$	0.05034
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00702	0.00724	$-[\mu +2(1-\mu ){\alpha }_c]$0.00702	0.00724	$-[\mu +2(1-\mu ){\alpha }_c]$0.00702	0.00724	$-[\mu +2(1-\mu ){\alpha }_c]0.00860$	0.07303
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01186	0.01456	$-[\mu +2(1-\mu ){\alpha }_c]$0.01186	0.01456	$-[\mu +2(1-\mu ){\alpha }_c]$0.01186	0.01456	$-[\mu +2(1-\mu ){\alpha }_c]0.01453$	0.07374
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01615	0.02194	$-[\mu +2(1-\mu ){\alpha }_c]$0.01615	0.02194	$-[\mu +2(1-\mu ){\alpha }_c]$0.01615	0.02194	$-[\mu +2(1-\mu ){\alpha }_c]0.01978$	0.07437
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.02012	0.02934	$-[\mu +2(1-\mu ){\alpha }_c]$0.02012	0.02934	$-[\mu +2(1-\mu ){\alpha }_c]$0.02012	0.02934	$-[\mu +2(1-\mu ){\alpha }_c]0.02463$	0.07496
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.02388	0.03682	$-[\mu +2(1-\mu ){\alpha }_c]$0.02388	0.03682	$-[\mu +2(1-\mu ){\alpha }_c]$0.02388	0.03682	$-[\mu +2(1-\mu ){\alpha }_c]0.02923$	0.07551

and

Table 6. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =1$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00120	0.02414	$-[\mu +2(1-\mu ){\alpha }_c]$0.00120	0.02414	$-[\mu +2(1-\mu ){\alpha }_c]$0.00120	0.02414	$-[\mu +2(1-\mu ){\alpha }_c]0.00120$	0.02414
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.02429	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.02429	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.02429	$-[\mu +2(1-\mu ){\alpha }_c]0.00241$	0.02429
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00363	0.02444	$-[\mu +2(1-\mu ){\alpha }_c]$0.00363	0.02444	$-[\mu +2(1-\mu ){\alpha }_c]$0.00363	0.02444	$-[\mu +2(1-\mu ){\alpha }_c]0.00363$	0.02444
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00486	0.02458	$-[\mu +2(1-\mu ){\alpha }_c]$0.00486	0.02458	$-[\mu +2(1-\mu ){\alpha }_c]$0.00486	0.02458	$-[\mu +2(1-\mu ){\alpha }_c]$0.00486	0.02458
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.00609	0.02473	$-[\mu +2(1-\mu ){\alpha }_c]$0.00609	0.02473	$-[\mu +2(1-\mu ){\alpha }_c]$0.00609	0.02473	$-[\mu +2(1-\mu ){\alpha }_c]$0.00609	0.02473
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.04829	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.04829	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.04829	$-[\mu +2(1-\mu ){\alpha }_c]$0.00241	0.04829
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00483	0.04858	$-[\mu +2(1-\mu ){\alpha }_c]$0.00483	0.04858	$-[\mu +2(1-\mu ){\alpha }_c]$0.00483	0.04858	$-[\mu +2(1-\mu ){\alpha }_c]$0.00483	0.04858
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00727	0.04887	$-[\mu +2(1-\mu ){\alpha }_c]$0.00727	0.04887	$-[\mu +2(1-\mu ){\alpha }_c]$0.00727	0.04887	$-[\mu +2(1-\mu ){\alpha }_c]$0.00727	0.04887
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00972	0.04917	$-[\mu +2(1-\mu ){\alpha }_c]$0.00972	0.04917	$-[\mu +2(1-\mu ){\alpha }_c]$0.00972	0.04917	$-[\mu +2(1-\mu ){\alpha }_c]$0.00972	0.04917
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01218	0.04946	$-[\mu +2(1-\mu ){\alpha }_c]$0.01218	0.04946	$-[\mu +2(1-\mu ){\alpha }_c]$0.01218	0.04946	$-[\mu +2(1-\mu ){\alpha }_c]$0.01218	0.04946
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00361	0.07243	$-[\mu +2(1-\mu ){\alpha }_c]$0.00361	0.07243	$-[\mu +2(1-\mu ){\alpha }_c]$0.00361	0.07243	$-[\mu +2(1-\mu ){\alpha }_c]$0.00361	0.07243
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00724	0.07287	$-[\mu +2(1-\mu ){\alpha }_c]$0.00724	0.07287	$-[\mu +2(1-\mu ){\alpha }_c]$0.00724	0.07287	$-[\mu +2(1-\mu ){\alpha }_c]$0.00724	0.07287
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01090	0.07331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01090	0.07331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01090	0.07331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01090	0.07331
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01457	0.07375	$-[\mu +2(1-\mu ){\alpha }_c]$0.01457	0.07375	$-[\mu +2(1-\mu ){\alpha }_c]$0.01457	0.07375	$-[\mu +2(1-\mu ){\alpha }_c]$0.01457	0.07375
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01827	0.07419	$-[\mu +2(1-\mu ){\alpha }_c]$0.01827	0.07419	$-[\mu +2(1-\mu ){\alpha }_c]$0.01827	0.07419	$-[\mu +2(1-\mu ){\alpha }_c]$0.01827	0.07419

, we show the $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ of ETHPM, HPM, RPSM, and CFADM between different values of x and t when the fractions $\alpha =0.5,0.75,1$, respectively. It is easy to see in the table that the value of $max|\widetilde{Res}(x,t)|$ increases with t when $\alpha$, x is fixed and the value of $max|\widetilde{Res}(x,t)|$ increases with x when $\alpha$, t is fixed.

In Figure 3, the picture shows the fuzzy approximate solution $\tilde{u}(x,t)$ of the Black–Scholes equation at the fractional parameter $\alpha =1$ and the auxiliary parameter $\mu =0.5$.

Example 3.

Consider the following fractional Black–Scholes option pricing equation

$$\begin{array}{c}\hfill \frac{{\partial }^{\alpha }\tilde{u}}{\partial t^{\alpha }}+\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\tilde{u}}{\partial x^2}+(r-\tau )x\frac{\partial \tilde{u}}{\partial x}-r\tilde{u}=0,\end{array}$$

(111)

with the initial condition

$$\begin{array}{c}\hfill \tilde{u}(x,0)=\tilde{c}max(Ax-B,0),\end{array}$$

(112)

$$\begin{array}{c}\hfill \tilde{c}=\left(0,1,2\right),\end{array}$$

(113)

From Definitions 7 and 8, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}& =-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right],\hfill \end{array}\end{array}$$

(114)

and,

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,0)\right]=[\mu +2(1-\mu ){\alpha }_c]max(Ax-B,0),\end{array}$$

(115)

for each $\mu ,{\alpha }_c\in [0,1]$ and

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=u^{gr}(\mu ,{\alpha }_c,x,t).\end{array}$$

(116)

Firstly, the ETHPM is used to solving the Example 3.

After using the Elzaki transform for the Equation (114), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E\left[\frac{{\partial }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial t^{\alpha }}\right]& =E[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right]].\hfill \end{array}\end{array}$$

(117)

Using the properties of the Elzaki transform, we get

$$\begin{array}{cc}\hfill v^{-\alpha }\left[E\right[\mathcal{H}\left[\tilde{u}(x,t)\right]& -v^2\mathcal{H}\left[\tilde{u}(x,0)\right]\left]\right]\hfill \\ \hfill & =E[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right]],\hfill \end{array}$$

(118)

$$\begin{array}{cc}\hfill E\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]& =v^{\alpha }\left[E\right[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right]\left]\right]+v^2\mathcal{H}\left[\tilde{u}(x,0)\right].\hfill \end{array}$$

(119)

By applying the inverse Elzaki transform for the Equation (119), we get

$$\begin{array}{cc}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]& =E^{-1}[v^{\alpha }\left[E\right[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ \hfill & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right]\left]\right]]+\mathcal{H}\left[\tilde{u}(x,0)\right],\hfill \end{array}$$

(120)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+p{E}^{-1}\left[v^{\alpha }E\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]\right]\right],\end{array}$$

(121)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]& =\mathcal{H}[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2{\tilde{u}}_n(x,t)}{\partial x^2}\hfill \\ & -(r-\tau )x\frac{\partial {\tilde{u}}_n(x,t)}{\partial x}+r{\tilde{u}}_n(x,t)].\hfill \end{array}\end{array}$$

(122)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max(Ax-B,0),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r-\tau )t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +\frac{\left(r{t}^{\alpha }\right)}{\mathsf{\Gamma }(\alpha +1)}max(Ax-B,0)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r^2-{\tau }^2){\left(t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}\hfill \\ & +\frac{{\left(r{t}^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}max(Ax-B,0)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =E^{-1}\left[v^{\alpha }E\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r^n-{\tau }^n){\left(t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}\hfill \\ & +\frac{{\left(r{t}^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}max(Ax-B,0)].\hfill \end{array}\end{array}$$

(123)

So that the solution $\mathcal{H}\left[\tilde{u}(x,t)\right]$ of the problem is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u^{gr}(\mu ,{\alpha }_c,x,t)& =\mathcal{H}\left[\tilde{u}(x,t)\right]\hfill \\ & =\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]],\hfill \end{array}\end{array}$$

(124)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{H}}^{-1}\left[u^{gr}(\mu ,{\alpha }_c,x,t)\right]& =[\mu ,2-\mu ][max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]].\hfill \end{array}\end{array}$$

(125)

By using the $\mu$-level sets representation theorem, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]]\hfill \\ & =(0,1,2)[max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]].\hfill \end{array}\end{array}$$

(126)

Secondly, the HPM is used to solving the Example 3.

$$\begin{array}{c}\hfill D_t^{\alpha }\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]\right]=p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right],\end{array}$$

(127)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}[{\tilde{\mathsf{\Psi }}}_n(x,t)]& =\mathcal{H}[-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}\hfill \\ & -(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right],\hfill \end{array}\end{array}$$

(128)

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]+J^{\alpha }\left[p\left[\sum _{n=0}^{\infty }{p}^n\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_n(x,t)\right]\right]\right].\end{array}$$

(129)

After comparing the similar power coefficients of p, the following results can be obtained

$$\begin{array}{c}\hfill p^0:\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]max(Ax-B,0),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^1:\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r-\tau )t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +\frac{\left(r{t}^{\alpha }\right)}{\mathsf{\Gamma }(\alpha +1)}max(Ax-B,0)],\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^2:\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r^2-{\tau }^2){\left(t^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}\hfill \\ & +\frac{{\left(r{t}^{\alpha }\right)}^2}{\mathsf{\Gamma }(2\alpha +1)}max(Ax-B,0)],\hfill \\ \hfill ⋮\end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p^n:\mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =J^{\alpha }\left[\mathcal{H}\left[{\tilde{\mathsf{\Psi }}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][-xmax(A,0)\frac{(r^n-{\tau }^n){\left(t^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}\hfill \\ & +\frac{{\left(r{t}^{\alpha }\right)}^n}{\mathsf{\Gamma }(n\alpha +1)}max(Ax-B,0)].\hfill \end{array}\end{array}$$

(130)

So, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]]\hfill \\ & =(0,1,2)[max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]].\hfill \end{array}\end{array}$$

(131)

Thirdly, the RPSM is used to solving the Example 3.

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{\alpha n}}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(132)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]=\sum _{n=0}^i\mathcal{H}\left[{\tilde{f}}_n\left(x\right)\right]\frac{t^{n\alpha }}{\mathsf{\Gamma }(1+n\alpha )},\end{array}$$

(133)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{f}}_0\left(x\right)\right]=\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]=\left[\mu +2(1-\mu ){\alpha }_c\right]max\left(Ax-B,0\right).\end{array}$$

(134)

The ith residual function as follows,

$$Re{s}_i^{gr}\left(x,t,\mu ,{\alpha }_c\right)=\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial t^{\alpha }}+\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_i(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_i(x,t)\right],$$

(135)

$$\begin{array}{c}\hfill D_t^{(i-1)\alpha }Re{s}_i^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0.\end{array}$$

(136)

Then, we obtain

$$\begin{array}{cc}\hfill Re{s}_1^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial t^{\alpha }}+\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\hfill \\ \hfill & =f_1^{gr}(x,\mu ,{\alpha }_c)+\frac{{\sigma }^2}{2}{x}^2\left[D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right]\hfill \\ \hfill & +(r-\tau )x\left(D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)\hfill \\ \hfill & -r\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right).\hfill \end{array}$$

(137)

By calculating $Re{s}_1^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we obtain

$$\begin{array}{c}\hfill f_1^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c]\left[rmax\left(Ax-B,0\right)-(r-\tau )xmax(A,0)\right],\end{array}$$

(138)

$$\begin{array}{c}\hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]=[\mu +2(1-\mu ){\alpha }_c]\left[max\left(Ax-B,0\right)+[rmax\left(Ax-B,0\right)-(r-\tau )xmax(A,0)]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right].\end{array}$$

(139)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Re{s}_2^{gr}\left(x,t,\mu ,{\alpha }_c\right)& =\frac{{\partial }^{\alpha }\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial t^{\alpha }}+\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\hfill \\ & =f_1^{gr}(x,\mu ,{\alpha }_c)+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +\frac{{\sigma }^2}{2}{x}^2\left[D_x^2\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x^2\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+D_x^2\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right]\hfill \\ & +(r-\tau )x\left[D_x\left[f_0^{gr}(x,\mu ,{\alpha }_c)\right]+D_x\left[f_1^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+D_x\left[f_2^{gr}(x,\mu ,{\alpha }_c)\right]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right]\hfill \\ & -r\left(f_0^{gr}(x,\mu ,{\alpha }_c)+f_1^{gr}(x,\mu ,{\alpha }_c)\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+f_2^{gr}(x,\mu ,{\alpha }_c)\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}\right).\hfill \end{array}\end{array}$$

(140)

By calculating $D_x^{\alpha }Re{s}_2^{gr}\left(x,0,\mu ,{\alpha }_c\right)=0$, we obtain

$$\begin{array}{c}\hfill f_2^{gr}(x,\mu ,{\alpha }_c)=[\mu +2(1-\mu ){\alpha }_c]\left[r^{2}max\left(Ax-B,0\right)-(r^2-{\tau }^2)xmax(A,0)\right],\end{array}$$

(141)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =[\mu +2(1-\mu ){\alpha }_c][max\left(Ax-B,0\right)+[rmax\left(Ax-B,0\right)-(r-\tau )xmax(A,0)]\frac{t^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & +[r^{2}max\left(Ax-B,0\right)-(r^2-{\tau }^2)xmax(A,0)]\frac{t^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}].\hfill \end{array}\end{array}$$

(142)

Continuing this way, one may find the values of $f_3^{gr}(x,\mu ,{\alpha }_c),f_4^{gr}(x,\mu ,{\alpha }_c),\dots$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]]\hfill \\ & =(0,1,2)[max(Ax-B,0)E_{\alpha }\left(r{t}^{\alpha }\right)\hfill \\ & -max\left(A,0\right)[E_{\alpha }\left(r{t}^{\alpha }\right)-E_{\alpha }\left(\tau t^{\alpha }\right)]].\hfill \end{array}\end{array}$$

(143)

Finally, the CFADM is used to solving the Example 3.

Let $L_{\alpha }={}^{CF}{D}_{\ast }^{\alpha }=\frac{{\partial }^{\alpha }}{\partial t^{\alpha }}$ be a linear operator

$$\begin{array}{c}\hfill {}^{CF}{D}_{\ast }^{\alpha }\mathcal{H}\left[\tilde{u}(x,t)\right]=-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}-(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right],\end{array}$$

(144)

$$\begin{array}{c}\hfill t^{1-\alpha }{\partial }_t\mathcal{H}\left[\tilde{u}(x,t)\right]=-\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}-(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}+r\mathcal{H}\left[\tilde{u}(x,t)\right].\end{array}$$

(145)

By the inverse of operator $L_{\alpha }$ which is $L_{\alpha }^{-1}={\int }_0^t\frac{1}{{\zeta }^{1-\alpha }}(.)d\zeta$, we get

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\mathcal{H}\left[\tilde{u}(x,0)\right]-L_{\alpha }^{-1}\left[\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[\tilde{u}(x,t)\right]}{\partial x}-r\mathcal{H}\left[\tilde{u}(x,t)\right]\right],\end{array}$$

(146)

$$\begin{array}{c}\hfill \mathcal{H}\left[\tilde{u}(x,t)\right]=\sum _{n=0}^{\infty }\mathcal{H}\left[{\tilde{u}}_n(x,t)\right].\end{array}$$

(147)

Also assumed in the Adomian decomposition method is that the nonlinear operator may be decomposed into an infinite polynomial series

$$\begin{array}{c}\hfill \mathcal{N}\left[\mathcal{H}\left[\tilde{u}(x,t)\right]\right]=\sum _{n=0}^{\infty }{A}_n,\end{array}$$

(148)

where $A_n\left[\mathcal{H}\left(\tilde{u}\right)\right]$ are Adomian polynomials, which are defined as

$$\begin{array}{c}\hfill A_n\mathcal{H}\left[\left({\tilde{u}}_0,{\tilde{u}}_1,{\tilde{u}}_2\dots {\tilde{u}}_n\right)\right]=\frac{1}{n!}\frac{d^n}{d{\lambda }^n}{\left[\mathcal{N}\left(\sum _{i=0}^n{\lambda }^i\mathcal{H}\left[{\tilde{u}}_i(x,t)\right]\right)\right]}_{\lambda =0},n=0,1,2,\cdots .\end{array}$$

(149)

So, by using the Adomian decomposition method in conformable sense, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]& =[\mu +2(1-\mu ){\alpha }_c]max\left(Ax-B,0\right),\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]& =-L_{\alpha }^{-1}\left[\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_0(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_0(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][rmax\left(Ax-B,0\right)-(r-\tau )xmax(A,0)]\frac{t^{\alpha }}{\alpha },\hfill \\ \hfill \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]& =-L_{\alpha }^{-1}\left[\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_1(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_1(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][r^{2}max\left(Ax-B,0\right)-(r^2-{\tau }^2)xmax(A,0)]\frac{{\left(t^{\alpha }\right)}^2}{2!{\alpha }^2},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}\left[{\tilde{u}}_3(x,t)\right]& =-L_{\alpha }^{-1}\left[\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_2(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_2(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][r^{3}max\left(Ax-B,0\right)-(r^3-{\tau }^3)xmax(A,0)]\frac{{\left(t^{\alpha }\right)}^3}{3!{\alpha }^3},\hfill \\ \hfill ⋮\\ \hfill \mathcal{H}\left[{\tilde{u}}_n(x,t)\right]& =-L_{\alpha }^{-1}\left[\frac{{\sigma }^2}{2}{x}^2\frac{{\partial }^2\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x^2}+(r-\tau )x\frac{\partial \mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]}{\partial x}-r\mathcal{H}\left[{\tilde{u}}_{n-1}(x,t)\right]\right]\hfill \\ & =[\mu +2(1-\mu ){\alpha }_c][r^{n}max\left(Ax-B,0\right)-(r^n-{\tau }^n)xmax(A,0)]\frac{{\left(t^{\alpha }\right)}^n}{n!{\alpha }^n}.\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{u}(x,t)& =\underset{\mu }{\cup }[\mu ,2-\mu ][max(Ax-B,0)e^{\frac{rt}{\alpha }}\hfill \\ & -max\left(A,0\right)(e^{\frac{rt}{\alpha }}-e^{\frac{\tau t}{\alpha }})]\hfill \\ & =(0,1,2)[max(Ax-B,0)e^{\frac{rt}{\alpha }}\hfill \\ & -max\left(A,0\right)(e^{\frac{rt}{\alpha }}-e^{\frac{\tau t}{\alpha }})].\hfill \end{array}\end{array}$$

(150)

In

Table 7. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.5,\tau =0.2,$ $r=0.25,A=1,B=10$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00405	0.00061	$-[\mu +2(1-\mu ){\alpha }_c]$0.00405	0.00061	$-[\mu +2(1-\mu ){\alpha }_c]$0.00405	0.00061	$-[\mu +2(1-\mu ){\alpha }_c]0.00729$	0.02339
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00605	0.00225	$-[\mu +2(1-\mu ){\alpha }_c]$0.00605	0.00225	$-[\mu +2(1-\mu ){\alpha }_c]$0.00605	0.00225	$-[\mu +2(1-\mu ){\alpha }_c]0.01093$	0.02490
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00772	0.00416	$-[\mu +2(1-\mu ){\alpha }_c]$0.00772	0.00416	$-[\mu +2(1-\mu ){\alpha }_c]$0.00772	0.00416	$-[\mu +2(1-\mu ){\alpha }_c]0.01399$	0.02609
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00923	0.00627	$-[\mu +2(1-\mu ){\alpha }_c]$0.00923	0.00627	$-[\mu +2(1-\mu ){\alpha }_c]$0.00923	0.00627	$-[\mu +2(1-\mu ){\alpha }_c]$0.01676	0.02712
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01063	0.00852	$-[\mu +2(1-\mu ){\alpha }_c]$0.01063	0.00852	$-[\mu +2(1-\mu ){\alpha }_c]$0.01063	0.00852	$-[\mu +2(1-\mu ){\alpha }_c]$0.01936	0.02803
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00811	0.00122	$-[\mu +2(1-\mu ){\alpha }_c]$0.00811	0.00122	$-[\mu +2(1-\mu ){\alpha }_c]$0.00811	0.00122	$-[\mu +2(1-\mu ){\alpha }_c]$0.01458	0.04677
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01210	0.00449	$-[\mu +2(1-\mu ){\alpha }_c]$0.01210	0.00449	$-[\mu +2(1-\mu ){\alpha }_c]$0.01210	0.00449	$-[\mu +2(1-\mu ){\alpha }_c]$0.02185	0.04980
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01544	0.00832	$-[\mu +2(1-\mu ){\alpha }_c]$0.01544	0.00832	$-[\mu +2(1-\mu ){\alpha }_c]$0.01544	0.00832	$-[\mu +2(1-\mu ){\alpha }_c]$0.02798	0.05219
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01845	0.01253	$-[\mu +2(1-\mu ){\alpha }_c]$0.01845	0.01253	$-[\mu +2(1-\mu ){\alpha }_c]$0.01845	0.01253	$-[\mu +2(1-\mu ){\alpha }_c]$0.03353	0.05424
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.02127	0.01704	$-[\mu +2(1-\mu ){\alpha }_c]$0.02127	0.01704	$-[\mu +2(1-\mu ){\alpha }_c]$0.02127	0.01704	$-[\mu +2(1-\mu ){\alpha }_c]$0.04191	0.06780
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.01216	0.00182	$-[\mu +2(1-\mu ){\alpha }_c]$0.01216	0.00182	$-[\mu +2(1-\mu ){\alpha }_c]$0.01216	0.00182	$-[\mu +2(1-\mu ){\alpha }_c]$0.02187	0.07016
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01815	0.00674	$-[\mu +2(1-\mu ){\alpha }_c]$0.01815	0.00674	$-[\mu +2(1-\mu ){\alpha }_c]$0.01815	0.00674	$-[\mu +2(1-\mu ){\alpha }_c]$0.03278	0.07470
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.02316	0.01249	$-[\mu +2(1-\mu ){\alpha }_c]$0.02316	0.01249	$-[\mu +2(1-\mu ){\alpha }_c]$0.02316	0.01249	$-[\mu +2(1-\mu ){\alpha }_c]$0.04197	0.07828
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.02768	0.01880	$-[\mu +2(1-\mu ){\alpha }_c]$0.02768	0.01880	$-[\mu +2(1-\mu ){\alpha }_c]$0.02768	0.01880	$-[\mu +2(1-\mu ){\alpha }_c]$0.05029	0.08136
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.03190	0.02556	$-[\mu +2(1-\mu ){\alpha }_c]$0.03190	0.02556	$-[\mu +2(1-\mu ){\alpha }_c]$0.03190	0.02556	$-[\mu +2(1-\mu ){\alpha }_c]$0.05808	0.08410

,

Table 8. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =0.75,\tau =0.2,$ $r=0.25,A=1,B=10$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.00128	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.00128	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.00128	$-[\mu +2(1-\mu ){\alpha }_c]0.00250$	0.02122
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00357	0.00292	$-[\mu +2(1-\mu ){\alpha }_c]$0.00357	0.00292	$-[\mu +2(1-\mu ){\alpha }_c]$0.00357	0.00292	$-[\mu +2(1-\mu ){\alpha }_c]0.00436$	0.02209
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00501	0.00472	$-[\mu +2(1-\mu ){\alpha }_c]$0.00501	0.00472	$-[\mu +2(1-\mu ){\alpha }_c]$0.00501	0.00472	$-[\mu +2(1-\mu ){\alpha }_c]0.00610$	0.02287
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00641	0.00666	$-[\mu +2(1-\mu ){\alpha }_c]$0.00641	0.00666	$-[\mu +2(1-\mu ){\alpha }_c]$0.00641	0.00666	$-[\mu +2(1-\mu ){\alpha }_c]$0.00779	0.02360
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.00779	0.00871	$-[\mu +2(1-\mu ){\alpha }_c]$0.00779	0.00871	$-[\mu +2(1-\mu ){\alpha }_c]$0.00779	0.00871	$-[\mu +2(1-\mu ){\alpha }_c]$0.00947	0.02431
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00409	0.00257	$-[\mu +2(1-\mu ){\alpha }_c]$0.00409	0.00257	$-[\mu +2(1-\mu ){\alpha }_c]$0.00409	0.00257	$-[\mu +2(1-\mu ){\alpha }_c]$0.00500	0.04244
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00715	0.00584	$-[\mu +2(1-\mu ){\alpha }_c]$0.00715	0.00584	$-[\mu +2(1-\mu ){\alpha }_c]$0.00715	0.00584	$-[\mu +2(1-\mu ){\alpha }_c]$0.00872	0.04417
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01001	0.00945	$-[\mu +2(1-\mu ){\alpha }_c]$0.01001	0.00945	$-[\mu +2(1-\mu ){\alpha }_c]$0.01001	0.00945	$-[\mu +2(1-\mu ){\alpha }_c]$0.01220	0.04574
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01281	0.01331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01281	0.01331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01281	0.01331	$-[\mu +2(1-\mu ){\alpha }_c]$0.01559	0.04721
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01558	0.01742	$-[\mu +2(1-\mu ){\alpha }_c]$0.01558	0.01742	$-[\mu +2(1-\mu ){\alpha }_c]$0.01558	0.01742	$-[\mu +2(1-\mu ){\alpha }_c]$0.01894	0.04861
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00614	0.00385	$-[\mu +2(1-\mu ){\alpha }_c]$0.00614	0.00385	$-[\mu +2(1-\mu ){\alpha }_c]$0.00614	0.00385	$-[\mu +2(1-\mu ){\alpha }_c]$0.00750	0.06366
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.01072	0.00877	$-[\mu +2(1-\mu ){\alpha }_c]$0.01072	0.00877	$-[\mu +2(1-\mu ){\alpha }_c]$0.01072	0.00877	$-[\mu +2(1-\mu ){\alpha }_c]$0.01308	0.06626
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.01502	0.01417	$-[\mu +2(1-\mu ){\alpha }_c]$0.01502	0.01417	$-[\mu +2(1-\mu ){\alpha }_c]$0.01502	0.01417	$-[\mu +2(1-\mu ){\alpha }_c]$0.01831	0.06861
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01922	0.01997	$-[\mu +2(1-\mu ){\alpha }_c]$0.01922	0.01997	$-[\mu +2(1-\mu ){\alpha }_c]$0.01922	0.01997	$-[\mu +2(1-\mu ){\alpha }_c]$0.02338	0.07081
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.02338	0.02613	$-[\mu +2(1-\mu ){\alpha }_c]$0.02338	0.02613	$-[\mu +2(1-\mu ){\alpha }_c]$0.02338	0.02613	$-[\mu +2(1-\mu ){\alpha }_c]$0.02841	0.07292

and

Table 9. $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ by ETHPM, HPM, RPSM, CFADM for $\alpha =1,\tau =0.2,$$r=0.25,A=1,B=10$.

		ETHPM		HPM		RPSM		CFADM
x	t	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$	$\mathcal{H}\left[\tilde{u}(x,t)\right]$	$max|\widetilde{Res}(x,t)|$
0.2	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00102	0.02051	$-[\mu +2(1-\mu ){\alpha }_c]$0.00102	0.02051	$-[\mu +2(1-\mu ){\alpha }_c]$0.00102	0.02051	$-[\mu +2(1-\mu ){\alpha }_c]$0.00102	0.02051
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00209	0.02102	$-[\mu +2(1-\mu ){\alpha }_c]$0.00209	0.02102	$-[\mu +2(1-\mu ){\alpha }_c]$0.00209	0.02102	$-[\mu +2(1-\mu ){\alpha }_c]$0.00209	0.02102
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00321	0.02155	$-[\mu +2(1-\mu ){\alpha }_c]$0.00321	0.02155	$-[\mu +2(1-\mu ){\alpha }_c]$0.00321	0.02155	$-[\mu +2(1-\mu ){\alpha }_c]$0.00321	0.02155
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00438	0.02209	$-[\mu +2(1-\mu ){\alpha }_c]$0.00438	0.02209	$-[\mu +2(1-\mu ){\alpha }_c]$0.00438	0.02209	$-[\mu +2(1-\mu ){\alpha }_c]$0.00438	0.02209
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.00559	0.02264	$-[\mu +2(1-\mu ){\alpha }_c]$0.00559	0.02264	$-[\mu +2(1-\mu ){\alpha }_c]$0.00559	0.02264	$-[\mu +2(1-\mu ){\alpha }_c]$0.00559	0.02264
0.4	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.04101	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.04101	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.04101	$-[\mu +2(1-\mu ){\alpha }_c]$0.00205	0.04101
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00418	0.04205	$-[\mu +2(1-\mu ){\alpha }_c]$0.00418	0.04205	$-[\mu +2(1-\mu ){\alpha }_c]$0.00418	0.04205	$-[\mu +2(1-\mu ){\alpha }_c]$0.00418	0.04205
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00642	0.04311	$-[\mu +2(1-\mu ){\alpha }_c]$0.00642	0.04311	$-[\mu +2(1-\mu ){\alpha }_c]$0.00642	0.04311	$-[\mu +2(1-\mu ){\alpha }_c]$0.00642	0.04311
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.00875	0.04419	$-[\mu +2(1-\mu ){\alpha }_c]$0.00875	0.04419	$-[\mu +2(1-\mu ){\alpha }_c]$0.00875	0.04419	$-[\mu +2(1-\mu ){\alpha }_c]$0.00875	0.04419
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01119	0.04529	$-[\mu +2(1-\mu ){\alpha }_c]$0.01119	0.04529	$-[\mu +2(1-\mu ){\alpha }_c]$0.01119	0.04529	$-[\mu +2(1-\mu ){\alpha }_c]$0.01894	0.04861
0.6	0.1	$-[\mu +2(1-\mu ){\alpha }_c]$0.00307	0.06152	$-[\mu +2(1-\mu ){\alpha }_c]$0.00307	0.06152	$-[\mu +2(1-\mu ){\alpha }_c]$0.00307	0.06152	$-[\mu +2(1-\mu ){\alpha }_c]$0.00307	0.06152
	0.2	$-[\mu +2(1-\mu ){\alpha }_c]$0.00628	0.06307	$-[\mu +2(1-\mu ){\alpha }_c]$0.00628	0.06307	$-[\mu +2(1-\mu ){\alpha }_c]$0.00628	0.06307	$-[\mu +2(1-\mu ){\alpha }_c]$0.00628	0.06307
	0.3	$-[\mu +2(1-\mu ){\alpha }_c]$0.00963	0.06466	$-[\mu +2(1-\mu ){\alpha }_c]$0.00963	0.06466	$-[\mu +2(1-\mu ){\alpha }_c]$0.00963	0.06466	$-[\mu +2(1-\mu ){\alpha }_c]$0.00963	0.06466
	0.4	$-[\mu +2(1-\mu ){\alpha }_c]$0.01313	0.06628	$-[\mu +2(1-\mu ){\alpha }_c]$0.01313	0.06628	$-[\mu +2(1-\mu ){\alpha }_c]$0.01313	0.06628	$-[\mu +2(1-\mu ){\alpha }_c]$0.01313	0.06628
	0.5	$-[\mu +2(1-\mu ){\alpha }_c]$0.01678	0.06793	$-[\mu +2(1-\mu ){\alpha }_c]$0.01678	0.06793	$-[\mu +2(1-\mu ){\alpha }_c]$0.01678	0.06793	$-[\mu +2(1-\mu ){\alpha }_c]$0.01678	0.06793

, we show the $\mathcal{H}\left[\tilde{u}(x,t)\right]$ and $max|\widetilde{Res}(x,t)|$ of ETHPM, HPM, RPSM, and CFADM between different values of x and t when the fractions $\alpha =0.5,0.75,1$ and the parameters $\tau =0.2,r=0.25,A=1,B=10$, respectively. It is easy to see in the table that the value of $max|\widetilde{Res}(x,t)|$ increases with t when $\alpha$, x is fixed, the value of $max|\widetilde{Res}(x,t)|$ increases with x when $\alpha$, t is fixed and the value of $max|\widetilde{Res}(x,t)|$ increases with $\alpha$ when x, t is fixed.

In Figure 4, the picture shows the fuzzy approximate solution $\tilde{u}(x,t)$ of the Black–Scholes equation at the fractional parameter $\alpha =1$ and the auxiliary parameters $\mu =0.5,$ $\tau =0.2,r=0.25,A=1,B=10$.

5. Conclusions

In this research, we handled the approximate solution of the fuzzy time-fractional Black–Scholes’ European option pricing equation by using the homotopy transforming method under gr-differentiability circumstances. According to the comparative analysis of the four methods, it can be seen that ETHPM doesn’t require fractional differentiation or integration operations, which reduces the complexity and the amount of computation. Our method is better suited to computer programming implementations.