Analysis of the Fractional Differential Equations Using Two Different Methods

Abstract

Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.

1. Introduction

The fractional calculus’ sense (FC) is presented after classical calculus, but after identifying the limitations of the classical one, many researchers weighed the notions of fractional calculus to comprehend the character systematically. Plenty of mathematicians promoted the vital establishment with the aid of new attributes and corresponding outcomes for FC [1,2,3,4,5,6]. Specifically, the particular functions are proposed to create novel non-integer integral and differential operators. These latter are presented by many people to investigate and symbolize different equations linked with phenomena [7,8,9,10,11,12,13,14,15]. The symmetric and anti-symmetric solitons of the fractional Schrödinger equation have been studied in [16]. In [17], the authors extended the Lie symmetry analysis to the time fractional generalized KdV equations. The Adomian decomposition technique for investigating the fractional KdV–Burgers equation was applied in [18]. Since the Chebyshev collocation technique is implemented for investigating the time-fractional nonlinear Klein–Gordon equation [19], a geometric method is applied for the Korteweg–de Vries equation [20]. A combination of FTIM and geometric method is applied for the fractional Burger–Huxley equation [21]. Additionally, in [22], a lie group approach is implemented for solving the fractional equation. Some other applications can be seen in [23,24,25,26,27,28,29,30,31].

Consider the following equation:

$$c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }u=\lambda u_{xx}+g(x,t),(x,t)\in \mathsf{\Omega },$$

(1)

$$u(x,t)=H(x,t),on\mathsf{\Gamma },$$

(2)

where c denotes the specific heat, c is the density, $\lambda$ describes the thermal conductivity coefficient, and ${}^C{\mathcal{D}}_{0^{+},t}^{\alpha }$ shows the Caputo fractional derivative. Equations of this type are used to describe the transport processes with a long memory.The fractional heat equation is one of the most well-known fractional partial differential equations that describe the physical phenomenon. In recent years, solving the fractional heat equation magnetized the attention of mathematicians because of its importance. Many methods have been worked to solve this problem. The higher-order numerical method is used for investigating the fractional heat equation [32]. Additionally, the Laplace homotopy technique is worked to solve this equation [33]. In [34], authors used one-step backward-forward algorithms for multi-dimensional backward heat conduction problems.

Motivated by the above works, in this paper, two numerical methods are worked to solve this equation. One is a combination of a specific type of fictitious time integration technique and the Runge–Kutta method. The other is the RKHSM. The main paper’s contributions are as follows:

• We present new results on the numerical simulation for the considered equation.
• We apply two effective numerical methods to obtain these new accurate results.
• The convergence analysis that confirms the theoretical parts of both methods is discussed.

Variable transformation of a time integration method, namely FTIM, was suggested by Atluri and Liu. Researchers used it for solving linear or nonlinear algebraic equations by defining the fictitious time and using it to derive a system of nonautonomous first-order ordinary differential equations that is equivalent to the original algebraic equations in an n-dimensional space. Some applications of this technique can be seen in [35,36,37,38,39].

In another aspect of this paper, as we mentioned before, we applied the RKHSM for solving the proposed equation.

Recently, the RKHSM has achieved great popularity and success. It became a powerful tool in treating different types of FPDEs, such as the fractional Bloch–Torrey equations [40] and fractional differential equations, including the ABC derivative [41], to name a few. See also [42,43,44,45,46,47] for more research about this method. The RKHSM has many advantages, such as its simplicity and flexibility in treating many fractional differential systems and the fact that it is a mesh-free method. The rest of the paper consists of the following: Section 2 recalls some essential concepts about fractional calculus and reproducing kernel theories. Section 3 and Section 4 are where we see the main theory of the FTIM and RKHSM, respectively, to build a numerical solution for the considered problem. Before finishing with the conclusion part, we validate the proposed methods through two examples.

2. Basic Definitions

Definition 1. 

The left-sided Riemannian–Liouville fractional integral of order $\mu \in {\mathbb{R}}_{+}$ of $f\in C_{\alpha },\alpha \ge -1$, is

$$I^{\mu }f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mu \right)}}\underset{0}{\overset{x}{\int }}{\displaystyle \frac{f\left(\eta \right)}{{(x-\eta )}^{1-\mu }}}\mathrm{d}\eta ,0<\mu ,x>0,I^{0}f\left(x\right)=f\left(x\right).$$

(3)

Definition 2. 

We write

$$f\left(x\right)\in C_{\alpha }^m,x>0,m\in N\cup 0,$$

provided

$$f^{\left(m\right)}\in C_{\alpha }.$$

Definition 3. 

Suppose a real function f (with $0<x$) is in the space $C_{\alpha }$, $\alpha \in \mathcal{R}$. By $p(>\alpha )$ such that $f\left(x\right)=x^p{f}_1\left(x\right)\in C[0,\infty ]$. Obviously, $C_{\alpha }\in C_{\beta }$ if $\beta \subset \alpha$.

Definition 4. 

Suppose $f\in C_{-1}^m,m\in N$, the Caputo derivative of f is

$$D^{\mu }f\left(x\right)=\left\{\begin{array}{c}[I^{m-\mu ]}f\left(x\right),\mu \in (m-1,m],\hfill \\ \\ \\ {\displaystyle \frac{d^m}{d{t}^m}}f\left(x\right),\mu =m.\hfill \end{array}\right.$$

(4)

$$I^{\mu }{I}^{\nu }f=I^{\mu +\nu },0\le \mu ,0\le \nu ,f\in C_{\alpha },0\le \alpha .$$

(5)

$$I^{\mu }{x}^{\xi }f={\displaystyle \frac{\mathsf{\Gamma }(\gamma +1)}{\mathsf{\Gamma }(\gamma +\mu +1)}}{x}^{\mu +\xi },0<\mu ,-1<\xi ,0<x.$$

(6)

Lemma 1. 

Assume $\alpha \in (m-1,m]$ and $f\in L_1[a,b].$ Then,

$$J_a^{\mu }f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{a}{\overset{x}{\int }}{(x-t)}^{1-\mu }f\left(t\right)\mathrm{d}t,D_a^{\alpha }{J}_a^{\alpha }f\left(x\right)=f\left(x\right),$$

(7)

and

$$D_a^{\alpha }{J}_a^{\alpha }f\left(x\right)=f\left(x\right)-\sum _{k=1}^{m-1}{f}^{\left(k\right)}\left(0\right){\displaystyle \frac{{(x-a)}^k}{k!}},x>0.$$

(8)

Definition 5. 

The fractional derivative of f in the Caputo sense is

$$D^{\alpha }f\left(x\right)=J^{m-\alpha }{D}^{m}f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(m-\alpha )}}\underset{0}{\overset{x}{\int }}{(x-t)}^{m-\alpha -1}{f}^m\left(t\right)\mathrm{d}t.$$

(9)

Definition 6. 

The Caputo time-fractional derivative operator is

$${}^C{\mathcal{D}}_t^{\alpha }u(x,t)=\left\{\begin{array}{c}\frac{1}{\mathsf{\Gamma }(m-\alpha )}{\int }_0^t{(t-z)}^{m-\alpha -1}\frac{{\partial }^{m}u(x,z)}{\partial z^m}\mathrm{d}z,\alpha \in (m-1,m),\hfill \\ \\ \\ \frac{{\partial }^{m}u(x,t)}{\partial t^m},\alpha =m,\hfill \end{array}\right.$$

(10)

for m to be the smallest integer that exceeds $\alpha .$ The space-fractional derivative with $\beta >0$ is described by

$${}^C{\mathcal{D}}_x^{\beta }u(x,t)=\left\{\begin{array}{c}\frac{1}{\mathsf{\Gamma }(m-\beta )}{\int }_0^x{(x-s)}^{m-\beta -1}\frac{{\partial }^{m}u(s,t)}{\partial s^m}\mathrm{d}s,m-1<\beta <m,\hfill \\ \\ \\ \frac{{\partial }^{m}u(x,t)}{\partial t^m},\beta =m.\hfill \end{array}\right.$$

(11)

Notations

(i)

We will write

$$AC[a,b]=\left\{f:[a,b]\to \mathbb{R}\left|f\mathrm{is}\mathrm{absolutely}\mathrm{continuous}\mathrm{on}\right[a,b]\right\}$$

to denote the collection of all absolutely continuous functions on $[a,b].$

(ii)

We write CCF to mean a completely continuous function.

Definition 7. 

Define the function space ${\mathfrak{W}}_2^3[a,b]$ by

$$\begin{array}{cc}\hfill {\mathfrak{W}}_2^3[a,b]=\left\{f\left(x\right)\right|& f^{\left(\mathfrak{j}\right)}\in AC[a,b],\mathfrak{j}=0,1,2,f^{\left(3\right)}\in L^2[a,b],\mathrm{and}f\left(a\right)=f\left(b\right)=0\}.\hfill \end{array}$$

Definition 8. 

If $f,g\in {\mathfrak{W}}_2^3[a,b],$ the inner product and norm of this space are described to be

$${⟨f,g⟩}_{{\mathfrak{W}}_2^3}=\sum _{\mathfrak{i}=0}^2{f}^{\left(\mathfrak{i}\right)}\left(a\right)g^{\left(\mathfrak{i}\right)}\left(a\right)+{\int }_a^b{f}^{\left(3\right)}\left(x\right)g^{\left(3\right)}\left(x\right)\mathrm{d}x,$$

(12)

and

$${∥f∥}_{{\mathfrak{W}}_2^3}=\sqrt{{⟨f,f⟩}_{{\mathfrak{W}}_2^3}}.$$

(13)

Theorem 1. 

The RK function of ${\mathfrak{W}}_2^3[0,1]$ is the function ${\mathtt{R}}_{\zeta }\left(x\right)$ described as

$${\mathtt{R}}_{\zeta }\left(x\right)=\left\{\begin{array}{cc}\hfill & \mathtt{r}(x,\zeta ),x\le \zeta ,\hfill \\ \hfill & \mathtt{r}(\zeta ,x),x>\zeta ,\hfill \end{array}\right.$$

(14)

with

$$\begin{array}{cc}\hfill \mathtt{r}(x,\zeta )=& \frac{3x\zeta }{13}-\frac{x{\zeta }^5}{156}+\frac{5x{\zeta }^4}{156}-\frac{5x{\zeta }^3}{78}-\frac{5x{\zeta }^2}{26}+\frac{21x^2{\zeta }^2}{104}-\frac{x^2{\zeta }^5}{624}+\frac{5x^2{\zeta }^4}{624}-\frac{5x^2{\zeta }^3}{312}-\frac{5x^2\zeta }{26}+\frac{7x^3{\zeta }^2}{104}-\frac{x^3{\zeta }^5}{1872}\hfill \\ \hfill & +\frac{5x^3{\zeta }^4}{1872}-\frac{5x^3{\zeta }^3}{936}-\frac{5x^3\left(\zeta \right)}{78}-\frac{x^4\left(\zeta \right)}{104}+\frac{x^4{\zeta }^5}{3744}-\frac{5x^4{\zeta }^4}{3744}+\frac{5x^4{\zeta }^3}{1872}+\frac{5x^4{\zeta }^2}{624}-\frac{x^5{\zeta }^5}{18720}+\frac{x^5{\zeta }^4}{3744}-\frac{x^5{\zeta }^3}{1872}\hfill \\ \hfill & -\frac{x^5{\zeta }^2}{624}-\frac{x^5\left(\zeta \right)}{156}+\frac{x^5}{120}.\hfill \end{array}$$

(15)

For the proof, see [43].

Definition 9. 

Define the function space ${\mathfrak{W}}_2^2[c,d]$ by

$$\begin{array}{cc}\hfill {\mathfrak{W}}_2^2[c,d]=\left\{f\left(t\right)\right|& f^{\left(\mathfrak{j}\right)}\in AC[c,d],\mathfrak{j}=0,1,f^{″}\in L^2[c,d],\mathrm{and}f\left(c\right)=0\}.\hfill \end{array}$$

Definition 10. 

If $f,g\in {\mathfrak{W}}_2^2[c,d],$ the inner product and norm are

$${⟨f,g⟩}_{{\mathfrak{W}}_2^2}=\sum _{\mathfrak{i}=0}^1{f}^{\left(\mathfrak{i}\right)}\left(c\right)g^{\left(\mathfrak{i}\right)}\left(c\right)+{\int }_c^d{f}^{″}\left(t\right)g^{″}\left(t\right)\mathrm{d}t,$$

(16)

$${∥f∥}_{{\mathfrak{W}}_2^2}=\sqrt{{⟨f,f⟩}_{{\mathfrak{W}}_2^2}}.$$

(17)

Theorem 2. 

The RK function of ${\mathfrak{W}}_2^2[c,d]$ is the function ${\mathtt{K}}_{\eta }\left(t\right)$ defined by

$${\mathtt{K}}_{\eta }\left(t\right)=\left\{\begin{array}{cc}\hfill & -\frac{1}{3}{c}^3+\frac{1}{2}{c}^2\eta +c^2-\eta c+\left(\frac{1}{2}{c}^2-\eta c-c+\eta \right)t+\frac{1}{2}\eta t^2-\frac{1}{6}{t}^3,t\le \eta ,\hfill \\ \hfill & -\frac{1}{3}{c}^3+\frac{1}{2}{c}^2\eta -\frac{1}{6}{\eta }^3+c^2-\eta c+\left(\frac{1}{2}{c}^2-\eta c+\frac{1}{2}{\eta }^2-c+\eta \right)t,t>\eta .\hfill \end{array}\right.$$

(18)

For the proof, see [43].

Definition 11. 

If $f,g\in {\mathfrak{W}}_2^1[a,b],$ the inner product and norm are

$${⟨f,g⟩}_{{\mathfrak{W}}_2^1}=f\left(a\right)g\left(a\right)+{\int }_a^b{f}^{\prime }\left(x\right)g^{\prime }\left(x\right)\mathrm{d}x,$$

(19)

and

$${∥f∥}_{{\mathfrak{W}}_2^1}=\sqrt{{⟨f,f⟩}_{{\mathfrak{W}}_2^1}}.$$

(20)

Theorem 3. 

The RK function of ${\mathfrak{W}}_2^1[a,b]$ is the function ${\mathtt{F}}_{\zeta }\left(x\right)$ defined by

$${\mathtt{F}}_{\zeta }\left(x\right)=\left\{\begin{array}{cc}\hfill & -a+1+t,t\le \zeta ,\hfill \\ \hfill & \zeta -a+1,t>\zeta .\hfill \end{array}\right.$$

(21)

For the proof, see [43].

Throughout $\mathsf{\Omega }=[a,b]\times [c,d].$

Definition 12. 

Define the binary function space ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)$ by

$$\begin{array}{cc}\hfill {\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)=\left\{u|\right.& \frac{{\partial }^3}{\partial x^2\partial t}uisCCFon\mathsf{\Omega },\frac{{\partial }^5}{\partial x^3\partial t^2}u\in L^2\left(\mathsf{\Omega }\right),\hfill \\ \hfill & \left.u(x,c)=u(a,t)=u(b,t)=0\right\}.\hfill \end{array}$$

Definition 13. 

If $u,v\in {\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right),$ the inner product and norm of this space are

$$\begin{array}{cc}\hfill {⟨u,v⟩}_{{\mathfrak{W}}_2^{(3,2)}}=& \sum _{\mathfrak{j}=0}^1{\int }_c^d\left[\frac{{\partial }^2}{\partial t^2}\frac{{\partial }^{\mathfrak{j}}}{\partial x^{\mathfrak{j}}}u(a,t)\frac{{\partial }^2}{\partial t^2}\frac{{\partial }^{\mathfrak{j}}}{\partial x^{\mathfrak{j}}}v(a,t)\right]\mathrm{d}t+{\int }_c^d\left[\frac{{\partial }^2}{\partial t^2}u(b,t)\frac{{\partial }^2}{\partial t^2}v(b,t)\right]\mathrm{d}t\hfill \\ \hfill & +\sum _{\mathfrak{j}=0}^1{⟨\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}u(x,c),\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}v(x,c)⟩}_{{\mathfrak{W}}_2^3}+{\int }_c^d{\int }_a^b\frac{{\partial }^3}{\partial x^3}\frac{{\partial }^2}{\partial t^2}u(x,t)\frac{{\partial }^3}{\partial x^3}\frac{{\partial }^2}{\partial t^2}v(x,t)\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(22)

and

$${∥u∥}_{{\mathfrak{W}}_2^{(3,2)}}=\sqrt{{⟨u,u⟩}_{{\mathfrak{W}}_2^{(3,2)}}}.$$

(23)

Theorem 4. 

The RK function of ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)$ is the function

$${\mathtt{Y}}_{(\zeta ,\eta )}(x,t)={\mathtt{R}}_{\zeta }\left(x\right){\mathtt{K}}_{\eta }\left(t\right).$$

(24)

Definition 14. 

Define the binary function space ${\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right)$ by

$${\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right)=\left\{u(x,t)|u(x,t)isCCFin\mathsf{\Omega },\frac{{\partial }^2}{\partial x\partial t}u(x,t)\in L^2\left(\mathsf{\Omega }\right)\right\}.$$

Definition 15. 

If $u,v\in {\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right),$ the inner product and norm of this space are

$$\begin{array}{cc}\hfill {⟨u,v⟩}_{{\mathfrak{W}}_2^{(1,1)}}=& {\int }_c^d\left[\frac{\partial }{\partial t}u(a,t)\frac{\partial }{\partial t}v(a,t)\right]\mathrm{d}t+{⟨u(x,c),v(x,c)⟩}_{{\mathfrak{W}}_2^1}\hfill \\ \hfill & +{\int }_c^d{\int }_a^b\left[\frac{\partial }{\partial x}\frac{\partial }{\partial t}u(x,t)\frac{\partial }{\partial x}\frac{\partial }{\partial t}v(x,t)\right]\mathrm{d}x\mathrm{d}t,\hfill \end{array}$$

(25)

and

$${∥u∥}_{{\mathfrak{W}}_2^{(1,1)}}=\sqrt{{⟨u,u⟩}_{{\mathfrak{W}}_2^{(1,1)}}}.$$

(26)

Theorem 5. 

The RK function of ${\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right)$ is the function

$${\mathtt{T}}_{(\zeta ,\eta )}(x,t)={\mathtt{F}}_{\zeta }\left(x\right){\mathtt{F}}_{\eta }\left(t\right).$$

(27)

3. The Fictitious Time Integration Method (FTIM)

Here, the FTIM is implemented. Consider the following equation:

$$c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }u+b\left(x\right)u_x+u_{xx}+g(x,t)=0,(x,t)\in \mathsf{\Omega }.$$

(28)

Using (10), $0<\alpha <1$ and $0<\beta \le 1$ for Equation (28), we have

$$\begin{array}{cc}\hfill & \frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{u_{\sigma }(x,\sigma )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma -b\left(x\right)u_x\hfill \\ \hfill & +u_{xx}+g(x,t)=0.\hfill \end{array}$$

(29)

To increase the stability of the technique, we propose a fictitious damping coefficient $\mu$ in Equation (29) by

$$\begin{array}{cc}\hfill & \frac{\mu cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{u_{\sigma }(x,\sigma )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma -\mu b\left(x\right)u_x\hfill \\ \hfill & +\mu u_{xx}+\mu g(x,t)=0.\hfill \end{array}$$

(30)

Imposing, in Equation (30), the transformation

$$\omega (x,t,\xi )={(1+\theta )}^{\lambda }u(x,t),0<\lambda \le 1,$$

(31)

yields

$$\begin{array}{cc}\hfill & \frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{{\omega }_{\sigma }(x,\sigma ,\theta )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x\right){\omega }_x(x,t,\theta )+{\omega }_{xx}(x,t,\theta ))+\mu g(x,t)=0.\hfill \end{array}$$

(32)

We consider

$$\frac{\partial \omega }{\partial \theta }=\lambda {(1+\theta )}^{\lambda -1}u(x,t).$$

(33)

Equation (32) will be

$$\begin{array}{cc}\hfill \frac{\partial \omega }{\partial \theta }& =\frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{{\omega }_{\sigma }(x,\sigma ,\theta )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x\right){\omega }_x(x,t,\theta )+{\omega }_{xx}(x,t,\theta ))+\mu g(x,t)+\lambda {(1+\theta )}^{\lambda -1}u.\hfill \end{array}$$

(34)

Equation (34) can be converted to a new class of PDE for $\omega$ by choosing $u=\omega /{(1+\theta )}^{\lambda }$:

$$\begin{array}{cc}\hfill \frac{\partial \omega }{\partial \theta }& =\frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{{\omega }_{\sigma }(x,\sigma ,\theta )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x\right){\omega }_x(x,t,\theta )+{\omega }_{xx}(x,t,\theta ))+\mu g(x,t)+\frac{\lambda \omega (x,t,\theta )}{1+\theta }.\hfill \end{array}$$

(35)

Using

$$\begin{array}{c}\hfill \frac{\partial }{\partial \theta }\left(\frac{\omega }{{(1+\theta )}^{\lambda }}\right)=\frac{{\omega }_{\theta }}{{(1+\theta )}^{\lambda }}-\frac{\lambda \omega }{{(1+\theta )}^{\lambda +1}},\end{array}$$

(36)

Implementing $1/{(1+\theta )}^{\lambda }$ for Equation (35), one obtains

$$\begin{array}{cc}\hfill \frac{\partial }{\partial \theta }\left(\frac{\omega }{{(1+\theta )}^{\lambda }}\right)& =\frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{{\omega }_{\sigma }(x,\sigma ,\theta )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x\right){\omega }_x(x,t,\theta )+{\omega }_{xx}(x,t,\theta ))+\mu g(x,t).\hfill \end{array}$$

(37)

By $u=\frac{\omega }{{(1+\theta )}^{\lambda }}$, we obtain

$$\begin{array}{cc}\hfill u_{\theta }& =\frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t{\displaystyle \frac{u_{\sigma }(x,\sigma ,\theta )}{{(t-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x\right)u_x(x,t,\theta )+u_{xx}(x,t,\theta ))+\mu g(x,t).\hfill \end{array}$$

(38)

Suppose that $u_i^j\left(\theta \right):=u(x_i,t_j,\theta )$ as the discrete values of u at a grid point $(x_i,t_j)$, and Equation (38) converts to

$$\begin{array}{cc}\hfill \frac{d}{d\xi }{u}_i^j\left(\theta \right)& =\frac{\mu }{{(1+\theta )}^{\lambda }}(\frac{cq}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^{t_j}{\displaystyle \frac{u_{\sigma }(x_i,\sigma ,\theta )}{{(t_j-\sigma )}^{\alpha }}}\mathrm{d}\sigma \hfill \\ \hfill & +b\left(x_i\right)u_x(x_i,t_j,\theta )+u_{xx}(x_i,t_j,\theta ))+\mu g(x_i,t_j),\hfill \end{array}$$

(39)

where

$$\begin{array}{c}\hfill {\int }_0^{t_j}\frac{u_{\sigma }(x_i,\sigma ,\theta )}{{(t_j-\sigma )}^{\alpha }}\mathrm{d}\sigma \approx \sum _{l=1}^{j-1}\frac{u(x_i,t_{l+1},\theta )-u(x_i,t_l,\theta )}{\mathsf{\Delta }t{(t_j-t_l)}^{\alpha }},\end{array}$$

(40)

where $t_j=j\mathsf{\Delta }t,$$x_i=a+i\mathsf{\Delta }x,$ and $\mathsf{\Delta }t=\frac{T}{n}$.

$\mathbf{u}={(u_1^1,u_1^2,\cdots ,u_m^n)}^T$, Equation (39) can be abstracted by

$${\mathbf{u}}^{\prime }=\mathbf{Q}(\mathbf{u},\xi ),\mathbf{u}\in {\mathbb{R}}^{m\times n},\xi \in \mathbb{R},M=m\times n,$$

(41)

where $\mathbf{Q}\in {\mathbb{R}}^M$ is a vector-valued function of $\mathbf{u}$ and $\theta$ and $\mathbf{u}$ is an M-dimensional vector. Now, we implement the group-preserving scheme (GPS) [48] to solve Equation (39) as

$${\mathbf{u}}_{s+1}={\mathbf{u}}_s+\frac{\left[cosh\left(\frac{\mathsf{\Delta }\theta \parallel {\mathbf{Q}}_s\parallel }{\parallel {\mathbf{u}}_s\parallel }\right)-1\right]{\mathbf{Q}}_s.{\mathbf{u}}_s+sinh\left(\frac{\mathsf{\Delta }\theta \parallel {\mathbf{Q}}_s\parallel }{\parallel {\mathbf{u}}_s\parallel }\right)\parallel {\mathbf{u}}_s\parallel \parallel {\mathbf{Q}}_s\parallel }{\parallel {\mathbf{Q}}_s{\parallel }^2}{\mathbf{Q}}_s={\mathbf{u}}_s+{\Xi }_s{\mathbf{Q}}_s.$$

(42)

Now, we employ the GPS by taking the initial value of $u_i^j\left(0\right)$ to solve Equation (39) from the initial fictitious time $\theta =0$ to a selected final fictitious time ${\theta }_f$. Additionally, the terminating criterion for this method is

$$\sqrt{\sum _{i,j=1}^{m,n}{[u_i^j(s+1)-u_i^j\left(s\right)]}^2}\le \epsilon ,$$

(43)

where $\epsilon$ is a picked convergence criterion. The solution of u will be obtained by

$$u_i^j=\frac{u_i^j\left({\theta }_0\right)}{{(1+{\theta }_0)}^{\lambda }},$$

(44)

where ${\theta }_0(\le {\theta }_f)$ satisfies the above criterion.

4. The Application of RKHSM

4.1. Methodology for RKHSM

In this part, the RKHSM is used to solve Problems (1) and (2), using the following way:

Step 1: Considering the following transformation

$$v=u-P,$$

(45)

where $P(x,t)=-\left(f_2\left(0\right)-f_2\left(t\right)\right)\frac{x}{b}-\left(f_1\left(0\right)-f_1\left(t\right)\right)\left(1-\frac{x}{b}\right)+u_0\left(x\right)$ for which $u(0,t)=f_1\left(t\right),u(b,t)=f_2\left(t\right),$ and $u(x,0)=u_0\left(x\right).$

Consequently, the new form of (1) and (2) is as follows:

$$c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }v-\lambda v_{xx}=\hslash (x,t),0\le x\le b,0\le t\le d,$$

(46)

With

$$v(x,t)=0on\mathsf{\Gamma },$$

(47)

where $\hslash \left(x,t\right)=\lambda P_{xx}(x,t)-c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }P(x,t)+g(x,t).$

Step 2: Defining a linear operator $\mathfrak{T}:{\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)\to {\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right)$ as follows

$$\begin{array}{cc}\mathfrak{T}:{\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)& \to {\mathfrak{W}}_2^{(1,1)}\left(\mathsf{\Omega }\right)\hfill \\ v& \to c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }v-\lambda v_{xx}.\hfill \end{array}$$

(48)

Lemma 2. 

The operator $\mathfrak{T}$ is a bounded linear.

Proof. 

We begin by checking directly that $\mathfrak{T}$ is bounded. So, we must show that

$${∥\mathfrak{T}v∥}_{{\mathfrak{W}}_2^{(1,1)}}\le \mathtt{C}{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}},\mathrm{with}\mathtt{C}>0.$$

(49)

We have

$$\begin{array}{cc}\hfill {∥\mathfrak{T}v(x,t)∥}_{{\mathfrak{W}}_2^{(1,1)}}^2=& {⟨\mathfrak{T}v(x,t),\mathfrak{T}v(x,t)⟩}_{{\mathfrak{W}}_2^{(1,1)}}\hfill \\ \hfill =& {\int }_c^d{\left[\frac{\partial }{\partial t}\mathfrak{T}v(a,t)\right]}^{2}dt+{⟨\mathfrak{T}v(x,c),\mathfrak{T}v(x,c)⟩}_{{\mathfrak{W}}_2^1}+{\int }_c^d{\int }_a^b{\left[\frac{\partial }{\partial x}\frac{\partial }{\partial t}\mathfrak{T}v(x,t)\right]}^{2}dxdt\hfill \\ \hfill =& {\left[\mathfrak{T}v(a,c)\right]}^2+{\int }_a^b{\left[\frac{\partial }{\partial x}\mathfrak{T}v(x,c)\right]}^{2}dx+{\int }_c^d{\left[\frac{\partial }{\partial t}\mathfrak{T}v(a,t)\right]}^{2}dt\hfill \\ \hfill & +{\int }_c^d{\int }_a^b{\left[\frac{\partial }{\partial x}\frac{\partial }{\partial t}\mathfrak{T}v(x,t)\right]}^{2}dxdt.\hfill \end{array}$$

(50)

In view of reproducing the property,

$$v(x,t)={⟨v(\diamond ,\ast ),{\mathtt{Y}}_{(x,t)}(\diamond ,\ast )⟩}_{{\mathfrak{W}}_2^{(3,2)}}.$$

(51)

In a similar way, we deduce

$$\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}v(x,t)={⟨v(\diamond ,\ast ),\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}{\mathtt{Y}}_{(x,t)}(\diamond ,\ast )⟩}_{{\mathfrak{W}}_2^{(3,2)}},\mathfrak{i},\mathfrak{j}\in \{0,1\}.$$

(52)

Applying the Schwarz inequality, we discover

$$\left|\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}v(x,t)\right|=\left|{⟨v(\diamond ,\ast ),\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}{\mathtt{Y}}_{(x,t)}(\diamond ,\ast )⟩}_{{\mathfrak{W}}_2^{(3,2)}}\right|\le {∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}{∥\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}{\mathtt{Y}}_{(x,t)}(\diamond ,\ast )∥}_{{\mathfrak{W}}_2^{(3,2)}}.$$

(53)

Since ${\mathtt{Y}}_{(x,t)}(\diamond ,\ast )$ is continuous, we consequently have

$$\left|\frac{{\partial }^{\mathfrak{i}}}{\partial x^{\mathfrak{i}}}\frac{{\partial }^{\mathfrak{j}}}{\partial t^{\mathfrak{j}}}\mathfrak{T}v(x,t)\right|\le {\mathtt{C}}_{\mathfrak{i},\mathfrak{j}}{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}},\mathfrak{i},\mathfrak{j}\in \{0,1\}.$$

(54)

Hence

$$\begin{array}{cc}\hfill {∥\mathfrak{T}v(x,t)∥}_{{\mathfrak{W}}_2^{(1,1)}}^2\le & {\mathtt{C}}_{0,0}^2{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{\int }_a^b{\mathtt{C}}_{1,0}^2{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2\mathrm{d}\xi +{\int }_c^d{\mathtt{C}}_{0,1}^2{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2\mathrm{d}\tau \hfill \\ \hfill & +{\int }_c^d{\int }_a^b{\mathtt{C}}_{1,1}^2{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2\mathrm{d}x\mathrm{d}t\hfill \\ \hfill \le & \left[{\mathtt{C}}_{0,0}^2+{\mathtt{C}}_{1,0}^2\left(b-a\right)+{\mathtt{C}}_{0,1}^2\left(d-c\right)+{\mathtt{C}}_{1,1}^2\left(b-a\right)\left(d-c\right)\right]{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2.\hfill \end{array}$$

(55)

Therefore,

$${∥\mathfrak{T}v(x,t)∥}_{{\mathfrak{W}}_2^{(1,1)}}^2\le \mathtt{C}{∥v∥}_{{\mathfrak{W}}_2^{(3,2)}}^2,$$

(56)

where $\mathfrak{C}={\mathtt{C}}_{0,0}^2+{\mathtt{C}}_{1,0}^2\left(b-a\right)+{\mathtt{C}}_{0,1}^2\left(d-c\right)+{\mathtt{C}}_{1,1}^2\left(b-a\right)\left(d-c\right).$ □

We apply, next, the operator $\mathfrak{T}$ in order to reformulate the problem (46) and (47) to be

$$\left\{\begin{array}{cc}\mathfrak{T}v(x,t)=\hslash \left(x,t\right),\hfill & (x,t)\in \mathsf{\Omega },\hfill \\ v(x,t)=0\hfill & \mathrm{on}\mathsf{\Gamma },\hfill \end{array}\right.$$

(57)

where $\hslash \left(x,t\right)=\lambda P_{xx}(x,t)-c{q}^C{\mathcal{D}}_{0^{+},t}^{\alpha }P(x,t)+g(x,t).$

Step 3: Construct the ${\left\{{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}\right\}}_{\mathfrak{i}=1}^{\infty }$ on ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right),$ providing this by using the Gram–Schmidt process:

$${\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)=\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}{\mathsf{\Theta }}_{\mathsf{k}}(x,t),0<{\aleph }_{\mathfrak{i}\mathfrak{i}},\mathfrak{i}=1,2,\cdots ,$$

(58)

where

• ${\mathsf{\Theta }}_{\mathfrak{i}}(x,t)={\mathfrak{T}}^{*}{\varrho }_{\mathfrak{i}}(x,t),$ in which ${\mathfrak{T}}^{*}$ denotes the adjoint of $\mathfrak{T}$ and ${\varrho }_{\mathfrak{i}}(x,t)={\mathtt{T}}_{(x_{\mathfrak{i}},t_{\mathfrak{i}})}(x,t)$ where ${\mathtt{T}}_{(x_{\mathfrak{i}},t_{\mathfrak{i}})}(x,t)$ is given by (27).
• The countable set ${\left\{\left(x_{\mathfrak{i}},t_{\mathfrak{i}}\right)\right\}}_{\mathfrak{i}=1}^{\infty }$ is dense in $\mathsf{\Omega }.$
• ${\left\{{\mathsf{\Theta }}_{\mathfrak{i}}\right\}}_{\mathfrak{i}=1}^{\infty }$ is a function system in ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)$ and the following shows the way that we can construct it:

  $$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{\mathfrak{i}}(x,t)& ={\mathfrak{T}}^{*}{\varrho }_{\mathfrak{i}}(x,t)={⟨{\mathfrak{T}}^{*}{\varrho }_{\mathfrak{i}}(\zeta ,\eta ),{\mathtt{Y}}_{(x,t)}(\zeta ,\eta )⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill & ={⟨{\varrho }_{\mathfrak{i}}(\zeta ,\eta ),{\mathfrak{T}}_{(\zeta ,\eta )}{\mathtt{Y}}_{(x,t)}(\zeta ,\eta )⟩}_{{\mathfrak{W}}_2^{(1,1)}}\hfill \\ \hfill & ={⟨{\mathtt{T}}_{({\zeta }_{\mathfrak{i}},{\eta }_{\mathfrak{i}})}(\zeta ,\eta ),{\mathfrak{T}}_{(\zeta ,\eta )}{\mathtt{Y}}_{(x,t)}(\zeta ,\eta )⟩}_{{\mathfrak{W}}_2^{(1,1)}}\hfill \\ \hfill & ={\mathfrak{T}}_{(\zeta ,\eta )}{\mathtt{Y}}_{(x,t)}{(\zeta ,\eta )|}_{(\zeta ,\eta )=(x_{\mathfrak{i}},t_{\mathfrak{i}})}\hfill \\ \hfill & ={\mathfrak{T}}_{(\zeta ,\eta )}{\mathtt{Y}}_{(\zeta ,\eta )}{(x,t)|}_{(\zeta ,\eta )=(x_{\mathfrak{i}},t_{\mathfrak{i}})}\hfill \\ \hfill & =\left\{c{q}^C{\mathcal{D}}_{0^{+},\eta }^{\alpha }{\mathtt{Y}}_{(\zeta ,\eta )}(x,t)-\lambda {\partial }_{{\zeta }^2}^2{\mathtt{Y}}_{(\zeta ,\eta )}(x,t)\right\}\left|{}_{(\zeta ,\eta )=(x_{\mathfrak{i}},t_{\mathfrak{i}})}.\right.\hfill \end{array}$$

  (59)
• ${\aleph }_{\mathfrak{i}\mathsf{k}}$ is the orthogonalization coefficients which are defined by

  $${\aleph }_{\mathfrak{i}\mathfrak{j}}=\left\{\begin{array}{cc}\frac{1}{∥{\mathsf{\Theta }}_1∥},\hfill & \mathrm{for}\mathfrak{i}=\mathfrak{j}=1,\hfill \\ \frac{1}{{\varsigma }_{\mathfrak{i}}},\hfill & \mathrm{for}\mathfrak{i}=\mathfrak{j}\ne 1,\hfill \\ -\frac{1}{{\varsigma }_{\mathfrak{i}}}\sum _{\mathsf{k}=\mathfrak{j}}^{\mathfrak{i}-1}{C}_{\mathfrak{i}\mathsf{k}}{\aleph }_{\mathsf{k}\mathfrak{j}},\hfill & \mathrm{for}\mathfrak{i}>\mathfrak{j},\hfill \end{array}\right.$$

  (60)

  where ${\varsigma }_{\mathfrak{i}}=\sqrt{{∥{\mathsf{\Theta }}_{\mathfrak{i}}∥}^2-{\sum }_{\mathsf{k}=1}^{\mathfrak{i}-1}{C}_{\mathfrak{i}\mathsf{k}}^2},C_{\mathfrak{i}\mathsf{k}}={⟨{\mathsf{\Theta }}_{\mathfrak{i}},{\overline{\mathsf{\Theta }}}_{\mathsf{k}}⟩}_{{\mathfrak{W}}_2^{(3,2)}}.$

Theorem 6. 

Assume ${\left\{\left(x_{\mathfrak{i}},t_{\mathfrak{i}}\right)\right\}}_{\mathfrak{i}=1}^{\infty }$ is dense; therefore, ${\left\{{\mathsf{\Theta }}_{\mathfrak{i}}\right\}}_{\mathfrak{i}=1}^{\infty }$ is the complete system of ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right).$

Proof. 

Clearly, ${\mathsf{\Theta }}_{\mathfrak{i}}(x,t)\in {\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right).$ Thus, for $v(x,t)\in {\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right),$

$${⟨v(x,t),{\mathsf{\Theta }}_{\mathfrak{i}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}=0,\mathfrak{i}=1,2,\cdots .$$

(61)

as

$${⟨v,{\mathsf{\Theta }}_{\mathfrak{i}}⟩}_{{\mathfrak{W}}_2^{(3,2)}}={⟨v(x,t),{\mathfrak{T}}^{*}{\varrho }_{\mathfrak{i}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}={⟨\mathfrak{T}v(x,t),{\varrho }_{\mathfrak{i}}(x,t)⟩}_{{\mathfrak{W}}_2^{(1,1)}}=\mathfrak{T}v(x_{\mathfrak{i}},t_{\mathfrak{i}})=0,$$

(62)

and due to the density of ${\left\{\left(x_{\mathfrak{i}},t_{\mathfrak{i}}\right)\right\}}_{\mathfrak{i}=1}^{\infty }$ in $\mathsf{\Omega }:$

$$\mathfrak{T}v(x,t)=0.$$

(63)

by applying ${\mathfrak{T}}^{-1},$

$$v(x,t)=0.$$

(64)

□

Step 4: The solution’s representation is given by

Theorem 7. 

Assume ${\left\{\left(x_{\mathfrak{i}},t_{\mathfrak{i}}\right)\right\}}_{\mathfrak{i}=1}^{\infty }$ is a dense set on Ω and (57) has a unique solution on ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right),$ then

$$v(x,t)=\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}\hslash (x_k,t_k){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)$$

(65)

is the solution of (57), and the solution of (1) and (2) is

$$u(x,t)=\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}\hslash (x_k,t_k){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)-P(x,t).$$

(66)

Proof. 

We know that the basis ${\left\{{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\right\}}_{\mathfrak{i}=1}^{\infty }$ is a complete orthonormal system in the space ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right),$ then

$$\begin{array}{ccc}\hfill v(x,t)& =\sum _{\mathfrak{i}=1}^{\infty }{⟨v(x,t),{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\hfill & \hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}{⟨v(x,t),{\mathsf{\Theta }}_{\mathsf{k}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}{⟨v(x,t),{\mathfrak{T}}^{*}{\rho }_{\mathsf{k}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}{⟨\mathfrak{T}v(x,t),{\varrho }_{\mathsf{k}}(x,t)⟩}_{{\mathfrak{W}}_2^{(1,1)}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}k}{⟨\mathfrak{T}v(x,t),{\mathtt{T}}_{(x_{\mathsf{k}},t_{\mathsf{k}})}(x,t)⟩}_{{\mathfrak{W}}_2^{(1,1)}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}k}\hslash (x_{\mathsf{k}},t_{\mathsf{k}}){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(\xi ,\tau ),\hfill \end{array}$$

(67)

with $\hslash (x_{\mathsf{k}},t_{\mathsf{k}})=\mathfrak{T}v(x_{\mathsf{k}},t_{\mathsf{k}}).$

On the other hand, (66) follows directly from $u=v-P(x,t).$ □

Remarks

1.

We have

$$v_n(x,t)=\sum _{\mathfrak{i}=1}^n\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}\hslash (x_k,t_k){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t).$$

(68)

2.

${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)$ is a Hilbert space. Then, we deduce

$$\sum _{\mathfrak{i}=1}^{\infty }\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}\hslash (x_k,t_k){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)<\infty .$$

(69)

4.2. Convergence Analysis

The approximate solution of (65) takes the form

$$v_n(x,t)=\sum _{\mathfrak{i}=1}^n{\mathsf{\Lambda }}_{\mathfrak{i}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),$$

(70)

where

$${\mathsf{\Lambda }}_{\mathfrak{i}}=\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}k}\hslash (x_{\mathsf{k}},t_{\mathsf{k}}).$$

(71)

Here, by letting $(x_1,t_1)=(a,c),$ it is possible to know the values of $v(x_1,t_1)$ from the initial and boundary conditions. In addition, $v_0(x_1,t_1)=v(x_1,t_1).$

Theorem 8. 

Suppose that ${∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}$ is bounded in (57), ${\left\{(x_{\mathfrak{i}},t_{\mathfrak{i}})\right\}}_{\mathfrak{i}=1}^{\infty }$ is dense on $\mathsf{\Omega },$ and the solution of (70) is unique. Then, $v_n$ converges to v and

$$v_n(x,t)=\sum _{\mathfrak{i}=1}^n{\mathsf{\Lambda }}_{\mathfrak{i}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t).$$

(72)

Proof. 

(i)

From (70), we know

$$v_{n+1}=v_n+{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}.$$

(73)

then,

$$\begin{array}{cc}\hfill {∥v_{n+1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2=& {∥v_n+{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2\hfill \\ \hfill =& {⟨v_n+{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1},v_n+{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill =& {⟨v_n,v_n⟩}_{{\mathfrak{W}}_2^{(3,2)}}+{⟨v_n,{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}⟩}_{{\mathfrak{W}}_2^{(3,2)}}+{⟨{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1},v_n⟩}_{{\mathfrak{W}}_2^{(3,2)}}+{⟨{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1},{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill =& {∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{⟨v_n,{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1}⟩}_{{\mathfrak{W}}_2^{(3,2)}}+{⟨{\mathsf{\Lambda }}_{n+1}{\overline{\mathsf{\Theta }}}_{n+1},v_n⟩}_{{\mathfrak{W}}_2^{(3,2)}}+,{\mathsf{\Lambda }}_{n+1}^2{⟨{\overline{\mathsf{\Theta }}}_{n+1},{\overline{\mathsf{\Theta }}}_{n+1}⟩}_{{\mathfrak{W}}_2^{(3,2)}}.\hfill \end{array}$$

(74)

Now, the orthogonality of ${\left\{{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t)\right\}}_{\mathfrak{i}=1}^{\infty }$ implies

$$\begin{array}{cc}\hfill {∥v_{n+1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2=& {∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{\mathsf{\Lambda }}_{n+1}^2\hfill \\ \hfill =& {∥v_{n-1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{\mathsf{\Lambda }}_n^2+{\mathsf{\Lambda }}_{n+1}^2\hfill \\ \hfill =& {∥v_{n-2}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{\mathsf{\Lambda }}_{n-1}^2+{\mathsf{\Lambda }}_n^2+{\mathsf{\Lambda }}_{n+1}^2\hfill \\ \hfill & ⋮\hfill \\ \hfill =& {∥v_1∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{\mathsf{\Lambda }}_2^2+{\mathsf{\Lambda }}_3^2+\cdots +{\mathsf{\Lambda }}_n^2+{\mathsf{\Lambda }}_{n+1}^2\hfill \\ \hfill =& {∥v_0∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+\sum _{\mathfrak{i}=1}^{n+1}{\mathsf{\Lambda }}_{\mathfrak{i}}^2,\hfill \end{array}$$

(75)

Hence,

$${∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}\le {∥v_{n+1}∥}_{{\mathfrak{W}}_2^{(3,2)}}.$$

(76)

The convergence of ${∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}$ follows directly from the boundedness of ${∥v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}$. So, there exists $ϝ$ such that

$$\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}^2=ϝ,$$

(77)

where the constant $ϝ$ is positive.

As a result,

$${\left\{{\mathsf{\Lambda }}_{\mathfrak{i}}^2\right\}}_{\mathfrak{i}=1}^{\infty }\in {\ell }^2.$$

As $\left(v_m-v_{m-1}\right)\perp \cdots \perp \left(v_{n+1}-v_n\right)$ and for $n<m$, we write

$$\begin{array}{cc}\hfill {∥v_m-v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2& ={∥v_m-v_{m-1}+v_{m-1}-\cdots +v_{n+1}-v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2\hfill \\ \hfill & ={∥v_m-v_{m-1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+{∥v_{m-1}-v_{m-2}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2+\cdots +{∥v_{n+1}-v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2.\hfill \end{array}$$

(78)

Furthermore,

$${∥v_m-v_{m-1}∥}_{{\mathfrak{W}}_2^{(3,2)}}^2={\mathsf{\Lambda }}_m^2.$$

(79)

Thus,

$${∥v_m-v_n∥}_{{\mathfrak{W}}_2^{(3,2)}}^2=\sum _{p=n+1}^m{\mathsf{\Lambda }}_p^2\to 0,\mathrm{as}n,m\to \infty .$$

(80)

The completeness of ${\mathfrak{W}}_2^{(3,2)}\left(\mathsf{\Omega }\right)$ allows us to deduce that $v_n\to \tilde{v}$ as $n\to \infty$.

(ii)

To prove this, let us take the limits in (70)

$$\tilde{v}(x,t)=\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t).$$

(81)

We apply the linear operator $\mathfrak{T}$ to (81)

$$\mathfrak{T}\tilde{v}(x,t)=\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}\mathfrak{T}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),$$

(82)

Hence,

$$\begin{array}{cc}\hfill \mathfrak{T}\tilde{v}(x_p,t_p)& =\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨\mathfrak{T}{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),{\varrho }_p(x,t)⟩}_{{\mathfrak{W}}_2^{(1,1)}}\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),{\mathfrak{T}}^{*}{\varrho }_p(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),{\mathsf{\Theta }}_p(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}.\hfill \end{array}$$

(83)

Thus,

$${\aleph }_{\mathfrak{j}p}\mathfrak{T}\tilde{v}(x_p,t_p)={\aleph }_{\mathfrak{j}p}\left[\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),{\mathsf{\Theta }}_p(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}\right],$$

(84)

and we take the summation ${\sum }_{p=1}^{\mathfrak{j}}$ to deduce

$$\begin{array}{cc}\hfill \sum _{p=1}^{\mathfrak{j}}{\aleph }_{\mathfrak{j}p}\mathfrak{T}\tilde{v}(x_p,t_p)=& \sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),\sum _{p=1}^{\mathfrak{j}}{\aleph }_{\mathfrak{j}p}{\mathsf{\Theta }}_p(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill & =\sum _{\mathfrak{i}=1}^{\infty }{\mathsf{\Lambda }}_{\mathfrak{i}}{⟨{\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t),{\overline{\mathsf{\Theta }}}_{\mathfrak{j}}(x,t)⟩}_{{\mathfrak{W}}_2^{(3,2)}}\hfill \\ \hfill & ={\mathsf{\Lambda }}_{\mathfrak{j}}.\hfill \end{array}$$

(85)

Observe then from (71) that

$$\mathfrak{T}\tilde{v}(x_p,t_p)=\hslash (x_p,t_p).$$

(86)

For all $(\zeta ,\eta )\in \mathsf{\Omega },$ it exists ${\left\{(x_{q\mathfrak{j}},t_{q\mathfrak{j}})\right\}}_{\mathfrak{j}=1}^{\infty }$ such that $(x_{q\mathfrak{j}},t_{q\mathfrak{j}})\to (\zeta ,\eta ),\mathrm{as}\mathfrak{j}\to \infty$.

It is well-known to us that

$$\mathfrak{T}\tilde{v}(x_{q\mathfrak{j}},t_{q\mathfrak{j}})=\hslash (x_{q\mathfrak{j}},t_{q\mathfrak{j}}).$$

(87)

Using the continuity of ℏ and letting $\mathfrak{j}\to \infty$ allows us to

$$\mathfrak{T}\tilde{v}(\zeta ,\eta )=\hslash (\zeta ,\eta ).$$

(88)

□

5. Numerical Experiments

We apply the proposed methods to solve some problems. In Example 1, we use RKHSM to solve the considered equation, and the GPS is considered for Example 2 to deal with the fractional convection–diffusion equation. Now, how to apply the RKHSM can be summarized in the following procedure:

• Step 1: Setting $n=p\times q;$
• Step 2: Setting ${\mathsf{\Theta }}_{\mathfrak{i}}(x_{\mathfrak{i}},t_{\mathfrak{i}})={\mathfrak{T}}_{(\zeta ,\eta )}{\mathtt{Y}}_{(x,t)}(\zeta ,\eta ){|}_{(\zeta ,\eta )=(x_{\mathfrak{i}},t_{\mathfrak{i}})};$
• Step 3: Calculating the orthogonalization coefficients ${\aleph }_{\mathfrak{i}\mathfrak{j}}$ using (60);
• Step 4: Setting ${\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x_{\mathfrak{i}},t_{\mathfrak{i}})={\sum }_{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}{\mathsf{\Theta }}_{\mathsf{k}}(x_{\mathfrak{i}},t_{\mathfrak{i}}),\mathfrak{i}=1,2,\cdots ,n;$
• Step 5: Choosing an initial guess $u_0(x_1,t_1);$
• Step 6: Setting $\mathfrak{i}=1;$
• Step 7: Setting ${\mathsf{\Lambda }}_{\mathfrak{i}}={\sum }_{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}k}\hslash (x_{\mathsf{k}},t_{\mathsf{k}});$
• Step 8:$u_{\mathfrak{i}}(x_{\mathfrak{i}},t_{\mathfrak{i}})={\sum }_{\ell =1}^{\mathfrak{i}}{\mathsf{\Lambda }}_{\ell }{\overline{\mathsf{\Theta }}}_{\ell }(x_{\ell },t_{\ell });$
• Step 9: If $\mathfrak{i}<n,$ set $\mathfrak{i}=\mathfrak{i}+1.$ Go to step 7. Else stop, Where $x_{\mathfrak{i}}=\frac{\mathfrak{i}}{p},\mathfrak{i}=1,2,\cdots ,p$ and $t_{\mathfrak{j}}=\frac{\mathfrak{j}}{q},\mathfrak{j}=1,2,\cdots ,q.$n is the grid points’ number.

Example 1. 

Considering the following problem with the fractional order $\alpha =0.5$:

$${}^C{\mathcal{D}}_{0^{+},t}^{\alpha }u=u_{xx}+g(x,t),(x,t)\in {[0,1]}^2,$$

(89)

where

$$u(x,t)=exp\left(x\right)x^2{(1-x)}^2{t}^{\alpha },$$

and

$$g(x,t)=e^x\left(\left(x\left(x^3+6x^2+x-8\right)+2\right)\left(-t^{\alpha }\right)-\frac{\pi {(x-1)}^2{x}^{2}csc\left(\pi \alpha \right)}{\mathsf{\Gamma }(-\alpha )}\right).$$

In this example, the RKHSM is tested with the standard grid points $x_{\mathfrak{i}}=\frac{\mathfrak{i}}{p},\mathfrak{i}=1,\cdots ,p$ and $t_{\mathfrak{j}}=\frac{\mathfrak{j}}{q},\mathfrak{j}=1,\cdots ,q$ with $p\times q=n=100.$ The comparison of (89) with (1) and (2) shows $\mathsf{\Omega }=[0,1]\times [0,1],cq=1,\lambda =1,$ and $u(0,t)=u(1,t)=u(x,0)=0.$ Therefore, as we see in Section 4, the approximate solution of (89) takes the form

$$v_n(x,t)=\sum _{\mathfrak{i}=1}^n\sum _{\mathsf{k}=1}^{\mathfrak{i}}{\aleph }_{\mathfrak{i}\mathsf{k}}g(x_k,t_k){\overline{\mathsf{\Theta }}}_{\mathfrak{i}}(x,t).$$

(90)

In Table 1, a numerical comparison between the obtained results via RKHSM with the exact solution, for $\alpha$ = 0.9, 0.8, 0.75, is given. These results clearly show that the approximate solution (using the RKHSM) converges to the exact solution. The results are in good agreement with each other, and this confirms the effectiveness of the RKHSM to solve this type of equation.

Table 1. Absolute errors of the RKHSM solution for Example 1.

	RKHSM-Absolute Error
(x,t)	$\mathbf{\alpha }=\mathbf{0}.\mathbf{9}$	$\mathbf{\alpha }=\mathbf{0}.\mathbf{8}$	$\mathbf{\alpha }=\mathbf{0}.\mathbf{75}$
(0.1, 0.1)	$1.3016\times {10}^{-3}$	$1.3446\times {10}^{-3}$	$1.9199\times {10}^{-3}$
(0.3, 0.3)	$1.0876\times {10}^{-4}$	$3.8615\times {10}^{-4}$	$8.1880\times {10}^{-4}$
(0.5, 0.5)	$5.9252\times {10}^{-5}$	$8.7214\times {10}^{-4}$	$1.3792\times {10}^{-3}$
(0.7, 0.7)	$5.9929\times {10}^{-4}$	$1.0365\times {10}^{-3}$	$1.6629\times {10}^{-3}$
(0.9,0.9)	$9.5379\times {10}^{-4}$	$1.6982\times {10}^{-4}$	$4.2782\times {10}^{-4}$

Example 2. 

Consider the problem (28) with $b\left(x\right)=x$ and fractional order α = 0.01 where

$$u(x,t)=x^2{t}^{2\alpha }{\displaystyle \frac{2\mathsf{\Gamma }(\alpha +1)}{\mathsf{\Gamma }(2\alpha +1)}},$$

and

$$g(x,t)=2t^2+2x^2+2.$$

We solve this problem by using the GPS with taking $\mathsf{\Delta }\theta =1e-10$, $m=n=20,$ and $\lambda =1$. Additionally, an initial guess of $u_i^j\left(0\right)=1e-1$ is taken. Figure 1 shows the numerical solution, exact solution, absolute error, and absolute error’s contour. Indeed, we present in Table 2 the values of absolute errors between the numerical solution (using the GPS) and the exact solution for Example 2. From this table’s results, it is clear that the error estimate confirms the accuracy of this new method, and Figure 1 shows that both graphs are very similar in their behavior.

Figure 1. Solution under applying GPS for the fractional convection-diffusion equation in Example 2.

Table 2. Comparison between the exact solution and FITM solution for Example 2.

(x, t)	Approximate	Exact	Absolute Error
(0.1, 0.1)	0.0215	0.0213	$1.5308\times {10}^{-4}$
(0.2, 0.2)	0.0893	0.0864	$2.9000\times {10}^{-3}$
(0.3, 0.3)	0.2016	0.1960	$5.6000\times {10}^{-3}$
(0.4, 0.4)	0.3133	0.3076	$5.7000\times {10}^{-3}$
(0.5, 0.5)	0.4958	0.4958	$3.9629\times {10}^{-8}$

6. Conclusions

In the current work, we successfully implemented two numerical schemes to gain approximate solutions to the considered problems. One is the FITM, which converted the original problem into a new one with one extra dimension. After that, we used GPS to solve the problem. The other is the RKHSM, which was used for the mentioned problem. The main steps for applying this method are defining an appropriate bounded linear operator and constructing an orthonormal function system of the appropriate RKHS. Indeed, the both methods are shown to have good convergence. Two examples were employed to show the capacity and reliability of the FITM and RKHSM. Our obtained results are compared with exact results and they are found to be in good agreement with each other. From the numerical results, it can be observed the suitability, ease, and effectiveness of the proposed approaches for solving such types of fractional partial differential equation. This research opens the way for the use of the two proposed methods to study the mentioned problem for various new fractional derivatives. As part of our purpose, we plan to apply the FTIM and RKHSM to multidimensional fractional partial differential equations, which will be new in the literature.