Conformable Double Laplace–Sumudu Iterative Method

Abstract

This research introduces a novel approach that combines the conformable double Laplace–Sumudu transform (CDLST) and the iterative method to handle nonlinear partial problems considering some given conditions, and we call this new approach the conformable Laplace–Sumudu iterative (CDLSI) method. Furthermore, we state and discuss the main properties and the basic results related to the proposed technique. The new method provides approximate series solutions that converge to a closed form of the exact solution. The advantage of using this method is that it produces analytical series solutions for the target equations without requiring discretization, transformation, or restricted assumptions. Moreover, we present some numerical applications to defend our results. The results demonstrate the strength and efficiency of the presented method in solving various problems in the fields of physics and engineering in symmetry with other methods.

1. Introduction

Fractional partial differential equations have recently emerged as crucial for simulating a variety of real-world applications in engineering and science, including mathematical biology, fluid dynamics, optics, electrical circuits, and quantum physics [1,2,3]. Several definitions of fractional derivatives and integrals have been discussed in the literature, including those by Riesz, Weyl, Riemann-Liouville, Caputo, Hadamard, and others. Numerous strange characteristics of these fractional derivatives, such as the fact that not all of them adhere to the product rules, quotient, and others, result in a variety of problems in engineering and physics applications. The authors Khalil et al. [4] proposed an interesting definition known as the conformable fractional derivative that meets most traditional characteristics of derivatives.

Recently, many researchers and mathematicians have developed new methods to obtain the solutions of conformable fractional partial differential equations, such as the exponential rational function method [5], the simplest equation method [6], the conformable Laplace transform method [7], the conformable double Sumudu transform [8], the modified double conformable Laplace transform [9], the double Shehu transform [10], the reduced differential transform method [11], the conformable double Laplace decomposition method [12], the conformable double Sumudu decomposition method [13], the conformable triple Laplace transform decomposition method [14], the conformable triple Laplace and Sumudu transforms decomposition method [15], and the conformable triple Laplace transform iterative method [16].

In recent years, a novel double integral transform approach known as the double Laplace–Sumudu transform method was effectively developed to handle various partial, integral, and fractional differential equations [17,18,19,20,21]. Unfortunately, similar to other integral transforms, this transformation has a problem when dealing with nonlinear problems. Therefore, researchers have discovered new techniques that combine transforms with numerical methods such as the variational iteration method, the decomposition method, the perturbation methods, and others [22,23,24,25,26,27,28,29].

Now, we present a conformable fractional nonlinear partial differential equation of the form

$$\begin{array}{c}\frac{{\partial }^{m\vartheta }}{\partial v^{m\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+\frac{{\partial }^{n\theta }}{\partial u^{n\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=g\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right),\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }>0, \\ 0<\theta , \vartheta \le 1, m,n\in \mathbb{N}\end{array}$$

(1)

Associated with (1), we consider the initial conditions (ICs)

$$\frac{{\partial }^{j\vartheta }\zeta \left(\frac{u^{\theta }}{\theta },0\right)}{\partial v^{j\vartheta }}=f_j\left(\frac{u^{\theta }}{\theta }\right), j=0,1,\dots ,m-1,$$

(2)

and the boundary conditions (BCs)

$$\frac{{\partial }^{k\theta }\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)}{\partial u^{k\theta }}=h_k\left(\frac{v^{\vartheta }}{\vartheta }\right), k=0,1,\dots ,n-1,$$

(3)

where $\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]$ is a nonlinear term and $g\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ is the source term in the form

$$g\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=g_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right).$$

The main goal of this study is to utilize the CDLS transform to solve nonlinear fractional partial differential equations. Our problem is basically that the CDLS transform cannot be applied to solve nonlinear problems directly. We need to combine it with one of the popular numerical approaches, such as the iteration method [30,31].

It is a simple and effective method with less computations in comparison to other numerical methods, and it is used in the literature to solve similar equations. Its convergence analysis and stability were proven in many articles [32,33,34,35].

In this research, we provide the CDLSI method for studying the applications expressed in Form (1) considering the conditions in (2) and (3). The CDLSI method is a new modification of the usual conformable double Laplace–Sumudu approach, in which we combine it with the iterative method [36,37,38,39]. The novelty of using the double Laplace–Sumudu transform approach along with the iterative method is that it offers a quick convergence to the exact solution without making any constrictive assumptions about the solution. It needs no discretization or differentiation in comparison to various numerical methods (see [40]). The purpose of this work is to offer a more expedient technique for using the CDLSI method to determine the precise solution of nonlinear conformable partial differential equations.

This study presents the basic definitions and properties of the conformable double Laplace–Sumudu transform. Then, we illustrate the main idea of the introduced approach, and finally we give the solutions of some numerical applications to show the applicability and accuracy of the proposed method.

2. Conformable Fractional Derivative

This section includes the basic properties and definitions of the conformable double Laplace–Sumudu transform.

Definition 1 [4].

Let$\zeta :\left(0,\infty \right)\to R$. Then, the${\vartheta }^{th}$order CFD of$\zeta$is defined by

$$\frac{d^{\vartheta }}{d{v}^{\vartheta }}\zeta \left(v\right)=\underset{\delta \to 0}{\lim }\frac{\zeta \left(v+\delta v^{1-\vartheta }\right)-\zeta \left(v\right)}{\delta },v>0, \delta \in \left(0,1\right].$$

(4)

Definition 2 [17].

Let$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right):R\times \left(0,\infty \right)\to R$. Then, the CPDs (conformable partial fractional derivatives) of order$\theta$and$\vartheta$of the function$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$are defined by

$$\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\underset{\epsilon \to 0}{\lim }\frac{\zeta \left(\frac{u^{\theta }}{\theta }+\epsilon u^{1-\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)}{\epsilon },$$

(5)

$$\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\underset{\delta \to 0}{\lim }\frac{\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }+\delta v^{1-\vartheta }\right)-\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)}{\delta },$$

(6)

where$\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }>0,0<\theta ,\vartheta \le 1,$and$\frac{{\partial }^{\theta }}{\partial u^{\theta }}$and$\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}$are called the fractional derivatives of order$\lambda$and$\mu$, respectively.

Theorem 1 [14,15,16].

Suppose$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$is a differentiable function at a point$\frac{u^{\theta }}{\theta }$, $\frac{v^{\vartheta }}{\vartheta }>0$, $0<\theta$, and$\vartheta \le 1$. Then,

$$\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=u^{-\theta +1}\frac{\partial \zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)}{\partial u},$$

(7)

$$\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=v^{-\vartheta +1}\frac{\partial \zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)}{\partial v}.$$

(8)

Example 1 [14,15,16].

Suppose$0<\theta ,\vartheta \le 1$, $c_1,c_2,m_1,m_2,\gamma ,$and$\eta \in ℝ;$then,

$$\begin{array}{cc}\hfill \frac{{\partial }^{\theta }}{\partial u^{\theta }}(c_1\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)& +c_2\psi \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right))\hfill \\ \hfill & =c_1\left(\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right)+c_2\left(\frac{{\partial }^{\theta }}{\partial u^{\theta }}\psi \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right),\hfill \end{array}$$

$$\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}\left(e^{\gamma \frac{u^{\theta }}{\theta }+\eta \frac{v^{\vartheta }}{\vartheta }}\right)=\eta e^{\gamma \frac{u^{\lambda }}{\lambda }+\eta \frac{v^{\mu }}{\mu }},$$

$$\frac{{\partial }^{\theta }}{\partial u^{\theta }}\left(e^{\gamma \frac{u^{\theta }}{\theta }+\eta \frac{v^{\vartheta }}{\vartheta }}\right)=\gamma e^{\gamma \frac{u^{\lambda }}{\lambda }+\eta \frac{v^{\mu }}{\mu }},$$

$$\frac{{\partial }^{\theta }}{\partial u^{\theta }}\left(\sin \left(\frac{u^{\theta }}{\theta }\right)\sin \left(\frac{v^{\vartheta }}{\vartheta }\right)\right)=\cos \left(\frac{u^{\theta }}{\theta }\right)\sin \left(\frac{v^{\vartheta }}{\vartheta }\right),$$

$$\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}\left(\sin \left(\frac{u^{\theta }}{\theta }\right)\sin \left(\frac{v^{\vartheta }}{\vartheta }\right)\right)=\sin \frac{u^{\theta }}{\theta }\cos \left(\frac{v^{\vartheta }}{\vartheta }\right).$$

3. Some Results and Theorem of the CDLST

Definition 3 [9].

Let$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$be a function of two variables$\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\in {ℝ}^{+}$. Then:

(i)

The CLT (conformable Laplace transform) of$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$with respect to$\frac{u^{\theta }}{\theta }$is denoted by$L_u^{\theta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\theta }\left(q,\frac{v^{\vartheta }}{\vartheta }\right)$and defined as

$$L_u^{\theta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\theta }\left(q,\frac{v^{\vartheta }}{\vartheta }\right)=\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-q\frac{u^{\theta }}{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)u^{\theta -1}du,u>0.$$

(9)

(ii)

The CST of$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$with respect to$\frac{v^{\vartheta }}{\vartheta }$is denoted by$S_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\vartheta }\left(\frac{u^{\theta }}{\theta },w\right)$and defined as

$$S_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\vartheta }\left(\frac{u^{\theta }}{\theta },w\right)=\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)v^{\vartheta -1}dv,v>0.$$

(10)

(iii)

The CDLST of$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$is denoted by$L_u^{\theta }{S}_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\theta ,\vartheta }\left(q,w\right)=\Omega \left(q,w\right)$and defined as

$$L_u^{\theta }{S}_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]={\Omega }_{\theta ,\vartheta }\left(q,w\right)=\Omega \left(q,w\right)=\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-q\frac{u^{\theta }}{\theta }-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)u^{\theta -1}{v}^{\vartheta -1}dudv,$$

(11)

where$q,w\in ℂ$are variables of Laplace and Sumudu, respectively.

Recall that $L_u^{\theta }{S}_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=S_v^{\vartheta }{L}_u^{\theta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=\Omega \left(q,w\right).$

The inverse conformable Laplace–Sumudu $L_{\theta }^{-1}{S}_{\vartheta }^{-1}\left[\Omega \left(q,w\right)\right]=\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ is defined by

$$L_{\theta }^{-1}{S}_{\vartheta }^{-1}\left[{\Omega }_{\theta ,\vartheta }\left(q,w\right)\right]=\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\frac{1}{2\pi i}\underset{\rho -i\infty }{\overset{\rho +i\infty }{{\displaystyle \int }}}{e}^{q\frac{u^{\theta }}{\theta }}\left(\frac{1}{2\pi i}\underset{\sigma -i\infty }{\overset{\sigma +i\infty }{{\displaystyle \int }}}\frac{1}{w}{e}^{\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}{\Omega }_{\theta ,\vartheta }\left(q,w\right)dw\right)dq.$$

(12)

Theorem 2 [18,19,20].

The CDLSTs of some functions are given below:

$$L_u^{\theta }{S}_v^{\vartheta }\left[c\right]=L_u{S}_v\left[c\right]=\frac{c}{q},c\in ℝ,q>0.$$

$$L_u^{\theta }{S}_v^{\vartheta }\left[{\left(\frac{u^{\theta }}{\theta }\right)}^m{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^n\right]=L_u{S}_v\left[{\left(u\right)}^m{\left(v\right)}^n\right]=\frac{m!n!w^n}{q^{m+1}},m,n\in {\mathbb{Z}}^{+}.$$

$$L_u^{\theta }{S}_v^{\vartheta }\left[e^{a\frac{u^{\theta }}{\theta }+b\frac{v^{\vartheta }}{\vartheta }}\right]=L_u{S}_v\left[e^{au+bv}\right]=\frac{1}{\left(q-a\right)\left(1-bw\right)}.$$

$$L_u^{\theta }{S}_v^{\vartheta }\left[\sin \left(a\frac{u^{\theta }}{\theta }\right)\sin \left(b\frac{v^{\vartheta }}{\vartheta }\right)\right]=L_u{S}_v\left[\sin \left(au\right)\sin \left(bv\right)\right]=\frac{a}{\left(q^2+a^2\right)}\frac{bw}{1+b^2{w}^2}.$$

Theorem 3 [18,19,20].

If a function$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$is on the interval$\left(0,U\right)$and$\left(0,V\right)$of an exponential order$a$and$b$, then the CDLST of$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$is well-defined for all$q$and$\frac{1}{w}$if$\mathrm{Re}[q]>a$and$\mathrm{Re}\left[\frac{1}{w}\right]>b.$

Theorem 4 .

If$L_u^{\lambda }{S}_v^{\mu }\left[\zeta \left(\frac{u^{\lambda }}{\lambda },\frac{v^{\mu }}{\mu }\right)\right]={\Omega }_{\lambda ,\mu }\left(q,w\right)=\Omega \left(q,w\right)$, then the CDLSTs of the CPDs of order$\theta$and$\vartheta$can be represented as follows:$0<\theta ,\vartheta \le 1$

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=q\Omega \left(q,w\right)-\Omega \left(0,w\right).$$

(13)

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{\vartheta }}{\partial v^{\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=\frac{1}{w}\Omega \left(q,w\right)-\frac{1}{w}\Omega \left(q,0\right).$$

(14)

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{2\theta }}{\partial u^{2\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=q^2\Omega \left(q,w\right)-q\Omega \left(0,w\right)-{\Omega }_u\left(0,w\right).$$

(15)

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{2\vartheta }}{\partial v^{2\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=\frac{1}{w^2}\Omega \left(q,w\right)-\frac{1}{w^2}\Omega \left(q,0\right)-\frac{1}{w}{\Omega }_v\left(q,0\right).$$

(16)

Proof. 

Proof of Result (13):

$$\begin{array}{c}{L}_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-q\frac{u^{\theta }}{\theta }-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}\frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}{u}^{\theta -1}{v}^{\vartheta -1}dudv\\ =\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}{v}^{\vartheta -1}\left(\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-q\frac{u^{\theta }}{\theta }}\frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}{u}^{\theta -1}du\right)dv.\end{array}$$

(17)

Since Theorem 1 states that $\frac{{\partial }^{\lambda }\zeta }{\partial u^{\lambda }}=u^{-\lambda +1}\frac{\partial \zeta }{\partial u}$, we use this result in Equation (18).

Therefore, Equation (17) becomes

$$\begin{array}{cc}\hfill L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{\theta }}{\partial u^{\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]& =\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}{v}^{\vartheta -1}\left(\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-q\frac{u^{\theta }}{\theta }}\frac{\partial \zeta }{\partial u}du\right)dv=\frac{1}{w}\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\frac{1}{w}\frac{v^{\vartheta }}{\vartheta }}{v}^{\vartheta -1}\left(-\zeta \left(0,v\right)+q\Omega \left(q,v\right)\right)dv\hfill \\ & =q\Omega \left(q,w\right)-\Omega \left(0,w\right).\hfill \end{array}$$

(18)

The proof of results (14–16) can be obtained in the same manner.

The above results can be extended as follows:

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{n\theta }}{\partial u^{n\theta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=q^n\Omega \left(q,w\right)-{\displaystyle \sum }_{k=0}^{n-1}{q}^{n-1-k}{S}_v^{\vartheta }\left[\frac{{\partial }^{k\theta }}{\partial u^{k\theta }}\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)\right],$$

(19)

where $\Omega \left(q,w\right)=L_u^{\theta }{S}_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=S_v^{\vartheta }{L}_u^{\theta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right].$

$$L_u^{\theta }{S}_v^{\vartheta }\left[\frac{{\partial }^{m\vartheta }}{\partial v^{m\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=w^{-m}\Omega \left(q,w\right)-{\displaystyle \sum }_{j=0}^{m-1}{w}^{-m+j}{L}_u^{\theta }\left[\frac{{\partial }^{j\vartheta }}{\partial v^{j\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },0\right)\right],$$

(20)

where $\Omega \left(q,w\right)=L_u^{\theta }{S}_v^{\vartheta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=S_v^{\vartheta }{L}_u^{\theta }\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right].$

The proof of (19) an (20) can be obtained by induction. □

4. Principle of CDLS Combined with Iterative Method

This section exercises the CDLST combined with the iterative method to investigate the solution of Equation (1). Here, we mention that the convergence analysis of the proposed method is well-known and was proven several times in the literature (see [30,31]). Applying the CDLST to Equation (1), we obtain

$$\begin{array}{cc}\hfill w^{-m}\Omega \left(q,w\right)& -{\displaystyle {\displaystyle \sum }_{j=0}^{m-1}}{w}^{-m+j}{L}_u^{\theta }\left[\frac{{\partial }^{j\vartheta }}{\partial v^{j\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },0\right)\right]+q^n\Omega \left(q,w\right)-{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}{q}^{n-1-k}{S}_v^{\vartheta }\left[\frac{{\partial }^{k\theta }}{\partial u^{k\theta }}\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)\right]\hfill \\ \hfill & +L_u^{\theta }{S}_v^{\vartheta }\left[\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right]=G_1\left(q,w\right)+L_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right].\hfill \end{array}$$

(21)

Using the conformable Laplace transform for the initial conditions in (2) and the conformable Sumudu transform for the conditions in (3), we obtain

$$L_u^{\theta }\left[\frac{{\partial }^{j\vartheta }}{\partial v^{j\vartheta }}\zeta \left(\frac{u^{\theta }}{\theta },0\right)\right]=F_j\left(q\right),j=0,1,\dots ,m-1,S_v^{\vartheta }\left[\frac{{\partial }^{k\theta }}{\partial u^{k\theta }}\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)\right]=H_k\left(w\right), k=0,1,\dots ,n-1.$$

(22)

By substituting (22) for (21), we have

$$\begin{array}{cc}\hfill w^{-m}\Omega \left(q,w\right)& -{\displaystyle \sum }_{j=0}^{m-1}{w}^{-m+j}{F}_j\left(q\right)+q^n\Omega \left(q,w\right)-{\displaystyle \sum }_{k=0}^{n-1}{q}^{n-1-k}{H}_k\left(w\right)\hfill \\ \hfill & =G_1\left(q,w\right)+L_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right].\hfill \end{array}$$

(23)

By simplifying Equation (23), we obtain

$$\begin{array}{cc}\hfill \Omega \left(q,w\right)=& {\left(w^{-m}+q^n\right)}^{-1}\left({\displaystyle \sum }_{j=0}^{m-1}{w}^{-m+j}{F}_j\left(q\right)+{\displaystyle \sum }_{k=0}^{n-1}{q}^{n-1-k}{H}_k\left(w\right)+G_1\left(q,w\right)\right)\hfill \\ \hfill & +{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right].\hfill \end{array}$$

(24)

Taking the inverse transform ${\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}$ of (24), we obtain

$$\begin{array}{c}\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}\left({\displaystyle \sum }_{j=0}^{m-1}{w}^{-m+j}{F}_j\left(q\right)+{\displaystyle \sum }_{k=0}^{n-1}{q}^{n-1-k}{H}_k\left(w\right)+G_1\left(q,w\right)\right)\right]\\ +{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right]\right].\end{array}$$

(25)

Now, using the iterative approach, by assuming

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\displaystyle \sum }_{i=0}^{\infty }{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$$

(26)

and substituting Equation (26) into Equation (25), we obtain

$$\begin{array}{c}{\displaystyle \sum _{i=0}^{\infty }}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}\left({\displaystyle \sum _{j=0}^{m-1}}{w}^{-m+j}{F}_j\left(q\right)+{\displaystyle \sum _{k=0}^{n-1}}{q}^{n-1-k}{H}_k\left(w\right)+G_1\left(q,w\right)\right)\right]\\ +{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right]\right].\end{array}$$

(27)

Note that $\mathcal{N}\left[\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]$ is a nonlinear term that can be decomposed into

$$\mathcal{N}\left[{\displaystyle \sum }_{i=0}^{\infty }{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]=\mathcal{N}\left[{\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]+{\displaystyle \sum }_{i=0}^{\infty }\left(\mathcal{N}\left[{\displaystyle \sum }_{r=0}^i{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]-\mathcal{N}\left[{\displaystyle \sum }_{r=0}^{i-1}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right)$$

(28)

By substituting Equation (28) into Equation (27), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=0}^{\infty }}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=& {\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}\left({\displaystyle \sum _{j=0}^{m-1}}{w}^{-m+j}{F}_j\left(q\right)+{\displaystyle \sum _{k=0}^{n-1}}{q}^{n-1-k}{H}_k\left(w\right)+G_1\left(q,w\right)\right)\right]\hfill \\ \hfill & +{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[{\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right]\right]\hfill \\ \hfill & -{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\displaystyle \sum _{i=0}^{\infty }}\left(\mathcal{N}\left[{\displaystyle \sum _{r=0}^i}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]-\mathcal{N}\left[{\displaystyle \sum _{r=0}^{i-1}}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right)\right]\right].\hfill \end{array}$$

(29)

Following that, we obtain the recurrence relations as follows:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}\left({\displaystyle \sum }_{j=0}^{m-1}{w}^{-m+j}{F}_j\left(q\right)+{\displaystyle \sum }_{k=0}^{n-1}{q}^{n-1-k}{H}_k\left(w\right)+G_1\left(q,w\right)\right)\right],$$

(30)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[g_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)-\mathcal{N}\left[{\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right]\right],$$

(31)

$$\begin{array}{cc}{\zeta }_{s+1}\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\hfill & \hfill -{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[{\left(w^{-m}+q^n\right)}^{-1}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\displaystyle \sum }_{i=0}^{\infty }\left(\mathcal{N}\left[{\displaystyle \sum }_{r=0}^i{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]-\mathcal{N}\left[{\displaystyle \sum }_{r=0}^{i-1}{\zeta }_i\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\right]\right)\right]\right],\\ & s\ge 1.\hfill \end{array}$$

(32)

As a result, we have the solution of Equation (1), which is stated as follows:

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+{\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+{\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)+\dots .$$

(33)

5. Numerical Applications

In this section, some examples of nonlinear conformable fractional partial differential equations are solved to demonstrate the performance and the efficiency of the CDLSI method. As a result of this section, we show below that we do not need many calculations in comparison to other numerical methods to obtain the exact solution. This is an important advantage of our technique.

Example 2 .

Consider the dissipative wave equation of the form

$$\begin{array}{c}\frac{{\partial }^{2\vartheta }\zeta }{\partial v^{2\vartheta }}-\frac{{\partial }^2\zeta }{\partial u^2}+\frac{\partial }{\partial v}\left(\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}\right)=2e^{-\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right)-2e^{-2\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(\frac{u^{\theta }}{\theta }\right),\\ 0<\theta , \vartheta \le 1\end{array}$$

(34)

with the ICs

$$\zeta \left(\frac{u^{\theta }}{\theta },0\right)=\sin \left(\frac{u^{\theta }}{\theta }\right), {\zeta }_v\left(\frac{u^{\theta }}{\theta },0\right)=-\sin \left(\frac{u^{\theta }}{\theta }\right),$$

(35)

and the BCs

$$\zeta \left(0,\frac{u^{\vartheta }}{\vartheta }\right)=0, {\zeta }_u\left(0,\frac{u^{\vartheta }}{\vartheta }\right)=e^{-\frac{u^{\vartheta }}{\vartheta }}.$$

(36)

This allows us to obtain the solution.

Operating the CDLST on (34), the conformable Laplace transform on (35), and the conformable Sumudu transform on (36), we obtain

$$\Omega \left(q,w\right)=\frac{1}{\left(q^2+1\right)\left(1+w\right)}-\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[2e^{-2\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(\frac{u^{\theta }}{\theta }\right)+\frac{\partial }{\partial v}\left(\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}\right)\right].$$

(37)

Taking the inverse ${\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}$ transform of (37), we obtain

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=e^{-\frac{v^{\vartheta }}{\vartheta }}\sin \frac{u^{\theta }}{\theta }-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[2e^{-2\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(\frac{u^{\theta }}{\theta }\right)+\frac{\partial }{\partial v}\left(\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}\right)\right]\right].$$

(38)

Now, applying the iterative method, we substitute (28) in (38) and, with the results in (30), (31), and (32), we obtain the following solution components:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=e^{-\frac{v^{\vartheta }}{\vartheta }}\sin \frac{u^{\theta }}{\theta },$$

(39)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[2e^{-2\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(\frac{u^{\theta }}{\theta }\right)+\frac{\partial }{\partial v}\left({\zeta }_0\frac{{\partial }^{\theta }{\zeta }_0}{\partial u^{\theta }}\right)\right]\right]=0,$$

(40)

$${\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[\frac{\partial }{\partial v}\left(\left({\zeta }_0+{\zeta }_1\right){\left({\zeta }_0+{\zeta }_1\right)}_u\right)-\frac{\partial }{\partial v}\left(\left({\zeta }_0\right){\left({\zeta }_0\right)}_u\right)\right]\right]=0.$$

(41)

As a result, we have the solution of Equation (34) as:

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=e^{-\frac{v^{\vartheta }}{\vartheta }}\sin \left(\frac{u^{\theta }}{\theta }\right),$$

(42)

The exact solution if $\theta =\vartheta =1$ is

$$\zeta \left(u,v\right)=e^{-v}\sin u.$$

(43)

The following figure (Figure 1) illustrates the 3D graph of the exact and approximate solutions of Example 2 at $\theta =\vartheta =1$. In Figure 2, we sketch the solutions of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ for Equation (42) at $\theta =\vartheta =1,0.9,0.8$.

Moreover, in

Table 1. The absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by the CDLSI method for Example 2 at different values of $\theta$, $\vartheta$, and $v$.

$\mathit{u}$	$\mathit{v}$	$\mathit{\theta }$	$\mathit{\vartheta }$	Exact	$\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}$	$\left|{\mathit{\zeta }}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\zeta }}_{\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}}\right|$
1	0.1	0.8	0.8	0.761394	0.778431	$1.7037\times {10}^{-2}$
	0.2			0.688938	0.672136	$1.68025\times {10}^{-2}$
	0.3			0.623377	0.588923	$3.44542\times {10}^{-2}$
1	0.1	0.9	0.9	0.761394	0.779205	$1.78107\times {10}^{-2}$
	0.2			0.688938	0.690302	$1.36333\times {10}^{-3}$
	0.3			0.623377	0.615339	$8.03807\times {10}^{-3}$

we present the absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by the CDLSI method at different values of $\theta$.

Example 3. 

Consider the KdV equation of the form

$$\frac{{\partial }^{\vartheta }\zeta }{\partial v^{\vartheta }}-\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}+\frac{{\partial }^3\zeta }{\partial u^3}=-e^{\frac{u^{\theta }}{\theta }}\left(\frac{v^{\vartheta }}{\vartheta }+1\right)+\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }}\left(-\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }}+1\right), 0<\theta \le 1,$$

(44)

with the IC

$$\zeta \left(\frac{u^{\theta }}{\theta },0\right)=1,$$

(45)

and the BCs

$$\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)=1-\frac{v^{\vartheta }}{\vartheta }, {\zeta }_u\left(0,\frac{v^{\vartheta }}{\vartheta }\right)={\zeta }_{uu}\left(0,\frac{v^{\vartheta }}{\vartheta }\right)=-\frac{v^{\vartheta }}{\vartheta }.$$

(46)

This allows us to obtain the solution.

By applying the same steps in Example 2, one can obtain the following solution components:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=1-\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }},$$

(47)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w}{1+w{q}^3}{L}_u^{\theta }{S}_v^{\vartheta }\left[\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }}\left(-\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }}+1\right)+\left({\zeta }_0\right)\left(\frac{{\partial }^{\theta }{\zeta }_0}{\partial u^{\theta }}\right)\right]\right]=0,$$

(48)

$${\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w}{1+w{q}^3}{L}_u^{\theta }{S}_v^{\vartheta }\left[\left({\zeta }_0+{\zeta }_1\right)\frac{\partial }{\partial u}\left({\zeta }_0+{\zeta }_1\right)-\left({\zeta }_0\right)\left(\frac{\partial {\zeta }_0}{\partial u}\right)\right]\right]=0,$$

(49)

As a result, we have the solution of Equation (44) as

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=1-\frac{v^{\vartheta }}{\vartheta }{e}^{\frac{u^{\theta }}{\theta }},$$

(50)

We can see that if $\theta =\vartheta =1$, then the exact solution is

$$\zeta \left(u,v\right)=1-v{e}^u.$$

(51)

The following figure (Figure 3) illustrates the 3D graph of the exact and approximate solutions of Example 3 at $\theta =\vartheta =1$. In Figure 4, we sketch the solutions of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ for Equation (50) at $\theta =\vartheta =1,0.9,0.8.$

Moreover, in

Table 2. The absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI for Example 3 at different values of $\theta$, $\vartheta$, and $v$.

$\mathit{u}$	$\mathit{v}$	$\mathit{\theta }$	$\mathit{\vartheta }$	Exact	$\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}$	$\left|{\mathit{\zeta }}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\zeta }}_{\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}}\right|$
1	0.1	0.8	0.8	0.728172	$0.308522$	$4.19649\times {10}^{-1}$
	0.2			0.456344	$-0.203932$	$2.52411\times {10}^{-1}$
	0.3			0.184515	$-0.665233$	$8.49749\times {10}^{-1}$
1	0.1	0.9	0.9	0.728172	$0.57508$	$1.53092\times {10}^{-1}$
	0.2			0.456344	$0.207072$	$2.49272\times {10}^{-1}$
	0.3			0.184515	$-0.142131$	$3.26647\times {10}^{-1}$

we present the absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI at different values of $\theta$, $\vartheta$, and $v$.

Example 4 .

Consider the Klein–Gordon equation of the form

$$\frac{{\partial }^{2\vartheta }\zeta }{\partial v^{2\vartheta }}-\frac{{\partial }^2\zeta }{\partial u^2}+{\zeta }^2=2{\left(\frac{u^{\theta }}{\theta }\right)}^2-2{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^2+{\left(\frac{u^{\theta }}{\theta }\right)}^4{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^4,0<\vartheta \le 1,$$

(52)

with the ICs

$$\zeta \left(\frac{u^{\theta }}{\theta },0\right)=0, {\zeta }_v\left(\frac{u^{\theta }}{\theta },0\right)=0,$$

(53)

and the BCs

$$\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)=0, {\zeta }_u\left(0,\frac{v^{\vartheta }}{\vartheta }\right)=0.$$

(54)

This allows us to obtain the solution.

By applying the CDLST to Equation (52), the conformable Laplace transform to (53), and the conformable Sumudu transform to (54), we obtain

$$\Omega \left(q,w\right)=\frac{4w^2}{q^3}+\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left(\frac{u^{\theta }}{\theta }\right)}^4{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^4-{\zeta }^2\right].$$

(55)

Taking the inverse ${\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}$ transform of (55), we obtain

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(\frac{u^{\theta }}{\theta }\right)}^2{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^2+{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left(\frac{u^{\theta }}{\theta }\right)}^4{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^4-{\zeta }^2\right]\right].$$

(56)

Now, by applying the iterative method, we substitute (26) in (56) and with the results in (30), (31), and (32) we obtain the following solution components:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(\frac{u^{\theta }}{\theta }\right)}^2{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^2,$$

(57)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left(\frac{u^{\theta }}{\theta }\right)}^4{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^4-{\left({\zeta }_0\right)}^2\right]\right]=0,$$

(58)

$${\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{1-q^2{w}^2}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left({\zeta }_0+{\zeta }_1\right)}^2-{\left({\zeta }_0\right)}^2\right]\right]=0.$$

(59)

As a result, we have the solution of Equation (52) as

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(\frac{u^{\theta }}{\theta }\right)}^2{\left(\frac{v^{\vartheta }}{\vartheta }\right)}^2.$$

(60)

We can see that if $\theta =\vartheta =1$, then the exact solution is

$$\zeta \left(u,v\right)=u^2{v}^2.$$

(61)

The following figure (Figure 5) illustrates the 3D graph of the exact and approximate solutions of Example 4 at $\theta =\vartheta =1$. In Figure 6, we sketch the solutions of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)\mathrm{for}\mathrm{Equation}\left(60\right)\mathrm{at}\theta =\vartheta =1,0.9,0.8.$

Moreover, in

Table 3. The absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI for Example 4 at different values of $\theta$, $\vartheta$, and $v$.

$\mathit{u}$	$\mathit{v}$	$\mathit{\theta }$	$\mathit{\vartheta }$	Exact	$\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}\mathit{M}$	$\left|{\mathit{\zeta }}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\zeta }}_{\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}\mathit{M}}\right|$
1	0.1	0.8	0.8	0.01	0.0613254	$5.13254\times {10}^{-2}$
	0.2			0.04	0.185904	$1.45904\times {10}^{-1}$
	0.3			0.09	0.355659	$2.65659\times {10}^{-1}$
1	0.1	0.9	0.9	0.01	0.0241563	$1.41563\times {10}^{-2}$
	0.2			0.04	0.084117	$4.4117\times {10}^{-2}$
	0.3			0.09	0.174521	$8.45212\times {10}^{-2}$

we present the absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI for Example 4 at different values of $\theta$, $\vartheta$, and $v$.

Example 5 .

Consider the telegraph equation of the form

$$\frac{{\partial }^{2\theta }\zeta }{\partial u^{2\theta }}=\frac{{\partial }^{2\vartheta }\zeta }{\partial v^{2\vartheta }}+2\frac{{\partial }^{\vartheta }\zeta }{\partial v^{\vartheta }}+{\zeta }^2+e^{\frac{u^{\theta }}{\theta }-2\frac{v^{\vartheta }}{\vartheta }}-e^{2\frac{u^{\theta }}{\theta }-4\frac{v^{\vartheta }}{\vartheta }},0<\theta ,\vartheta \le 1,$$

(62)

with the Ics

$$\zeta \left(\frac{u^{\theta }}{\theta },0\right)=e^{\frac{u^{\theta }}{\theta }}, {\zeta }_v\left(\frac{u^{\theta }}{\theta },0\right)=-2e^{\frac{u^{\theta }}{\theta }},$$

(63)

and the BCs

$$\zeta \left(0,\frac{v^{\vartheta }}{\vartheta }\right)={\zeta }_u\left(0,\frac{v^{\vartheta }}{\vartheta }\right)=e^{-2\frac{v^{\vartheta }}{\vartheta }}.$$

(64)

Applying the same steps as in Example 5, we obtain the following solution components:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=e^{\frac{u^{\theta }}{\theta }-2\frac{v^{\vartheta }}{\vartheta }},$$

(65)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{\left(w^2{q}^2-2w-1\right)}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left({\zeta }_0\right)}^2-e^{2\frac{u^{\theta }}{\theta }-4\frac{v^{\vartheta }}{\vartheta }}\right]\right]=0,$$

(66)

$${\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)={\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[\frac{w^2}{\left(w^2{q}^2-2w-1\right)}{L}_u^{\theta }{S}_v^{\vartheta }\left[{\left({\zeta }_0+{\zeta }_1\right)}^2-{\left({\zeta }_0\right)}^2\right]\right]=0.$$

(67)

As a result, we have the solution of Equation (62) as

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=e^{\frac{u^{\theta }}{\theta }-2\frac{v^{\vartheta }}{\vartheta }}.$$

(68)

We can see that if$\theta =\vartheta =1$, then the exact solution is

$$\zeta \left(u,v\right)=e^{u-2v}.$$

(69)

The following figure (Figure 7) illustrates the 3D graph of the exact and approximate solutions of Example 5 at $\theta =\vartheta =1$. In Figure 8, we sketch the solutions of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ for Equation (68) at $\theta =\vartheta =1,0.9,0.8.$

Moreover, in

Table 4. The absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI for Example 5 at different values of $\theta$, $\vartheta$, and $v$.

$\mathit{u}$	$\mathit{v}$	$\mathit{\theta }$	$\mathit{\vartheta }$	Exact	$\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}$	$\left|{\mathit{\zeta }}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\zeta }}_{\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}}\right|$
1	0.1	0.8	0.8	2.22554	2.3485	$1.22959\times {10}^{-1}$
	0.2			1.82212	1.75091	$7.12096\times {10}^{-2}$
	0.3			1.491823	1.34421	$1.47616\times {10}^{-1}$
1	0.1	0.9	0.9	2.22554	2.29642	$7.08757\times {10}^{-2}$
	0.2			1.82212	1.80229	$1.98285\times {10}^{-2}$
	0.3			1.491823	1.43211	$5.97162\times {10}^{-2}$

we present the absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI at different values of $\theta$, $\vartheta$, and $v$.

Example 6. 

We consider the following nonlinear fractional equation in the sense of CFD:

$$\frac{{\partial }^{\vartheta }\zeta }{\partial v^{\vartheta }}+\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}=0, 0<\theta ,\vartheta \le 1$$

(70)

with the Ics

$$\zeta \left(\frac{u^{\theta }}{\theta },0\right)=\sin \frac{u^{\theta }}{\theta },$$

(71)

and the BCs

$$\zeta \left(0,\frac{u^{\vartheta }}{\vartheta }\right)=0.$$

(72)

This allows us to obtain the solution.

Operating the CDLST on (70) and the conformable Laplace transform on (71), we obtain

$$\Omega \left(q,w\right)=\frac{1}{q^2+1}-w{L}_u^{\theta }{S}_v^{\vartheta }\left[\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}\right].$$

(73)

Taking the inverse ${\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}$ transform of (73), we obtain

$$\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\sin \frac{u^{\theta }}{\theta }-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[w{L}_u^{\theta }{S}_v^{\vartheta }\left[\zeta \frac{{\partial }^{\theta }\zeta }{\partial u^{\theta }}\right]\right].$$

(74)

Now, applying the iterative method, we substitute (28) in (74) and with the results in (30), (31), and (32) we obtain the following solution components:

$${\zeta }_0\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=\sin \frac{u^{\theta }}{\theta },$$

(75)

$${\zeta }_1\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)=-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[w{L}_u^{\theta }{S}_v^{\vartheta }\left[\left({\zeta }_0\right)\left(\frac{{\partial }^{\theta }{\zeta }_0}{\partial u^{\theta }}\right)\right]\right]=-\frac{v^{\vartheta }}{2\vartheta }\sin \left(2\frac{u^{\theta }}{\theta }\right) ,$$

(76)

$$\begin{array}{cc}\hfill {\zeta }_2\left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)& =-{\left(L_u^{\theta }\right)}^{-1}{\left(S_v^{\vartheta }\right)}^{-1}\left[w{L}_u^{\theta }{S}_v^{\vartheta }\left[\left(\left({\zeta }_0+{\zeta }_1\right){\left({\zeta }_0+{\zeta }_1\right)}_u\right)-\left(\left({\zeta }_0\right){\left({\zeta }_0\right)}_u\right)\right]\right]\hfill \\ & =\frac{v^{2\vartheta }}{2{\vartheta }^2}\left(\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(2\frac{u^{\theta }}{\theta }\right)+\frac{1}{2}\cos \left(\frac{u^{\theta }}{\theta }\right)\sin \left(2\frac{u^{\theta }}{\theta }\right) \right)-\frac{v^{3\vartheta }}{3!{\vartheta }^3}\cos \left(2\frac{u^{\theta }}{\theta }\right)\sin \left(2\frac{u^{\theta }}{\theta }\right).\hfill \end{array}$$

(77)

As a result, we have the solution of Equation (70) as:

$$\begin{array}{cc}\hfill \zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)& =\sin \left(\frac{u^{\theta }}{\theta }\right)-\frac{v^{\vartheta }}{2\vartheta }\sin \left(2\frac{u^{\theta }}{\theta }\right)+\frac{v^{2\vartheta }}{2{\vartheta }^2}\left(\sin \left(\frac{u^{\theta }}{\theta }\right)\cos \left(2\frac{u^{\theta }}{\theta }\right)+\frac{1}{2}\cos \left(\frac{u^{\theta }}{\theta }\right)\sin \left(2\frac{u^{\theta }}{\theta }\right) \right)\hfill \\ & -\frac{v^{3\vartheta }}{3!{\vartheta }^3}\cos \left(2\frac{u^{\theta }}{\theta }\right)\sin \left(2\frac{u^{\theta }}{\theta }\right).\hfill \end{array}$$

(78)

The exact solution if $\theta =\vartheta =1$ is

$$\zeta \left(u,v\right)=\sin \left(u\right)-\frac{v}{2}\sin (2u)+\frac{v^2}{2}\left(\sin \left(u\right)\cos \left(2u\right)+\frac{1}{2}\cos \left(u\right)\sin \left(2u\right) \right)-\frac{v^3}{3!}\cos \left(2u\right)\sin \left(2u\right).$$

(79)

The following figure (Figure 9) illustrates the 3D graph of the exact and approximate solutions of Example 6 at $\theta =\vartheta =1$. In Figure 10, we sketch the solutions of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ for Equation (78) at $\theta =\vartheta =1,0.9,0.8.$

Moreover, in

Table 5. The absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI for Example 6 at different values of $\theta$, $\vartheta$, and $v$.

$\mathit{u}$	$\mathit{v}$	$\mathit{\theta }$	$\mathit{\vartheta }$	Exact	$\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}\mathit{M}$	$\left|{\mathit{\zeta }}_{\mathit{E}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}-{\mathit{\zeta }}_{\mathit{C}\mathit{D}\mathit{L}\mathit{S}\mathit{I}\mathit{M}}\right|$
1	0.1	0.8	0.8	0.795547	0.877256	$8.17092\times {10}^{-2}$
	0.2			0.748955	0.809433	$6.04778\times {10}^{-2}$
	0.3			0.702075	0.739109	$3.70339\times {10}^{-2}$
1	0.1	0.9	0.9	0.795547	0.837204	$4.16575\times {10}^{-2}$
	0.2			0.748955	0.781333	$3.23776\times {10}^{-2}$
	0.3			0.702075	0.725031	$2.29557\times {10}^{-2}$

we present the absolute error of $\zeta \left(\frac{u^{\theta }}{\theta },\frac{v^{\vartheta }}{\vartheta }\right)$ obtained by CDLSI at different values of $\theta$, $\vartheta$, and $v$.

6. Results and Discussion

This part discusses the suggested method’s accuracy and applicability by comparing the approximate and exact solutions using graphs and tables. Figure 1, Figure 3, Figure 5, Figure 7 and Figure 9 depict the 3D plot solutions of Examples 2–6 obtained by the present method in comparison with the exact solutions at $\theta =\vartheta =1$. These figures reveal that the approximate solutions obtained by the CDLSI method are almost identical to the exact solutions. Figure 2, Figure 4, Figure 6, Figure 8 and Figure 10 present comparisons of line plots of the approximate solutions from the proposed method and the exact solutions of Examples 2–6 for different values of $u$ and fractional-order values $\theta$ and $\vartheta$. As we can see from the figures, the numerical solutions become close to the exact solutions when $\theta ,\vartheta \to 1.$ A comparative study between the exact and approximate solutions of each example in terms of absolute error at $u=1$ and at $\theta =\vartheta =0.8,0.9$ for different values of $v=0.1,0.2,0.3$ is provided in Table 1, Table 2, Table 3, Table 4 and Table 5. As observed from the figures and tables, it is confirmed that our method of solution converges quickly towards an exact solution.

7. Conclusions

The definition of the CDLST was introduced in this article. First, we applied the CDLST to a few particular functions. Following that, some theorems and properties connected to the CDLST were presented and proven. In order to gain precise solutions of a wide class of nonlinear conformable partial differential equations in the sense of conformable derivatives, we used the CDLST combined with the iterative approach to demonstrate the applicability and effectiveness of the suggested approach. For the strength of the outcomes, we sketched the solutions and compared them to the exact solutions in the integer case. We concluded that the suggested approach is effective, appropriate, trustworthy, and adequate to achieve the exact solutions of nonlinear conformable partial differential equations. In addition, the CDLSI method’s computations use less computing power than those of other similar techniques.