Solution of Time-Fractional Partial Differential Equations via the Natural Generalized Laplace Transform Decomposition Method

Abstract

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is developed for a broad class of time-fractional system equations. In particular, we check the validity and effectiveness of the approach by providing three illustrative examples, confirming its accuracy and applicability in solving both linear and nonlinear fractional problems.

1. Introduction

Recently, researchers have paid considerable attention to fractional calculus due to its wide-ranging applications in science and engineering. New developments in this field have emerged across disciplines such as physics, mathematics and engineering, highlighting the growing importance of fractional models in the analysis of real-world problems. In particular, both classical (nonlocal) and local discrete interpretations of fractional calculus have become active areas of exploration.

The literature offers a broad spectrum of analytical and approximate techniques for solving fractional partial differential equations (FPDEs). These include the fractional complex transformation method [1], the homotopy perturbation method [2], and extended homotopy techniques for solving FPDEs of arbitrary order over finite domains [3]. In [4], the Adomian Decomposition Method (ADM) was employed to solve fractional diffusion and wave equations. Other prominent methods, such as the Variational Iteration Method (VIM) and ADM, have been effectively applied to various nonlinear FPDEs [5]. Furthermore, the Yang transform, when combined with the decomposition method, has been successfully used to address fractional-order diffusion equations [6].

Additional methods have tackled specific models, such as the Caudrey–Dodd–Gibbon (CDG) equation using non-singular kernel derivatives, and the modified fifth-order KdV (fKdV) equation using the shifted Jacobi collocation method to address space–time FPDEs with variable coefficients [7,8]. Laplace-type integral transforms have been examined in [9], and applied to differential equations with variable coefficients [10]. In [11], the G-Laplace transform combined with ADM was used to analyze nonlinear equations such as the time-fractional Burgers’ equation and its coupled system.

In related recent works, the authors in [12] obtained the solutions of integral–differential equations by computational and analytical methods, combining techniques from functional analysis and fixed-point theory. The same authors studied the stability and numerical solutions for second-order ordinary differential equations, with application in mechanical systems, by applying the Picard–Lindelöf and fixed-point theorems in [13]. The authors in [14] applied a synergistic approach combining computational modeling and machine learning techniques to examine the intricacies of animal decision-making in T-maze environments.

Motivated by these developments, the present study aims to solve time-fractional system partial differential equations by employing the Natural Generalized Laplace Transform (NGLT) in conjunction with the Adomian Decomposition Method. This hybrid approach enables both exact and approximate solutions. The structure of the paper is as follows: Section 2 introduces essential definitions. Section 3 presents the definitions and properties of the NGLT required in this work. In Section 4, we formulate the proposed method for solving time-fractional partial differential equations. Section 5 provides illustrative examples to validate the method. Finally, Section 6 concludes the study with key findings and observations.

2. Preliminaries

In this section, we recall some fundamental definitions of the Caputo fractional derivative and the generalized integral transform:

Definition 1.

The left-sided Caputo fractional derivative of a function g, where $g\in C_{-1}^m$ and $m\in \mathbb{N}\cup \left\{0\right\}$, is defined in [15,16] as

$$D_{\nu }^{\delta }g\left(\nu \right)={\displaystyle \frac{{\partial }^{\delta }g\left(\nu \right)}{\partial {\nu }^{\delta }}}=\left\{\begin{array}{l}{j}^{n-\delta }\left[{\displaystyle \frac{{\partial }^{n}g\left(\nu \right)}{\partial {\nu }^n}}\right],n-1<\delta <n,n\in \mathbb{N}\\ {\displaystyle \frac{{\partial }^{n}f\left(\nu \right)}{\partial {\nu }^n}},\delta =n\end{array}\right.,$$

Definition 2

([17]). The partial fractional integrals and Caputo derivatives of a function $g\left(\mu ,\nu \right)$, where $\left(\mu ,\nu \right)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$, are given by

$$D_{\nu }^{\delta }g\left(\mu ,\nu \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(n-\delta \right)}}{\int }_0^{\nu }{\left(\nu -\lambda \right)}^{n-\delta -1}{\displaystyle \frac{{\partial }^n}{\partial {\lambda }^n}}g\left(\mu ,\lambda \right)d\lambda ,$$

(1)

where $n-1<\delta \le n$ and $\delta >0$.

Definition 3

([18]). The Riemann–Liouville fractional integral of a well defined function $g\left(\nu \right)$ of order δ is denoted as

$$j^{\delta }g\left(\nu \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\delta \right)}}\underset{0}{\overset{v}{\int }}{\left(v-z\right)}^{\delta -1}g\left(z\right)dz,0<\delta ,v,$$

where $\mathsf{\Gamma }\left(\delta \right)$ is the Gamma function.

Definition 4

([10,19]). Let $g\left(\nu \right)$ be an integrable function for all $\nu \ge 0$. The generalized integral transform $G_{\alpha }$ of the function $g\left(\nu \right)$ is given by

$$G\left(s\right)=G_{\alpha }\left(g\right)=s^{\alpha }\underset{0}{\overset{\infty }{\int }}g\left(\nu \right)e^{-{\displaystyle \frac{\nu }{s}}}d\nu ,$$

(2)

for $s\in \mathbb{C}$ and $\alpha \in \mathbb{Z}$.

3. Main Results

This section presents several definitions related to the Natural Generalized Laplace Transform (NGLT). We establish its existence condition and demonstrate the transform of the Caputo fractional derivative $D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)$.

Definition 5.

The Natural Generalized Laplace Transform (NGLT) of the function $g(\mu ,\nu )$ is defined as

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]& =& G\left(p;u,s\right)={\displaystyle \frac{s^{\alpha }}{u}}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu -{\displaystyle \frac{1}{s}}\nu }g(\mu ,\nu )d\nu d\mu ,\hfill \\ \hfill \mathrm{Re}\left(s\right),\mathrm{Re}\left(p\right)>0,\mathrm{Re}\left(u\right)>0.\end{array}$$

(3)

where $\alpha \in \mathbb{Z}$, $p,u,s\in \mathbb{C}$. The symbol ${\mathbf{N}}_{\mu }^{+}{G}_{\nu }$ denotes the transform with respect to μ and ν, respectively, and the function $G\left(p;u,s\right)$ represents the NGLT of $g\left(\mu ,\nu \right)$.

Definition 6.

The inverse Natural Generalized Laplace Transform (INGLT) is defined as

$$N_{p,u}^{-1}{G}_s^{-1}\left(G\left(p;u,s\right)\right)=g\left(\mu ,\nu \right)={\displaystyle \frac{1}{{\left(2\pi j\right)}^2}}{\int }_{\tau -j\infty }^{\tau +j\infty }{\int }_{\nu -j\infty }^{\nu +j\infty }{e}^{{\displaystyle \frac{p}{u}}\mu +{\displaystyle \frac{1}{s}}\nu }G\left(p;u,s\right)dsd\rho ,$$

where $N_{p,u}^{-1}{G}_s^{-1}$ denotes the INGLT.

The NGLTs of the functions $f\left(\mu ,\nu \right)=e^{\mu +\nu }$ and $g\left(\mu ,\nu \right)={\mu }^m{\nu }^m$ are given by the following:

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]& =& {\displaystyle \frac{s^{\alpha +1}}{\left(p-u\right)\left(1-s\right)}}\hfill \\ \hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]& =& m!{\displaystyle \frac{u^m}{p^{m+1}}}m!s^{\alpha +m+1}\hfill \end{array}$$

Definition 7.

The NGLT of the partial derivative ${\displaystyle \frac{{\partial }^n\omega \left(\mu ,\nu \right)}{\partial {\nu }^n}}$ is given by the following:

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\nu \right)}{\partial {\nu }^n}}\right]={\displaystyle \frac{\mathsf{\Phi }\left(p;u,s\right)}{s^n}}-s^{\alpha }\sum _{k=1}^n{\displaystyle \frac{1}{s^{n-k}}}{\mathbf{N}}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{k-1}\omega \left(\mu ,0\right)}{\partial {\nu }^{k-1}}}\right]$$

(4)

where $\mathsf{F}\left(p;u,s\right)$ is the (NGLT) of the function $\omega \left(\mu ,\nu \right).$

Notation 1.

The following definition can be obtained from (NGLT).

1. 

Set $\alpha =-1,p=1$, we gained double Sumudu transform

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]=G\left(u,s\right)={\displaystyle \frac{1}{us}}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{1}{u}}\mu -{\displaystyle \frac{1}{s}}\nu }g(\mu ,\nu )d\nu d\mu ,$$

2. 

Set $\alpha =0,u=1$ and $s={\displaystyle \frac{1}{s}},$ one can get double Laplace transform as

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]=G\left(p,s\right)={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-p\mu -s\nu }g(\mu ,\nu )d\nu d\mu ,$$

3. 

Set $\alpha =0,u=1$ and $s=\omega ,$ we derive the Laplace-Yang Transform

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[g\left(\mu ,\nu \right)\right]=G\left(p,\omega \right)={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-p\mu -{\displaystyle \frac{1}{\omega }}}g(\mu ,\nu )d\nu d\mu ,$$

Theorem 8.

The NGLT of the Caputo fractional derivative $D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)$ is given by the following:

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)\right]={\displaystyle \frac{\mathsf{F}\left(p;u,s\right)}{s^{\delta }}}-s^{\alpha }\sum _{k=1}^n{\displaystyle \frac{1}{s^{\delta -k}}}{\mathbf{N}}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{k-1}f\left(\mu ,0\right)}{\partial {\nu }^{k-1}}}\right]$$

(5)

where $n-1<\delta \le n$ and $\delta >0$.

Proof. 

By applying (3), we have

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)\right]& =& {\displaystyle \frac{s^{\alpha }}{u}}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu -{\displaystyle \frac{1}{s}}\nu }{D}_{\nu }^{\delta }\omega \left(\mu ,\nu \right)d\nu d\mu \hfill \\ & =& {\displaystyle \frac{s^{\alpha }}{u}}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu -{\displaystyle \frac{1}{s}}\nu }{\displaystyle \frac{1}{\mathsf{\Gamma }\left(n-\delta \right)}}{\int }_0^{\nu }{\left(\nu -\lambda \right)}^{n-\delta -1}{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}d\lambda d\nu d\mu \hfill \\ & =& {\displaystyle \frac{s^{\alpha }}{u\mathsf{\Gamma }\left(n-\delta \right)}}{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu -{\displaystyle \frac{1}{s}}\nu }{\int }_{\nu }^{\infty }{\left(\nu -\lambda \right)}^{n-\delta -1}{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}d\nu d\mu d\lambda ,\hfill \\ & & \hfill \end{array}$$

(6)

Now, substituting $\tau =\nu -\lambda$ in (6), we get

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)\right]& =& {\displaystyle \frac{s^{\alpha }}{u\mathsf{\Gamma }\left(n-\delta \right)}}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu }{\int }_0^{\infty }{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}{e}^{-{\displaystyle \frac{\lambda }{s}}}\left[{\int }_0^{\infty }{\tau }^{n-\delta -1}{e}^{-{\displaystyle \frac{\tau }{s}}}d\tau \right]d\lambda d\mu ,\hfill \\ & & \hfill \end{array}$$

(7)

The integral in brackets is a Gamma integral and is evaluated as

$${\int }_0^{\infty }{\tau }^{n-\delta -1}{e}^{-{\displaystyle \frac{\tau }{s}}}d\tau ={\displaystyle \frac{\mathsf{\Gamma }\left(n-\delta \right)}{{\left(\frac{1}{s}\right)}^{n-\delta }}},$$

so (7) becomes

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)\right]& =& {\displaystyle \frac{s^{\alpha }}{u\mathsf{\Gamma }\left(n-\delta \right)}}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu }{\int }_0^{\infty }{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}{e}^{-{\displaystyle \frac{\lambda }{s}}}\left[{\displaystyle \frac{\mathsf{\Gamma }\left(n-\delta \right)}{{\left(\frac{1}{s}\right)}^{n-\delta }}}\right]d\lambda d\mu ,\hfill \\ & & \hfill \end{array}$$

(8)

Rewriting (8), we obtain

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \left(\mu ,\nu \right)\right]& =& s^{n-\delta }\left[{\displaystyle \frac{s^{\alpha }}{u}}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu }{\int }_0^{\infty }{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}{e}^{-{\displaystyle \frac{\lambda }{s}}}d\lambda d\mu \right],\hfill \\ & & \hfill \end{array}$$

(9)

The inner integral is defined by

$$\begin{array}{ccc}\hfill {\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\nu \right)}{\partial {\nu }^n}}\right]& =& {\displaystyle \frac{s^{\alpha }}{u}}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{p}{u}}\mu }{\int }_0^{\infty }{\displaystyle \frac{{\partial }^n\omega \left(\mu ,\lambda \right)}{\partial {\lambda }^n}}{e}^{-{\displaystyle \frac{\lambda }{s}}}d\lambda d\mu \hfill \\ & =& {\displaystyle \frac{\mathsf{\Phi }\left(p;u,s\right)}{s^n}}-s^{\alpha }\sum _{k=1}^n{\displaystyle \frac{1}{s^{n-k}}}{\mathbf{N}}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{k-1}\omega \left(\mu ,0\right)}{\partial {\nu }^{k-1}}}\right],\hfill \end{array}$$

(10)

Substituting (10) into (9), we obtain

$${\mathbf{N}}_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }f\left(\mu ,\nu \right)\right]={\displaystyle \frac{\mathsf{F}\left(p;u,s\right)}{s^{\delta }}}-s^{\alpha }\sum _{k=1}^n{\displaystyle \frac{1}{s^{\delta -k}}}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{k-1}\omega \left(\mu ,0\right)}{\partial {\nu }^{k-1}}}\right].$$

This completes the proof. □

4. Preparation of the Natural Generalized Laplace Transform for Systems of Fractional Partial Differential Equations

In order to obtain the solution of time-fractional partial differential equations in the Caputo sense, we present the foundational idea of the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). For this purpose, we consider a general system of fractional partial differential equations subject to initial conditions as follows:

$$\begin{array}{ccc}\hfill D_{\nu }^{{\delta }_j}{\omega }_j\left(\stackrel{*}{\mu },\nu \right)& =& f\left(\stackrel{*}{\mu },\nu \right)+L_j\stackrel{*}{\omega }+N\stackrel{*}{\omega },n_j-1<{\delta }_j<n_j,j=1,2,\dots \hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{\left(n_j-1\right)}{\omega }_j\left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{n_j-1}}}& =& f_{m_j}\left(\stackrel{*}{\mu }\right),n_j=1,2,\dots ,m_j-1,n_j\in \mathbb{N}\hfill \end{array}$$

(12)

Here, $L_j$ and $N_j$ denote linear and nonlinear operators, respectively. Let $\stackrel{*}{\omega }=\omega \left(\stackrel{*}{\mu },\nu \right)$, where $f\left(\stackrel{*}{\mu },\nu \right)$ is a known function, and $D_{\nu }^{{\delta }_j}$ denotes the Caputo fractional derivative of order ${\delta }_j$. We define the following:

$$\omega \left(\stackrel{*}{\mu },\nu \right)=\left({\omega }_1\left(\stackrel{*}{\mu },\nu \right),{\omega }_2\left(\stackrel{*}{\mu },\nu \right),\dots ,{\omega }_n\left(\stackrel{*}{\mu },\nu \right)\right),$$

where $\stackrel{*}{\mu }=\left({\mu }_1,{\mu }_2,\dots ,{\mu }_n\right)\in {\mathbb{R}}^n$.

To solve (11), we apply the NGLT to (11) and the standard GLT to (12). Using Theorem 1, we obtain the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{N_{\mu }^{+}{G}_{\nu }\left[{\omega }_j\left(\stackrel{*}{\mu },\nu \right)\right]}{s^{{\delta }_j}}}& =& s^{\alpha }\sum _{m_j=1}^{n_j-1}{\displaystyle \frac{1}{s^{{\delta }_j-k}}}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{m_j-1}\omega \left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{m_j-1}}}\right]\hfill \\ & & +N_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)+L_j\stackrel{*}{\omega }+N\stackrel{*}{\omega }\right].\hfill \end{array}$$

(13)

Multiplying both sides of the above equation by $s^{{\delta }_j}$ for $j=1,2,3,\dots ,n$, we obtain the following:

$$\begin{array}{ccc}\hfill N_{\mu }^{+}{G}_{\nu }\left[{\omega }_j\left(\stackrel{*}{\mu },\nu \right)\right]& =& \sum _{m_j=1}^{n_j-1}{s}^{\alpha +m_j}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{m_j-1}\omega \left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{m_j-1}}}\right]\hfill \\ & & +s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)+L_j\stackrel{*}{\omega }+N\stackrel{*}{\omega }\right].\hfill \end{array}$$

(14)

Applying the inverse NGLT to both sides of (14) yields the following:

$$\begin{array}{ccc}\hfill {\omega }_j\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[\sum _{m_j=1}^{n_j-1}{s}^{\alpha +m_j}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{m_j-1}\omega \left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{m_j-1}}}\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)+L_j\stackrel{*}{\omega }+N\stackrel{*}{\omega }\right]\right].\hfill \end{array}$$

(15)

The solution of (11) can be expressed using the Adomian decomposition method as a series:

$${\omega }_j\left(\stackrel{*}{\mu },\nu \right)=\sum _{j=0}^{\infty }{\omega }_{jj}\left(\stackrel{*}{\mu },\nu \right),j=1,2,\dots ,n.$$

(16)

From the above, the solution can be written as the following:

$$\begin{array}{ccc}\hfill \sum _{j=0}^{\infty }{\omega }_{jj}\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[\sum _{m_j=1}^{n_j-1}{s}^{\alpha +m_j}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{m_j-1}\omega \left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{m_j-1}}}\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[\sum _{j=0}^{\infty }{L}_j\stackrel{*}{{\omega }_j}+\sum _{j=0}^{\infty }N{\stackrel{*}{\omega }}_j\right]\right].\hfill \end{array}$$

(17)

The equation derived above leads us to the recursive formulation of the components of the solution series. The zeroth-order term is given by the following:

$${\omega }_{j0}\left(\stackrel{*}{\mu },\nu \right)=N_{p;u}^{-1}{G}_s^{-1}\left[\sum _{m_j=1}^{n_j-1}{s}^{\alpha +m_j}{N}_{\mu }^{+}\left[{\displaystyle \frac{{\partial }^{m_j-1}\omega \left(\stackrel{*}{\mu },0\right)}{\partial {\nu }^{m_j-1}}}\right]+s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[f\left(\stackrel{*}{\mu },\nu \right)\right]\right],$$

and the rest of the terms for $j\ge 1$ are defined as follows:

$$\begin{array}{ccc}\hfill {\omega }_{j1}\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[L_j\stackrel{*}{{\omega }_0}+N\stackrel{*}{{\omega }_0}\right]\right],\hfill \\ \hfill {\omega }_{j2}\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[L_j\stackrel{*}{{\omega }_1}+N\stackrel{*}{{\omega }_1}\right]\right],\hfill \\ \hfill {\omega }_{j3}\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[L_j\stackrel{*}{{\omega }_2}+N\stackrel{*}{{\omega }_2}\right]\right],\hfill \\ & & ⋮\hfill \\ \hfill {\omega }_{jj}\left(\stackrel{*}{\mu },\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{{\delta }_j}{N}_{\mu }^{+}{G}_{\nu }\left[L_j\stackrel{*}{{\omega }_{j-1}}+N\stackrel{*}{{\omega }_{j-1}}\right]\right].\hfill \end{array}$$

(18)

Therefore, the approximate solution to (11) is obtained by substituting the expressions in (18) into the series representation in (16), giving us the following:

$${\omega }_j\left(\stackrel{*}{\mu },\nu \right)=\underset{n\to \infty }{lim}{\omega }_{jj}\left(\stackrel{*}{\mu },\nu \right)={\omega }_{j0}\left(\stackrel{*}{\mu },\nu \right)+{\omega }_{j1}\left(\stackrel{*}{\mu },\nu \right)+{\omega }_{j2}\left(\stackrel{*}{\mu },\nu \right)+\cdots +{\omega }_{jn}\left(\stackrel{*}{\mu },\nu \right),$$

where $j=1,2,\dots ,n$.

• Convergence:

Theorem 9.

Let A be a Banach space. The series solution of (18) converges to an element $S_j\in A$ for $j=1,2,\dots ,n$, if there exists a constant ${\nu }_j$, with $0\le {\nu }_j<1$, such that

$$∥{\omega }_{jn}(\stackrel{*}{\mu },\nu )∥\le {\nu }_j∥{\omega }_{j(n-1)}(\stackrel{*}{\mu },\nu )∥$$

for all $n\in \mathbb{N}$.

Proof. 

Define the partial sums of the series in (18) as follows:

$$\begin{array}{ccc}\hfill S_{j0}& =& {\omega }_{j0}(\stackrel{*}{\mu },\nu ),\hfill \\ \hfill S_{j1}& =& {\omega }_{j0}(\stackrel{*}{\mu },\nu )+{\omega }_{j1}(\stackrel{*}{\mu },\nu ),\hfill \\ \hfill S_{j2}& =& {\omega }_{j0}(\stackrel{*}{\mu },\nu )+{\omega }_{j1}(\stackrel{*}{\mu },\nu )+{\omega }_{j2}(\stackrel{*}{\mu },\nu ),\hfill \\ & ⋮& \\ \hfill S_{jn}& =& \sum _{k=0}^n{\omega }_{jk}(\stackrel{*}{\mu },\nu ).\hfill \end{array}$$

We aim to show that the sequence $\left\{S_{jn}\right\}$ is a Cauchy sequence in the Banach space A. To estimate the difference between consecutive terms,

$$\begin{array}{cc}\hfill ∥S_{j(n+1)}-S_{jn}∥& =∥{\omega }_{j(n+1)}(\stackrel{*}{\mu },\nu )∥\hfill \\ \hfill & \le {\nu }_j∥{\omega }_{jn}(\stackrel{*}{\mu },\nu )∥\hfill \\ \hfill & \le {\nu }_j^2∥{\omega }_{j(n-1)}(\stackrel{*}{\mu },\nu )∥\hfill \\ \hfill & \le \cdots \hfill \\ \hfill & \le {\nu }_j^{n+1}∥{\omega }_{j0}(\stackrel{*}{\mu },\nu )∥.\hfill \end{array}$$

Now consider $n\ge m$. By the triangle inequality,

$$\begin{array}{cc}\hfill ∥S_{jn}-S_{jm}∥& =∥\sum _{k=m+1}^n\left(S_{jk}-S_{j(k-1)}\right)∥\hfill \\ \hfill & \le \sum _{k=m+1}^n∥S_{jk}-S_{j(k-1)}∥\hfill \\ \hfill & \le \sum _{k=m+1}^n{\nu }_j^k∥{\omega }_{j0}(\stackrel{*}{\mu },\nu )∥\hfill \\ \hfill & ={\nu }_j^{m+1}\left(1+{\nu }_j+\cdots +{\nu }_j^{n-m-1}\right)∥{\omega }_{j0}(\stackrel{*}{\mu },\nu )∥\hfill \\ \hfill & \le {\nu }_j^{m+1}\left({\displaystyle \frac{1-{\nu }_j^{n-m}}{1-{\nu }_j}}\right)∥{\omega }_{j0}(\stackrel{*}{\mu },\nu )∥.\hfill \end{array}$$

Since $0\le {\nu }_j<1$, it follows that $1-{\nu }_j^{n-m}\le 1$, and thus we get the following:

$$∥S_{jn}-S_{jm}∥\le {\displaystyle \frac{{\nu }_j^{m+1}}{1-{\nu }_j}}∥{\omega }_{j0}(\stackrel{*}{\mu },\nu )∥.$$

As ${\omega }_{j0}(\stackrel{*}{\mu },\nu )$ is bounded, we conclude that $∥S_{jn}-S_{jm}∥\to 0$ as $n,m\to \infty$. Therefore, $\left\{S_{jn}\right\}$ is a Cauchy sequence in the Banach space A, and the series solution of (18) converges. This completes the proof. □

5. Applications

Here, we provide three examples of systems of time-fractional partial differential equations and solve them using the Natural Generalized Laplace Transform Decomposition Method. Before we present the examples, we note the following identities:

$$\begin{array}{ccc}\hfill sinh\left(\mu \right)& =& \mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots ,\hfill \\ \hfill cosh\left(\mu \right)& =& 1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\hfill \end{array}$$

(19)

Also, the symbols $\delta$ and $\beta$ are of the fractional-order in the range between $(0,1]$.

Example 1.

Consider the linear time-fractional partial differential equation with initial conditions given by the following:

$$\begin{array}{ccc}\hfill D_{\nu }^{\delta }\omega -{\varphi }_{\mu }+\omega +\varphi & =& 0,\hfill \\ \hfill D_{\nu }^{\delta }\varphi -{\omega }_{\mu }+\omega +\varphi & =& 0,0<\delta \le 1\hfill \end{array}$$

(20)

associated with the initial conditions:

$$\omega \left(\mu ,0\right)=sinh\left(\mu \right),\varphi \left(\mu ,0\right)=cosh\left(\mu \right).$$

(21)

Taking the (NGLT) of (20), we get the following:

$$\begin{array}{ccc}\hfill N_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \right]& =& N_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }-\omega -\varphi \right],\hfill \\ \hfill N_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\varphi \right]& =& N_{\mu }^{+}{G}_{\nu }\left[{\omega }_{\mu }-\omega -\varphi \right],\hfill \end{array}$$

(22)

Using Theorem 1, the left-hand side of (22) becomes the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{F}\left(p;u,s\right)}{s^{\delta }}}-s^{\alpha -\delta +1}\omega \left(p;u,0\right)& =& N_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }-\omega -\varphi \right],\hfill \\ \hfill {\displaystyle \frac{\mathsf{\Phi }\left(p;u,s\right)}{s^{\delta }}}-s^{\alpha -\delta +1}\varphi \left(p;u,0\right)& =& N_{\mu }^{+}{G}_{\nu }\left[{\omega }_{\mu }-\omega -\varphi \right],\hfill \end{array}$$

(23)

where $\mathsf{F}\left(p;u,s\right)$ and $\mathsf{\Phi }\left(p;u,s\right)$ are the (NGLT) of $\omega \left(\mu ,\nu \right)$ and $\varphi \left(\mu ,\nu \right)$, and $\omega \left(p;u,0\right),\varphi \left(p;u,0\right)$ are the Natural transforms of $\omega \left(\mu ,0\right)$ and $\varphi \left(\mu ,0\right)$, respectively.

By using the Natural transform of (21) along with (19) and organizing the results, we obtain the following:

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(p,s\right)& =& s^{\alpha +1}\left[{\displaystyle \frac{u}{p^2}}+{\displaystyle \frac{u^2}{p^3}}+{\displaystyle \frac{u^3}{p^4}}+\dots \right]\hfill \\ & & +s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }-\omega -\varphi \right],\hfill \\ \hfill \mathsf{\Phi }\left(p,s\right)& =& s^{\alpha +1}\left[{\displaystyle \frac{1}{p}}+{\displaystyle \frac{u^3}{p^4}}+{\displaystyle \frac{u^5}{p^6}}+\dots \right]\hfill \\ & & +s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\omega }_{\mu }-\omega -\varphi \right].\hfill \end{array}$$

(24)

Employing (INGLT) for (24), we get

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& \mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots .\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }-\omega -\varphi \right]\right],\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& 1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\omega }_{\mu }-\omega -\varphi \right]\right],\hfill \end{array}$$

(25)

The series solutions of (20) are defined as follows:

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& \sum _{n=0}^{\infty }{\omega }_n\left(\mu ,\nu \right),n=1,2,\dots \hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& \sum _{n=0}^{\infty }{\varphi }_n\left(\mu ,\nu \right),n=1,2,\dots \hfill \end{array}$$

(26)

by replacing (26) into (25), we obtain

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\omega }_n\left(\mu ,\nu \right)& =& \mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots .\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\sum _{n=0}^{\infty }\left({\varphi }_{n\mu }-{\omega }_n-{\varphi }_n\right)\right]\right],\hfill \\ \hfill \sum _{n=0}^{\infty }{\varphi }_n\left(\mu ,\nu \right)& =& 1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\sum _{n=0}^{\infty }\left({\omega }_{n\mu }-{\omega }_n-{\varphi }_n\right)\right]\right],\hfill \end{array}$$

(27)

from (27), we introduce the iteration relation as follows:

$$\begin{array}{ccc}\hfill {\omega }_0\left(\mu ,\nu \right)& =& \mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots =sinh\mu ,\hfill \\ \hfill {\varphi }_0\left(\mu ,\nu \right)& =& 1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots =cosh\mu ,\hfill \end{array}$$

(28)

and

$$\begin{array}{ccc}\hfill {\omega }_{n+1}\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\varphi }_{n\mu }-{\omega }_n-{\varphi }_n\right)\right]\right],\hfill \\ \hfill {\varphi }_{n+1}\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\omega }_{n\mu }-{\omega }_n-{\varphi }_n\right)\right]\right],\hfill \end{array}$$

(29)

at $n=0,$ (29) becomes

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\varphi }_{0\mu }-{\omega }_0-{\varphi }_0\right)\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left(-{\varphi }_0\right)\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right)\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\alpha +\delta +1}\left({\displaystyle \frac{1}{p}}+{\displaystyle \frac{u^3}{p^4}}+{\displaystyle \frac{u^5}{p^6}}+\dots \right)\right],\hfill \\ & =& -\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right){\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}},\hfill \\ & =& -{\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varphi }_1\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\omega }_{0\mu }-{\omega }_0-{\varphi }_0\right)\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\omega }_0\right)\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\alpha +\delta +1}\left(p^{\alpha +2}+p^{\alpha +4}+p^{\alpha +6}+p^{\alpha +8}+\dots \right)\right],\hfill \\ & =& -\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right){\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}},\hfill \\ & =& -{\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right),\hfill \end{array}$$

By setting $n=1$ in (29), we obtain

$$\begin{array}{ccc}\hfill {\omega }_2\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\varphi }_{1\mu }-{\omega }_1-{\varphi }_1\right)\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left({\varphi }_1\right)\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\alpha +2\delta +1}\left({\displaystyle \frac{u}{p^2}}+{\displaystyle \frac{u^2}{p^3}}+{\displaystyle \frac{u^3}{p^4}}+\dots .\right)\right],\hfill \\ & =& {\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right),\hfill \end{array}$$

Similarly,

$${\varphi }_2\left(\mu ,\nu \right)={\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right),$$

At $n=2$, we have

$$\begin{array}{ccc}\hfill {\omega }_3\left(\mu ,\nu \right)& =& -{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right),\hfill \\ \hfill {\varphi }_3\left(\mu ,\nu \right)& =& -{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right).\hfill \end{array}$$

The series solution of (20) is given by

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& {\omega }_0+{\omega }_1+{\omega }_2+\dots ,\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& {\varphi }_0+{\varphi }_1+{\varphi }_2+\dots ,\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& \left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)\hfill \\ & & -{\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right)\hfill \\ & & +{\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)\hfill \\ & & -{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right)+\dots ,\hfill \\ \hfill \omega \left(\mu ,\nu \right)& =& \left(1+{\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}+{\displaystyle \frac{{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}+\dots \right)sinh\left(\mu \right)\hfill \\ & & -\left({\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}+{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}+{\displaystyle \frac{{\nu }^{5\delta }}{\mathsf{\Gamma }\left(5\delta +1\right)}}+\dots \right)cosh\left(\mu \right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \varphi \left(\mu ,\nu \right)& =& \left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right)\hfill \\ & & -{\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)\hfill \\ & & +{\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\left(1+{\displaystyle \frac{{\mu }^2}{2!}}+{\displaystyle \frac{{\mu }^4}{4!}}+{\displaystyle \frac{{\mu }^6}{6!}}+\dots .\right)\hfill \\ & & -{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\left(\mu +{\displaystyle \frac{{\mu }^3}{3!}}+{\displaystyle \frac{{\mu }^5}{5!}}+{\displaystyle \frac{{\mu }^7}{7!}}+\dots \right)+\dots ,\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& \left(1+{\displaystyle \frac{{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}+{\displaystyle \frac{{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}+\dots \right)cosh\left(\mu \right)\hfill \\ & & -\left({\displaystyle \frac{{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}+{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}+{\displaystyle \frac{{\nu }^{5\delta }}{\mathsf{\Gamma }\left(5\delta +1\right)}}+\dots \right)sinh\left(\mu \right).\hfill \end{array}$$

By setting $\delta =1$, the exact solution of (20) is given by the following:

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& cosh\left(\nu \right)sinh\mu \hfill \\ & & -sinh\left(\nu \right)cosh\mu ,\hfill \\ \hfill \omega \left(\mu ,\nu \right)& =& sinh\left(\mu -\nu \right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \varphi \left(\mu ,\nu \right)& =& cosh\left(\nu \right)cosh\mu \hfill \\ & & -sinh\left(\nu \right)sinh\mu ,\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& cosh\left(\mu -\nu \right).\hfill \end{array}$$

The results obtained agree with [20]. Figure 1 presents the 3D plots of the approximate solutions of $\omega (\mu ,\nu )$ and $\varphi (\mu ,\nu )$ for different values of δ.

In the following example, we solve a nonlinear time-fractional partial differential equation.

Example 2.

Consider the nonlinear time-fractional partial differential equation with initial conditions given by the following:

$$\begin{array}{ccc}\hfill D_{\nu }^{\delta }\omega +\varphi {\omega }_{\mu }+\omega & =& 1,\hfill \\ \hfill D_{\nu }^{\delta }\varphi -{\varphi }_{\mu }\omega -\varphi & =& 1,0<\delta \le 1,\hfill \end{array}$$

(30)

subject to the initial conditions

$$\omega \left(\mu ,0\right)=e^{\mu },\varphi \left(\mu ,0\right)=e^{-\mu }.$$

(31)

Applying the Natural Generalized Laplace Transform (NGLT) to Equation (30), we obtain

$$\begin{array}{ccc}\hfill N_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\omega \right]& =& N_{\mu }^{+}{G}_{\nu }\left[1-\omega \right]-N_{\mu }^{+}{G}_{\nu }\left[\varphi {\omega }_{\mu }\right],\hfill \\ \hfill N_{\mu }^{+}{G}_{\nu }\left[D_{\nu }^{\delta }\varphi \right]& =& N_{\mu }^{+}{G}_{\nu }\left[1+\varphi \right]+N_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }\omega \right],\hfill \end{array}$$

(32)

By employing Theorem 1, the left-hand side of Equation (32) becomes

$$\begin{array}{ccc}\hfill {\displaystyle \frac{N_{\mu }^{+}{G}_{\nu }\left[\omega \left(\mu ,\nu \right)\right]}{s^{\delta }}}-s^{\alpha -\delta +1}{N}_{\mu }^{+}\left[\omega \left(\mu ,0\right)\right]& =& N_{\mu }^{+}{G}_{\nu }\left[1-\omega \right]-N_{\mu }^{+}{G}_{\nu }\left[\varphi {\omega }_{\mu }\right],\hfill \\ \hfill {\displaystyle \frac{N_{\mu }^{+}{G}_{\nu }\left[\varphi \left(\mu ,\nu \right)\right]}{s^{\delta }}}-s^{\alpha -\delta +1}{N}_{\mu }^{+}\left[\varphi \left(\mu ,0\right)\right]& =& N_{\mu }^{+}{G}_{\nu }\left[1+\varphi \right]+N_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }\omega \right],\hfill \end{array}$$

(33)

Applying the Generalized Laplace Transform (GLT) to (31) and simplifying, we obtain

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(p;u,s\right)& =& {\displaystyle \frac{s^{\alpha +1}}{p-u}}+s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-\omega \right]\hfill \\ & & -s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\varphi {\omega }_{\mu }\right],\hfill \\ \hfill \mathsf{\Phi }\left(p;u,s\right)& =& {\displaystyle \frac{s^{\alpha +1}}{p+u}}+s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1+\varphi \right]\hfill \\ & & +s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }\omega \right].\hfill \end{array}$$

(34)

Using the inverse NGLT (INGLT) on (34), we obtain the following:

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& e^{\mu }-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-\omega \right]\right]\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\varphi {\omega }_{\mu }\right]\right],\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& e^{-\mu }+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1+\varphi \right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{\mu }\omega \right]\right].\hfill \end{array}$$

(35)

The NGLT Decomposition Method (NGLTDM) assumes a series solution for (30) of the form:

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& \sum _{n=0}^{\infty }{\omega }_n\left(\mu ,\nu \right),n=1,2,\dots \hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& \sum _{n=0}^{\infty }{\varphi }_n\left(\mu ,\nu \right),n=1,2,\dots \hfill \end{array}$$

(36)

Additionally, we assume that the nonlinear terms $\varphi {\omega }_{\mu }$ and ${\varphi }_{\mu }\omega$ are decomposed as

$$\varphi {\omega }_{\mu }=\sum _{n=0}^{\infty }{A}_n,{\varphi }_{\mu }\omega =\sum _{n=0}^{\infty }{B}_n,$$

(37)

where

$$\begin{array}{ccc}\hfill A_0& =& {\varphi }_0{\omega }_{0\mu },A_1={\varphi }_0{\omega }_{1\mu }+{\varphi }_{1\mu }{\omega }_{0\mu },\hfill \\ \hfill A_2& =& {\varphi }_0{\omega }_{2\mu }+{\varphi }_1{\omega }_{1\mu }+{\varphi }_2{\omega }_{0\mu },\hfill \\ \hfill A_3& =& {\varphi }_0{\omega }_{3\mu }+{\varphi }_1{\omega }_{2\mu }+{\varphi }_2{\omega }_{1\mu }+{\varphi }_3{\omega }_{0\mu ,}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill B_0& =& {\varphi }_{0\mu }{\omega }_0,B_1={\varphi }_{0\mu }{\omega }_1+{\varphi }_{1\mu }{\omega }_0,\hfill \\ \hfill B_2& =& {\varphi }_{0\mu }{\omega }_2+{\varphi }_{1\mu }{\omega }_1+{\varphi }_{2\mu }{\omega }_0,\hfill \\ \hfill B_3& =& {\varphi }_{0\mu }{\omega }_3+{\varphi }_{1\mu }{\omega }_2+{\varphi }_{2\mu }{\omega }_1+{\varphi }_{3\mu }{\omega }_0.\hfill \end{array}$$

By substituting (37) and (36) into (35), one obtains the following:

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\omega }_n\left(\mu ,\nu \right)& =& e^{\mu }+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-\sum _{n=0}^{\infty }{\omega }_n\right]\right]\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\sum _{n=0}^{\infty }{A}_n\right]\right],\hfill \\ \hfill \sum _{n=0}^{\infty }{\varphi }_n\left(\mu ,\nu \right)& =& e^{-\mu }+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1+\sum _{n=0}^{\infty }{\varphi }_n\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[\sum _{n=0}^{\infty }{B}_n\right]\right].\hfill \end{array}$$

(38)

From (38), we introduce the iteration relations as follows:

$${\omega }_0\left(\mu ,\nu \right)=e^{\mu },{\varphi }_0\left(\mu ,\nu \right)=e^{-\mu },$$

and

$$\begin{array}{ccc}\hfill {\omega }_{n+1}\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-{\omega }_n\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[A_n\right]\right],\hfill \\ \hfill {\varphi }_{n+1}\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1+{\varphi }_n\right]\right]+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[B_n\right]\right].\hfill \end{array}$$

(39)

At $n=0$, (38) becomes

$$\begin{array}{ccc}\hfill {\omega }_1\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-{\omega }_0\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[A_0\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1-e^{\mu }\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[e^{\mu }\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{s^{\alpha +\delta +1}}{\left(p-u\right)}}\right],\hfill \\ & =& -{\displaystyle \frac{e^{\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\varphi }_1\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[1+{\varphi }_0\right]\right]+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[B_0\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_q^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_y{G}_{\nu }\left[-{\varphi }_0\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[e^{-\mu }\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{s^{\alpha +\delta +1}}{(p+u)}}\right],\hfill \\ & =& {\displaystyle \frac{e^{-\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}.\hfill \end{array}$$

At $n=1$,

$$\begin{array}{ccc}\hfill {\omega }_2\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\omega }_1\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[A_1\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\omega }_1\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{0\mu }{\omega }_1+{\varphi }_{1\mu }{\omega }_0\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{e^{\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\right]\right],\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{s^{\alpha +2\delta +1}}{(p-u)}}\right],\hfill \\ & =& {\displaystyle \frac{e^{\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}.\hfill \end{array}$$

And

$$\begin{array}{ccc}\hfill {\varphi }_2\left(\mu ,\nu \right)& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_1\right]\right]+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[B_1\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_1\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{e^{-\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{s^{\alpha +2\delta +1}}{(p+u)}}\right],\hfill \\ & =& {\displaystyle \frac{e^{-\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}.\hfill \end{array}$$

In the same manner, at $n=2$, we have

$$\begin{array}{ccc}\hfill {\omega }_3& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\omega }_2\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[A_2\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\omega }_2\right]\right],\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_0{\omega }_{2\mu }+{\varphi }_1{\omega }_{1\mu }+{\varphi }_2{\omega }_{0\mu }\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\displaystyle \frac{e^{\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\right]\right]\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{2e^{\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}-{\displaystyle \frac{e^{\mu }{\nu }^{2\delta }}{{\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\right]\right],\hfill \\ \hfill {\omega }_3& =& -{\displaystyle \frac{e^{\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}+{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varphi }_3& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_2\right]\right]+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[B_2\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_2\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{0\mu }{\omega }_2+{\varphi }_{1\mu }{\omega }_1+{\varphi }_{2\mu }{\omega }_0\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{e^{-\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\displaystyle \frac{2{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}+{\displaystyle \frac{{\nu }^{2\delta }}{{\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\right]\right],\hfill \\ \hfill {\varphi }_3& =& {\displaystyle \frac{e^{-\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}+{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\omega }_4& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\omega }_3\right]\right]-N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[A_3\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\omega }_3\right]\right],\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_0{\omega }_{3\mu }+{\varphi }_1{\omega }_{2\mu }+{\varphi }_2{\omega }_{1\mu }+{\varphi }_3{\omega }_{0\mu }\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{3e^{\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\right]\right]\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{e^{\mu }{\nu }^{2\delta }}{{\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\right]\right]\hfill \\ & & -N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\right]\right],\hfill \\ \hfill {\omega }_4& =& {\displaystyle \frac{3e^{\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\hfill \\ & & +{\displaystyle \frac{2{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varphi }_4& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_3\right]\right]+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[B_3\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_2\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\varphi }_{0\mu }{\omega }_3+{\varphi }_{1\mu }{\omega }_2+{\varphi }_{2\mu }{\omega }_1+{\varphi }_{3\mu }{\omega }_0\right]\right],\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{3e^{-\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{-\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{3\delta }}{{\left(\mathsf{\Gamma }\left(3\delta +1\right)\right)}^2}}\right]\right]\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\delta }{N}_{\mu }^{+}{G}_{\nu }\left[{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\right]\right],\hfill \\ \hfill {\varphi }_4& =& {\displaystyle \frac{3e^{-\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{-\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}+{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}.\hfill \end{array}$$

The approximate solution of (30) is given by

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& {\omega }_0+{\omega }_1+{\omega }_2+\cdots ,\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& {\varphi }_0+{\varphi }_1+{\varphi }_2+\cdots ,\hfill \end{array}$$

hence, we obtain

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& e^{\mu }-{\displaystyle \frac{e^{\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}+{\displaystyle \frac{e^{\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}-{\displaystyle \frac{e^{\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\hfill \\ & & +{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\hfill \\ & & +{\displaystyle \frac{3e^{\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\hfill \\ & & -{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}+{\displaystyle \frac{2{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}+\cdots ,\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& e^{-\mu }+{\displaystyle \frac{e^{-\mu }{\nu }^{\delta }}{\mathsf{\Gamma }\left(\delta +1\right)}}+{\displaystyle \frac{e^{-\mu }{\nu }^{2\delta }}{\mathsf{\Gamma }\left(2\delta +1\right)}}\hfill \\ & & +{\displaystyle \frac{e^{-\mu }{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}+{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{3\delta }}{\mathsf{\Gamma }\left(3\delta +1\right)}}\hfill \\ & & +{\displaystyle \frac{3e^{-\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}-{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right)e^{-\mu }{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}\hfill \\ & & +{\displaystyle \frac{\mathsf{\Gamma }\left(2\delta +1\right){\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right){\left(\mathsf{\Gamma }\left(\delta +1\right)\right)}^2}}-{\displaystyle \frac{2{\nu }^{4\delta }}{\mathsf{\Gamma }\left(4\delta +1\right)}}+\cdots \hfill \end{array}$$

By setting $\delta =1$, the exact solution of (30) is given by

$$\begin{array}{ccc}\hfill \omega \left(\mu ,\nu \right)& =& \left(1-\nu +{\displaystyle \frac{{\nu }^2}{2!}}-{\displaystyle \frac{{\nu }^3}{3!}}+\cdots \right)e^{\mu }=e^{\mu -\nu },\hfill \\ \hfill \varphi \left(\mu ,\nu \right)& =& \left(1+\nu +{\displaystyle \frac{{\nu }^2}{2!}}+{\displaystyle \frac{{\nu }^3}{3!}}+\cdots \right)e^{-\mu }=e^{-\mu +\nu }.\hfill \end{array}$$

The results we have obtained are consistent with those in [20]. Figure 2 shows the 3D plots of the approximate solutions of $\omega (\mu ,\nu )$ and $\varphi (\mu ,\nu )$ corresponding to different choices of δ.

Example 3.

Consider the nonlinear time-fractional partial differential equation with the initial conditions given by the following:

$$\begin{array}{ccc}\hfill D_{\nu }^{\beta }\psi -{\omega }_{\chi }{\varphi }_{\nu }-{\displaystyle \frac{1}{2}}{\omega }_{\nu }{\psi }_{\chi \chi }& =& -4\chi \nu ,\hfill \\ \hfill D_{\nu }^{\beta }\varphi -{\omega }_{\nu }{\psi }_{\chi \chi }& =& 6\nu ,\hfill \\ \hfill D_{\nu }^{\beta }\omega -{\psi }_{\chi \chi }-{\varphi }_{\chi }{\omega }_{\nu }& =& 4\chi \nu -2\nu -2,0<\beta \le 1\hfill \end{array}$$

(40)

subject to the initial conditions:

$$\psi \left(\chi ,0\right)={\chi }^2+1,\varphi \left(\chi ,0\right)={\chi }^2-1,\omega \left(\chi ,0\right)={\chi }^2-1.$$

(41)

Applying the Natural Generalized Laplace Transform (NGLT) to Equation (40), we obtain the following:

$$\begin{array}{ccc}\hfill N_{\chi }^{+}{G}_{\nu }\left[D_{\nu }^{\beta }\psi \right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\chi }{\varphi }_{\nu }+{\displaystyle \frac{1}{2}}{\omega }_{\nu }{\psi }_{\chi \chi }-4\chi \nu \right],\hfill \\ \hfill N_{\chi }^{+}{G}_{\nu }\left[D_{\nu }^{\beta }\varphi \right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\nu }{\psi }_{\chi \chi }+6\nu \right],\hfill \\ \hfill N_{\chi }^{+}{G}_{\nu }\left[D_{\nu }^{\beta }\omega \right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\psi }_{\chi \chi }+{\varphi }_{\chi }{\omega }_{\nu }+4\chi \nu -2\nu -2\right].\hfill \end{array}$$

(42)

Using Theorem 1, the left-hand side of Equation (42) becomes the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\Psi }\left(p;u,s\right)}{s^{\beta }}}-s^{\alpha -\beta +1}{N}_{\chi }^{+}\left[\psi \left(\chi ,0\right)\right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\chi }{\varphi }_{\nu }+{\displaystyle \frac{1}{2}}{\omega }_{\nu }{\psi }_{\chi \chi }-4\chi \nu \right],\hfill \\ \hfill {\displaystyle \frac{\mathsf{\Phi }\left(p;u,s\right)}{s^{\beta }}}-s^{\alpha -\beta +1}{N}_{\chi }^{+}\left[\varphi \left(\chi ,0\right)\right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\nu }{\psi }_{\chi \chi }+6\nu \right],\hfill \\ \hfill {\displaystyle \frac{W\left(p;u,s\right)}{s^{\beta }}}-s^{\alpha -\beta +1}{N}_{\chi }^{+}\left[\omega \left(\chi ,0\right)\right]& =& N_{\chi }^{+}{G}_{\nu }\left[{\psi }_{\chi \chi }+{\varphi }_{\chi }{\omega }_{\nu }+4\chi \nu -2\nu -2\right].\hfill \end{array}$$

(43)

By applying the Generalized Laplace Transform (GLT) to Equation (41) and arranging the resulting expressions, we obtain the following:

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(p;u,s\right)& =& \left[{\displaystyle \frac{2u^2}{p^3}}+{\displaystyle \frac{1}{p}}\right]s^{\alpha +1}\hfill \\ & & +s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\chi }{\varphi }_{\nu }+{\displaystyle \frac{1}{2}}{\omega }_{\nu }{\psi }_{\chi \chi }-4\chi \nu \right],\hfill \\ \hfill \mathsf{\Phi }\left(p;u,s\right)& =& \left[{\displaystyle \frac{2u^2}{p^3}}-{\displaystyle \frac{1}{p}}\right]s^{\alpha +1}\hfill \\ & & +s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\nu }{\psi }_{\chi \chi }+6\nu \right],\hfill \\ \hfill W\left(p;u,s\right)& =& \left[{\displaystyle \frac{2u^2}{p^3}}-{\displaystyle \frac{1}{p}}\right]s^{\alpha +1}\hfill \\ & & +s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\psi }_{\chi \chi }+{\varphi }_{\chi }{\omega }_{\nu }+4\chi \nu -2\nu -2\right].\hfill \end{array}$$

(44)

Next, by applying the inverse NGLT to Equation (44), we obtain the following:

$$\begin{array}{ccc}\hfill \psi \left(\chi ,\nu \right)& =& {\chi }^2+1\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\chi }{\varphi }_{\nu }+{\displaystyle \frac{1}{2}}{\omega }_{\nu }{\psi }_{\chi \chi }-4\chi \nu \right]\right],\hfill \\ \hfill \varphi \left(\chi ,\nu \right)& =& {\chi }^2-1\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\omega }_{\nu }{\psi }_{\chi \chi }+6\nu \right]\right],\hfill \\ \hfill \omega \left(\chi ,\nu \right)& =& {\chi }^2-1\hfill \\ & & +N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left[{\psi }_{\chi \chi }+{\varphi }_{\chi }{\omega }_{\nu }+4\chi \nu -2\nu -2\right]\right].\hfill \end{array}$$

(45)

The Natural Generalized Laplace Transform Decomposition Method (NGLTDM) assumes that the approximate solution of Equation (40) is given by the following:

$$\begin{array}{ccc}\hfill \psi \left(\chi ,\nu \right)& =& \sum _{n=0}^{\infty }{\psi }_n\left(\chi ,\nu \right),n=1,2,\dots ,\hfill \\ \hfill \varphi \left(\chi ,\nu \right)& =& \sum _{n=0}^{\infty }{\varphi }_n\left(\chi ,\nu \right),n=1,2,\dots ,\hfill \\ \hfill \omega \left(\chi ,\nu \right)& =& \sum _{n=0}^{\infty }{\omega }_n\left(\chi ,\nu \right),n=1,2,\dots .\hfill \end{array}$$

(46)

In addition, we assume that the nonlinear terms ${\varphi }_{\chi }{\omega }_{\nu }$, ${\varphi }_{\nu }{\omega }_{\chi }$, ${\psi }_{\chi }{\omega }_{\nu }$, ${\psi }_{\nu }{\omega }_{\chi }$, ${\psi }_{\chi }{\varphi }_{\nu }$, and ${\psi }_{\nu }{\varphi }_{\chi }$ are denoted as follows:

$${\omega }_{\chi }{\varphi }_{\nu }=\sum _{n=0}^{\infty }{A}_n,{\omega }_{\nu }{\psi }_{\chi \chi }=\sum _{n=0}^{\infty }{B}_n,{\varphi }_{\chi }{\omega }_{\nu }=\sum _{n=0}^{\infty }{C}_n,$$

(47)

where the terms $A_n$, $B_n$, and $C_n$ are defined by the following:

$$\begin{array}{ccc}\hfill A_0& =& {\omega }_{0\chi }{\varphi }_{0\nu },\hfill \\ \hfill A_1& =& {\omega }_{0\chi }{\varphi }_{1\nu }+{\omega }_{1\chi }{\varphi }_{0\nu },\hfill \\ \hfill A_2& =& {\omega }_{0\chi }{\varphi }_{2\nu }+{\omega }_{1\chi }{\varphi }_{1\nu }+{\omega }_{2\chi }{\varphi }_{0\nu },\hfill \\ \hfill A_3& =& {\omega }_{0\chi }{\varphi }_{3\nu }+{\omega }_{1\chi }{\varphi }_{2\nu }+{\omega }_{2\chi }{\varphi }_{1\nu }+{\omega }_{3\chi }{\varphi }_{0\nu },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_0& =& {\omega }_{0\nu }{\psi }_{0\chi \chi },\hfill \\ \hfill B_1& =& {\omega }_{0\nu }{\psi }_{1\chi \chi }+{\omega }_{1\nu }{\psi }_{0\chi \chi },\hfill \\ \hfill B_2& =& {\omega }_{0\nu }{\psi }_{2\chi \chi }+{\omega }_{1\nu }{\psi }_{1\chi \chi }+{\omega }_{2\nu }{\psi }_{0\chi \chi },\hfill \\ \hfill B_3& =& {\omega }_{0\nu }{\psi }_{3\chi \chi }+{\omega }_{1\nu }{\psi }_{2\chi \chi }+{\omega }_{2\nu }{\psi }_{1\chi \chi }+{\omega }_{3\nu }{\psi }_{0\chi \chi },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill C_0& =& {\varphi }_{0\chi }{\omega }_{0\nu },\hfill \\ \hfill C_1& =& {\varphi }_{0\chi }{\omega }_{1\nu }+{\varphi }_{1\chi }{\omega }_{0\nu },\hfill \\ \hfill C_2& =& {\varphi }_{0\chi }{\omega }_{2\nu }+{\varphi }_{1\chi }{\omega }_{1\nu }+{\varphi }_{2\chi }{\omega }_{0\nu },\hfill \\ \hfill C_3& =& {\varphi }_{0\chi }{\omega }_{3\nu }+{\varphi }_{1\chi }{\omega }_{2\nu }+{\varphi }_{2\chi }{\omega }_{1\nu }+{\varphi }_{3\chi }{\omega }_{0\nu }.\hfill \end{array}$$

By substituting Equations (46) and (47) into Equation (45), we obtain the following expression:

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\psi }_n(\chi ,\nu )& =& {\chi }^2+1+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(\sum _{n=0}^{\infty }\left(A_n+{\displaystyle \frac{1}{2}}{B}_n\right)-4\chi \nu \right)\right],\hfill \\ \hfill \sum _{n=0}^{\infty }{\varphi }_n(\chi ,\nu )& =& {\chi }^2-1+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(\sum _{n=0}^{\infty }{C}_n+6\nu \right)\right],\hfill \\ \hfill \sum _{n=0}^{\infty }{\omega }_n(\chi ,\nu )& =& {\chi }^2-1+N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(\sum _{n=0}^{\infty }\left({\psi }_{n\chi \chi }+B_n\right)+4\chi \nu -2\nu -2\right)\right].\hfill \end{array}$$

From Equation (48), we derive the following iterative scheme:

$${\psi }_0(\chi ,\nu )={\chi }^2+1,{\varphi }_0(\chi ,\nu )={\chi }^2-1,{\omega }_0(\chi ,\nu )={\chi }^2-1,$$

and for $n=0,1,2,\dots$, the recursive relations are given by the following:

$$\begin{array}{ccc}\hfill {\psi }_{n+1}(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(A_n+{\displaystyle \frac{1}{2}}{B}_n-4\chi \nu \right)\right],\hfill \\ \hfill {\varphi }_{n+1}(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(C_n+6\nu \right)\right],\hfill \\ \hfill {\omega }_{n+1}(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\psi }_{n\chi \chi }+B_n+4\chi \nu -2\nu -2\right)\right].\hfill \end{array}$$

(48)

Putting $n=0$ in Equation (38), we obtain

$$\begin{array}{ccc}\hfill {\psi }_1(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(A_0+{\displaystyle \frac{1}{2}}{B}_0-4\chi \nu \right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(-4\chi \nu \right)\right]\hfill \\ & =& -N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{4u{s}^{\alpha +\beta +2}}{p^2}}\right]\hfill \\ & =& -{\displaystyle \frac{4\chi {\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}.\hfill \end{array}$$

Similarly, for ${\varphi }_1(\chi ,\nu )$,

$$\begin{array}{ccc}\hfill {\varphi }_1(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(C_0+6\nu \right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(6\nu \right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{s^{\alpha +\beta +2}}{p}}\right]\hfill \\ & =& {\displaystyle \frac{6{\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}.\hfill \end{array}$$

For ${\omega }_1(\chi ,\nu )$,

$$\begin{array}{ccc}\hfill {\omega }_1(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\psi }_{0\chi \chi }+B_0+4\chi \nu -2\nu -2\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(4\chi \nu -2\nu \right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{4u{s}^{\alpha +\beta +2}}{p^2}}-{\displaystyle \frac{2s^{\alpha +\beta +2}}{p}}\right]\hfill \\ & =& {\displaystyle \frac{4\chi {\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}-{\displaystyle \frac{2{\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}.\hfill \end{array}$$

At $n=1,$ we have

$$\begin{array}{ccc}\hfill {\psi }_2(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(A_1+{\displaystyle \frac{1}{2}}{B}_1\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\displaystyle \frac{16\chi {\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}-{\displaystyle \frac{2{\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{16u{s}^{\alpha +2\beta +1}}{p^2}}-{\displaystyle \frac{2s^{\alpha +2\beta +1}}{p}}\right]\hfill \\ & =& {\displaystyle \frac{16\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{2{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\varphi }_2(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left(C_1\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\displaystyle \frac{8\chi {\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}-{\displaystyle \frac{4{\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{8u{s}^{\alpha +2\beta +1}}{p^2}}-{\displaystyle \frac{4s^{\alpha +2\beta +1}}{p}}\right]\hfill \\ & =& {\displaystyle \frac{8\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{4{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\omega }_2(\chi ,\nu )& =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\psi }_{1\chi \chi }+B_1\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[s^{\beta }{N}_{\chi }^{+}{G}_{\nu }\left({\displaystyle \frac{8{\chi }^2{\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}-{\displaystyle \frac{4\chi {\nu }^{\beta }}{\mathsf{\Gamma }(\beta +1)}}\right)\right]\hfill \\ & =& N_{p;u}^{-1}{G}_s^{-1}\left[{\displaystyle \frac{16u^2{s}^{\alpha +2\beta +1}}{p^3}}-{\displaystyle \frac{4u{s}^{\alpha +2\beta +1}}{p^2}}\right]\hfill \\ & =& {\displaystyle \frac{8{\chi }^2{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{4\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}.\hfill \end{array}$$

In the same way, at $n=2$, we have

$$\begin{array}{ccc}\hfill {\psi }_3& =& {\displaystyle \frac{24\mathsf{\Gamma }(2\beta +1){\nu }^{3\beta }}{\mathsf{\Gamma }(3\beta +1)\mathsf{\Gamma }(\beta +1)\mathsf{\Gamma }(\beta +2)}}+{\displaystyle \frac{24{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{12\chi {\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}},\hfill \\ \hfill {\varphi }_3& =& {\displaystyle \frac{16{\chi }^3{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{8{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}},\hfill \\ \hfill {\omega }_3& =& {\displaystyle \frac{16{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{8\chi {\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}.\hfill \end{array}$$

The series solution of Equation (40) is given by

$$\begin{array}{ccc}\hfill \psi (\chi ,\nu )& =& {\psi }_0+{\psi }_1+{\psi }_2+\cdots ,\hfill \\ \hfill \varphi (\chi ,\nu )& =& {\varphi }_0+{\varphi }_1+{\varphi }_2+\cdots ,\hfill \\ \hfill \omega (\chi ,\nu )& =& {\omega }_0+{\omega }_1+{\omega }_2+\cdots ,\hfill \end{array}$$

thus

$$\begin{array}{ccc}\hfill \psi (\chi ,\nu )& =& {\chi }^2+1-{\displaystyle \frac{4\chi {\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}+{\displaystyle \frac{16\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{2{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}\hfill \\ & & +{\displaystyle \frac{24\mathsf{\Gamma }(2\beta +1){\nu }^{3\beta }}{\mathsf{\Gamma }(3\beta +1)\mathsf{\Gamma }(\beta +1)\mathsf{\Gamma }(\beta +2)}}+{\displaystyle \frac{24{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{12\chi {\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}+\cdots ,\hfill \\ \hfill \varphi (\chi ,\nu )& =& {\chi }^2-1+{\displaystyle \frac{6{\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}+{\displaystyle \frac{8\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{4{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}\hfill \\ & & +{\displaystyle \frac{16{\chi }^3{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{8{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}+\cdots ,\hfill \\ \hfill \omega (\chi ,\nu )& =& {\chi }^2-1+{\displaystyle \frac{4\chi {\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}-{\displaystyle \frac{2{\nu }^{\beta +1}}{\mathsf{\Gamma }(\beta +2)}}+{\displaystyle \frac{8{\chi }^2{\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}-{\displaystyle \frac{4\chi {\nu }^{2\beta }}{\mathsf{\Gamma }(2\beta +1)}}\hfill \\ & & +{\displaystyle \frac{16{\chi }^2{\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}-{\displaystyle \frac{8\chi {\nu }^{3\beta -1}}{\mathsf{\Gamma }\left(3\beta \right)}}+\cdots .\hfill \end{array}$$

By putting $\beta =1$, the exact solution of Equation (30) is given by the following:

$$\begin{array}{ccc}\hfill \psi (\chi ,\nu )& =& {\chi }^2-{\nu }^2+1,\hfill \\ \hfill \varphi (\chi ,\nu )& =& {\chi }^2+{\nu }^2-1,\hfill \\ \hfill \omega (\chi ,\nu )& =& {\chi }^2-{\nu }^2-1.\hfill \end{array}$$

6. Conclusions

This study introduces a combined approach—the Mixed Decomposition Method coupled with the Natural Generalized Laplace Transform method—forming a robust technique referred to as the NGLTDM. This method has been successfully applied to solve time-fractional systems of both linear and nonlinear partial differential equations. The NGLTDM is an analytical method that requires only initial conditions and is applicable to equations involving fractional as well as integer order derivatives in time. A significant advantage of this approach is its fast convergence and computational efficiency. The effectiveness and applicability of the method are demonstrated through three illustrative problems. Additionally, the convergence properties of the technique have been thoroughly discussed.