Yang-Laplace Decomposition Method for Nonlinear System of Local Fractional Partial Differential Equations

Fractional partial differential equations is a fundamental tool for the analysis of physical phenomena, such as, electromagnetic, acoustics, viscoelasticity, electrochemistry, and others. These physical and other phenomena are expressed by fractional partial differential equations, which have been solved by several numerical-analytical methods ([13], [21], [25]). Among them, one of the most popular methods is the so-called Adomian decomposition method (ADM), which has been developed between the 1970s and 1990s by George Adomian ([1], [2], [3], [4], [5]).

With the new concepts of fractional derivative and fractional integral, as well as local fractional derivative and local fractional integral, researchers were able to use the ADM method to solve these new types of equations or systems which include, local fractional differential equations, local fractional partial differential equations and local fractional integro-differential equations ([6], [7], [8], [9], [10]).

Some more results were obtained by combining fractional differential operators with some known transforms, such as the Young-Laplace transform method and the Sumudu transform method, thus obtaining a solution of fractional equations or systems, even in the nonlinear case. Among these works we find the local fractional Laplace decomposition method [11] and the local fractional Sumudu decomposition method ([12], [14]). The theory and applications of local fractional derivative and integral operators, has been defined and deeply investigated by Xiao-Jun Yang (see e.g.[15], [16]).

Based on the concept of local fractional operator and on the Yang-Laplace decomposition method, Jassim has recently proposed a method [11] to solve linear local fractional partial differential equations. In this paper, we will generalize the Jassim method and extend it to solve nonlinear systems of local fractional partial differential equations. The Yang-Laplace decomposition will enable us to obtain exact solutions for nonlinear system of local fractional partial differential equations. Two examples are also given to show the effectiveness of this method.

This paper has been organized as follows: In Section 2 some basic definitions and properties of local fractional calculus and local fractional Yang-Laplace transform method are given. In section 3, we present an analysis of the modified method. In section 4 we apply the modified method (LFLDM) to nonlinear systems to obtain the analytical solution. Two applications are given in the same section.

Basic definitions

In this section, we will present the basic concepts of fractional local calculus, and in particular the local fractional derivative, local fractional integral, and local fractional Yang-Laplace transform.

2.1

Local fractional derivative

Definition 2.1

The local fractional derivative of Φ(ϰ) of order σ at ϰ = ϰ₀ is defined as ([15], [16])

$\begin{matrix} Φ^{(σ)} (ϰ) = {\frac{d^{σ} Φ}{d ϰ^{σ}}|}_{ϰ = ϰ_{0}} = lim_{ϰ \to ϰ_{0}} \frac{Δ^{σ} (Φ (ϰ) - Φ (ϰ_{0}))}{(ϰ - ϰ_{0})^{σ}}, \end{matrix}$ $$\begin{array}{} \displaystyle \Phi ^{(\sigma )}(\varkappa )=\left. \frac{d^{\sigma }\Phi }{d\varkappa^{\sigma }}\right\vert _{\varkappa =\varkappa _{0}}=\lim_{\varkappa\rightarrow \varkappa_{0} }\frac{\Delta ^{\sigma}(\Phi (\varkappa )-\Phi (\varkappa _{0}))}{(\varkappa -\varkappa_{0})^{\sigma }}, \end{array}$$(1)

where

$\begin{matrix} Δ^{σ} (Φ (ϰ) - Φ (ϰ_{0})) ≅ Γ (1 + σ) [(Φ (ϰ) - Φ (ϰ_{0}))] . \end{matrix}$ $$\begin{array}{} \displaystyle \Delta ^{\sigma }(\Phi (\varkappa )-\Phi (\varkappa _{0}))\cong \Gamma(1+\sigma )\left[ (\Phi (\varkappa )-\Phi (\varkappa _{0}))\right] . \end{array}$$(2)

This operator can be extended to the interval (α, β) so that for any ϰ ∈ (α, β), we can define

$\begin{matrix} Φ^{(σ)} (ϰ) = D_{ϰ}^{σ} Φ (ϰ), \end{matrix}$ $$\begin{array}{} \displaystyle \Phi ^{(\sigma )}(\varkappa )=D_{\varkappa }^{\sigma }\Phi (\varkappa ), \end{array}$$

denoted by

$\begin{matrix} Φ (ϰ) \in D_{ϰ}^{σ} (α, β) . \end{matrix}$ $$\begin{array}{} \displaystyle \Phi (\varkappa )\in D_{\varkappa }^{\sigma }(\alpha ,\beta ). \end{array}$$

For the local fractional derivative of high order

$\begin{matrix} Φ^{(m σ)} (ϰ) = \overset{m t i m e s}{\overset{⏞}{D_{ϰ}^{(σ)} \dots D_{ϰ}^{(σ)} Φ (ϰ)},} \end{matrix}$ $$\begin{array}{} \displaystyle \Phi ^{(m\sigma )}(\varkappa )=\overset{m\text{ }times}{\overbrace{D_{\varkappa }^{(\sigma )}\cdots D_{\varkappa }^{(\sigma )}\Phi (\varkappa )},} \end{array}$$(3)

and local fractional partial derivative of high order

$\begin{matrix} \frac{\partial^{m σ} Φ (ϰ, τ)}{\partial ϰ^{m σ}} = \overset{m t i m e s}{\overset{⏞}{\frac{\partial^{σ}}{\partial ϰ^{σ}} \dots \frac{\partial^{σ}}{\partial ϰ^{σ}} Φ (ϰ, τ)} .} \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial ^{m\sigma }\Phi (\varkappa , \tau)}{\partial \varkappa ^{m\sigma }}=\overset{m\text{ }times}{\overbrace{\frac{\partial ^{\sigma }}{\partial \varkappa ^{\sigma }}\cdots \frac{\partial ^{\sigma }}{\partial \varkappa^{\sigma }}\Phi (\varkappa , \tau)}.} \end{array}$$(4)

2.2

Local fractional integral

Definition 2.2

The local fractional integral of Φ(ϰ) of order σ in the interval [α, β] is defined as ([15], [16])

$\begin{matrix} \begin{aligned} _{α} I_{β}^{(σ)} Φ (ϰ) & = & \frac{1}{Γ (1 + σ)} \int_{α}^{β} Φ (τ) (d τ)^{σ} \\ = & \frac{1}{Γ (1 + σ)} lim_{Δ τ ⟶ 0} \sum_{j = 0}^{N - 1} f (τ_{j}) (Δ τ_{j})^{σ}, \end{aligned} \end{matrix}$ $$\begin{array}{} \begin{split} \displaystyle _{\alpha }I_{\beta }^{(\sigma )}\Phi (\varkappa ) &=&\frac{1}{\Gamma(1+\sigma )}\int_{\alpha }^{\beta }\Phi (\tau )(d\tau )^{\sigma } \notag \\ &=&\frac{1}{\Gamma (1+\sigma )}\lim_{\Delta \tau \longrightarrow 0}\sum_{j=0}^{N-1}f(\tau _{j})(\Delta \tau _{j})^{\sigma }, \end{split} \end{array}$$(5)

where Δτ_j = τ_j+1 – τ_j, Δτ = max{Δτ₀, Δτ₁, Δτ₂, ⋯} and [τ_j, τ_j+1], τ₀ = α, τ_N = β, is a partition of the interval [α, β].

For any ϰ ∈ (α, β), there exists

$\begin{matrix} _{α} I_{ϰ}^{(σ)} Φ (ϰ), \end{matrix}$ $$\begin{array}{} \displaystyle _{\alpha }I_{\varkappa }^{(\sigma )}\Phi (\varkappa ), \end{array}$$

denoted by

$\begin{matrix} Φ (ϰ) \in I_{ϰ}^{(σ)} (α, β) . \end{matrix}$ $$\begin{array}{} \displaystyle \Phi (\varkappa )\in I_{\varkappa }^{(\sigma )}(\alpha ,\beta ). \end{array}$$

2.3

Some properties of the local fractional operators

The local fractional operators fulfill some fundamental equations. In particular, starting from the Mittag-Leffler function we have the following

Definition 2.3

The Mittag-Leffler function, the hyperbolic sine and hyperbolic cosine are defined as ([15], [16], [17])

$\begin{matrix} E_{σ} (ϰ^{σ}) = \sum_{m = 0}^{+ \infty} \frac{ϰ^{m σ}}{Γ (1 + m σ)}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle E_{\sigma }(\varkappa ^{\sigma })=\sum_{m=0}^{+\infty }\frac{\varkappa ^{m\sigma }}{\Gamma (1+m\sigma )},{ \, \, }0 \lt \sigma \leqslant 1, \end{array}$$(6)

$\begin{matrix} E_{σ} (ϰ^{σ}) E_{σ} (υ^{σ}) = E_{σ} (ϰ + υ)^{σ}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle E_{\sigma }(\varkappa ^{\sigma })E_{\sigma }(\upsilon ^{\sigma })=E_{\sigma}(\varkappa +\upsilon )^{\sigma },{ \, \, }0 \lt \sigma \leqslant 1, \end{array}$$(7)

$\begin{matrix} E_{σ} (ϰ^{σ}) E_{σ} (- υ^{σ}) = E_{σ} (ϰ - υ)^{σ}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle E_{\sigma }(\varkappa ^{\sigma })E_{\sigma }(-\upsilon ^{\sigma })=E_{\sigma}(\varkappa -\upsilon )^{\sigma },{ \, \, }0 \lt \sigma \leqslant 1, \end{array}$$(8)

$\begin{matrix} \sin_{σ} (ϰ^{σ}) = \sum_{m = 0}^{+ \infty} (- 1)^{m} \frac{ϰ^{(2 m + 1) σ}}{Γ (1 + (2 m + 1) σ)}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle \sin _{\sigma }(\varkappa ^{\sigma })=\sum_{m=0}^{+\infty }(-1)^{m}\frac{\varkappa ^{(2m+1)\sigma }}{\Gamma (1+(2m+1)\sigma )},{ \, \, }0 \lt \sigma \leqslant 1, \end{array}$$(9)

$\begin{matrix} \cos_{σ} (ϰ^{σ}) = \sum_{m = 0}^{+ \infty} (- 1)^{m} \frac{ϰ^{2 m σ}}{Γ (1 + 2 m σ)}, 0 < σ ⩽ 1. \end{matrix}$ $$\begin{array}{} \displaystyle \cos _{\sigma }(\varkappa ^{\sigma })=\sum_{m=0}^{+\infty }(-1)^{m}\frac{\varkappa ^{2m\sigma }}{\Gamma (1+2m\sigma )},{ \, \, }0 \lt \sigma \leqslant 1. \end{array}$$(10)

By using the local fractional derivative (1) and the definitions (2.3) it can be easily shown that ([15], [16])

$\begin{matrix} \frac{d^{σ} ϰ^{m σ}}{d ϰ^{σ}} = \frac{Γ (1 + m σ) ϰ^{(m - 1) σ}}{Γ (1 + (m - 1) σ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d^{\sigma }\varkappa ^{m\sigma }}{d\varkappa ^{\sigma }}=\frac{\Gamma (1+m\sigma )\varkappa ^{(m-1)\sigma }}{\Gamma (1+(m-1)\sigma )}. \end{array}$$(11)

$\begin{matrix} \frac{d^{σ}}{d ϰ^{σ}} E_{σ} (ϰ^{σ}) = E_{σ} (ϰ^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d^{\sigma }}{d\varkappa ^{\sigma }}E_{\sigma }(\varkappa ^{\sigma})=E_{\sigma }(\varkappa ^{\sigma }). \end{array}$$(12)

$\begin{matrix} \frac{d^{σ}}{d ϰ^{σ}} \sin_{σ} (ϰ^{σ}) = \cos_{σ} (ϰ^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d^{\sigma }}{d\varkappa ^{\sigma }}\sin _{\sigma }(\varkappa ^{\sigma})=\cos _{\sigma }(\varkappa ^{\sigma }). \end{array}$$(13)

$\begin{matrix} \frac{d^{σ}}{d ϰ^{σ}} \cos_{σ} (ϰ^{σ}) = - \sin_{σ} (ϰ^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d^{\sigma }}{d\varkappa ^{\sigma }}\cos _{\sigma }(\varkappa ^{\sigma})=-\sin _{\sigma }(\varkappa ^{\sigma }). \end{array}$$(14)

$\begin{matrix} _{0} I_{ϰ}^{(σ)} \frac{ϰ^{m σ}}{Γ (1 + m σ)} = \frac{ϰ^{(m + 1) σ}}{Γ (1 + (m + 1) σ)} . \end{matrix}$ $$\begin{array}{} \displaystyle _{0}I_{\varkappa }^{(\sigma )}\frac{\varkappa ^{m\sigma }}{\Gamma (1+m\sigma)}=\frac{\varkappa ^{(m+1)\sigma }}{\Gamma (1+(m+1)\sigma )}. \end{array}$$(15)

2.4

Local fractional Yang-Laplace transform

We present here the definition of local fractional Yang-Laplace transform (denoted in this paper by ^LFL_σ[.]) and some properties concerning this transformation.

Definition 2.4

Let

$\begin{matrix} \frac{1}{Γ (1 + σ)} \int_{0}^{\infty} |w (ϰ, τ)| {(d τ)}^{σ} < k < \infty, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{\Gamma (1+\sigma )}\int_{0}^{\infty }\left\vert w(\varkappa ,\tau )\right\vert \left( d\tau \right) ^{\sigma } \lt k \lt \infty \qquad , \end{array}$$

the Yang-Laplace transform of w(ϰ, τ) is defined as ([18], [19], [20]):

$\begin{matrix} ^{L F} L_{σ} \{w (ϰ, τ)\} = W (ϰ, s) := \frac{1}{Γ (1 + σ)} \int_{0}^{\infty} E_{σ} (- s^{σ} τ^{σ}) w (ϰ, τ) (d τ)^{σ}; 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ w(\varkappa ,\tau )\right\} =\ W(\varkappa ,s):=\frac{1}{\Gamma (1+\sigma )}\int_{0}^{\infty }E_{\sigma }(-s^{\sigma }\tau^{\sigma })w(\varkappa ,\tau )(d\tau )^{\sigma };\ 0 \lt \sigma \leqslant 1, \end{array}$$(16)

where the integral converges and s ∈ ℝ.

Definition 2.5

The inverse Yang-Laplace transforms of W(ϰ, τ) is defined as [19]:

$\begin{matrix} ^{L F} L_{σ}^{- 1} \{W (ϰ, s)\} = w (ϰ, τ) := \frac{1}{(2 π)^{σ}} \int_{β - i \infty}^{β + i \infty} E_{σ} (s^{σ} τ^{σ}) W (x, s) (d s)^{σ}; 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }^{-1}\left\{ W(\varkappa ,s)\right\} =\, w(\varkappa ,\tau ):=\frac{1}{(2\pi )^{\sigma }}\int_{\beta -i\infty }^{\beta +i\infty}E_{\sigma }(s^{\sigma }\tau ^{\sigma })W(x,s)(ds)^{\sigma };\, 0 \lt \sigma\leqslant 1, \end{array}$$(17)

where s^σ = β^σ + i^σ ∞^σ, (Re(s) = β > 0) and i^σ is the fractal imaginary unit.

The Yang-Laplace transform is a linear operator ([18], [20])

$\begin{matrix} ^{L F} L_{σ} \{a f (ϰ) + b g (ϰ)\} = a^{L F} L_{σ} (f (ϰ)) + b^{L F} L_{σ} (g (ϰ)), \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ af(\varkappa )+bg(\varkappa )\right\} =a^{LF}L_{\sigma}(f(\varkappa ))+b^{LF}L_{\sigma }(g(\varkappa )), \end{array}$$(18)

moreover the following properties hold ([18], [20])

$\begin{matrix} ^{L F} L_{σ} \{f^{(n σ)} (ϰ)\} = s^{n σ}^{L F} L_{σ} \{f (ϰ)\} - \sum_{k = 1}^{n} s^{(k - 1) σ} f (0) - f^{((n - k) σ)} (0), \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ f^{\left( n\sigma \right) }(\varkappa )\right\}=s^{n\sigma }{\, }^{LF}L_{\sigma }\left\{ f(\varkappa )\right\}-\sum_{k=1}^{n}s^{(k-1)\sigma }f(0)-f^{((n-k)\sigma )}(0), \end{array}$$(19)

$\begin{matrix} lim_{x ⟶ 0} f (ϰ) = lim_{σ ⟶ \infty} s^{σ} F (s), \end{matrix}$ $$\begin{array}{} \displaystyle \lim_{x\longrightarrow 0}f(\varkappa )=\lim_{\sigma \longrightarrow \infty}s^{\sigma }F(s), \end{array}$$(20)

$\begin{matrix} lim_{x ⟶ \infty} f (ϰ) = lim_{σ ⟶ 0} s^{σ} F (s), \end{matrix}$ $$\begin{array}{} \displaystyle \lim_{x\longrightarrow \infty }f(\varkappa )=\lim_{\sigma \longrightarrow0}s^{\sigma }F(s), \end{array}$$(21)

$\begin{matrix} ^{L F} L_{σ} \{ϰ^{k σ} E_{σ} (a^{σ} ϰ^{σ})\} = (\frac{Γ (1 + k σ)}{(s - a)^{(k + 1) σ}}, \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ \varkappa ^{k\sigma }E_{\sigma }(a^{\sigma }\varkappa^{\sigma })\right\} =(\frac{\Gamma (1+k\sigma )}{(s-a)^{(k+1)\sigma }}, \end{array}$$(22)

$\begin{matrix} ^{L F} L_{σ} \{\sin_{σ} (a^{σ} ϰ^{σ})\} = \frac{a^{σ}}{s^{2 σ} + a^{2 σ}}, \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ \sin _{\sigma }(a^{\sigma }\varkappa ^{\sigma})\right\} =\, \frac{a^{\sigma }}{s^{2\sigma }+a^{2\sigma }}, \end{array}$$(23)

$\begin{matrix} ^{L F} L_{σ} \{\cos_{σ} (a^{σ} ϰ^{σ})\} = \frac{s^{σ}}{s^{2 σ} + a^{2 σ}}, \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ \cos _{\sigma }(a^{\sigma }\varkappa ^{\sigma})\right\} =\ \frac{s^{\sigma }}{s^{2\sigma }+a^{2\sigma }}, \end{array}$$(24)

$\begin{matrix} ^{L F} L_{σ} \{ϰ^{k σ}\} = \frac{Γ (1 + k σ)}{s^{(k + 1) σ}} . \end{matrix}$ $$\begin{array}{} \displaystyle ^{LF}L_{\sigma }\left\{ \varkappa ^{k\sigma }\right\} =\, \frac{\Gamma(1+k\sigma )}{s^{(k+1)\sigma }}. \end{array}$$(25)

Solution of a nonlinear fractional order differential system

Let us consider a general nonlinear system with local fractional derivative:

$\begin{matrix} \{\begin{matrix} \frac{\partial^{σ} X}{\partial τ^{σ}} + \frac{\partial^{σ} T}{\partial ϰ^{σ}} + N_{σ, 1} (X, T) + R_{σ, 1} (X, T) = φ (ϰ, τ), \\ \frac{\partial^{σ} T}{\partial τ^{σ}} + \frac{\partial^{σ} X}{\partial ϰ^{σ}} + N_{σ, 2} (X, T) + R_{σ, 2} (X, T) = ψ (ϰ, τ), \end{matrix} \end{matrix}$ $$\begin{array}{} \ \left\{ \begin{array}{c} \frac{\partial ^{\sigma }X}{\partial \tau ^{\sigma }}+\frac{\partial^{\sigma }T}{\partial \varkappa ^{\sigma }}+N_{\sigma ,1}(X,T)+R_{\sigma,1}(X,T)=\varphi (\varkappa ,\tau ), \\[0.1cm] \frac{\partial ^{\sigma }T}{\partial \tau ^{\sigma }}+\frac{\partial^{\sigma }X}{\partial \varkappa ^{\sigma }}+N_{\sigma ,2}(X,T)+R_{\sigma,2}(X,T)=\psi (\varkappa ,\tau ), \end{array} \right. \end{array}$$(26)

where $\begin{matrix} \frac{\partial^{σ}}{\partial {(\cdot)}^{σ}} \end{matrix}$ $\begin{array}{} \displaystyle \frac{\partial ^{\sigma }}{\partial \left( \cdot \right) ^{\sigma }} \end{array}$ denotes linear local fractional derivative operator of order σ, R_σ,1, R_σ,2 are the linear local fractional operators, N_σ,1, N_σ,2 represent the nonlinear local fractional operators, and φ(ϰ, τ), ψ(ϰ, τ) are two given functions.

We will search an analytical solution of this system by the following steps.

Step 1

First we apply the local Yang-Laplace transform to both sides of each equation in system (26), so that:

$\begin{matrix} \{\begin{matrix} ^{L F} L_{σ} [\frac{\partial^{σ} X}{\partial τ^{σ}}] +^{L F} L_{σ} [\frac{\partial^{σ} T}{\partial ϰ^{σ}}] +^{L F} L_{σ} [N_{σ, 1} (X, T)] +^{L F} L_{σ} [R_{σ, 1} (X, T)] =^{L F} L_{σ} [φ (ϰ, τ)] \\ ^{L F} L_{σ} [\frac{\partial^{σ} T}{\partial τ^{σ}}] +^{L F} L_{σ} [\frac{\partial^{σ} X}{\partial ϰ^{σ}}] +^{L F} L_{σ} [N_{σ, 2} (X, T)] +^{L F} L_{σ} [R_{σ, 2} (X, T)] =^{L F} L_{σ} [ψ (ϰ, τ)] \end{matrix} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X}{\partial \tau ^{\sigma }}\right] +\text{ }^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }T}{\partial \varkappa ^{\sigma }}\right] +^{LF}L_{\sigma }\left[ N_{\sigma ,1}(X,T)\right] +^{LF}L_{\sigma }\left[ R_{\sigma ,1}(X,T)\right] =\text{ }^{LF}L_{\sigma }\left[ \varphi (\varkappa ,\tau )\right] \\[0.1cm] ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }T}{\partial \tau ^{\sigma }}\right] +\text{ }^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X}{\partial \varkappa ^{\sigma }}\right] +^{LF}L_{\sigma }\left[ N_{\sigma ,2}(X,T)\right] +^{LF}L_{\sigma }\left[ R_{\sigma ,2}(X,T)\right] =\text{ }^{LF}L_{\sigma }\left[ \psi (\varkappa ,\tau )\right] \end{array} \right. . \end{array}$$(27)

According to the properties of this transform, we have:

$\begin{matrix} \{\begin{matrix} ^{L F} L_{σ} [X] = X (ϰ, 0) + s^{- σ} (^{L F} L_{σ} [φ (ϰ, τ)]) - s^{- σ} (L F L_{σ} [\frac{\partial^{σ} T}{\partial ϰ^{σ}} + N_{σ, 1} (X, T) + R_{σ, 1} (X, T)]) \\ ^{L F} L_{σ} [T] = T (ϰ, 0) + s^{- σ} (^{L F} L_{σ} [ψ (ϰ, τ)]) - s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} X}{\partial ϰ^{σ}} + N_{σ, 2} (X, T) + R_{σ, 2} (X, T)]) \end{matrix} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} ^{LF}L_{\sigma }\left[ X\right] =X(\varkappa ,0)+s^{-\sigma }\left( ^{LF}L_{\sigma}\left[ \varphi (\varkappa ,\tau )\right] \right) -s^{-\sigma }\left(LFL_{\sigma }\left[ \frac{\partial ^{\sigma }T}{\partial \varkappa ^{\sigma }}+N_{\sigma ,1}(X,T)+R_{\sigma ,1}(X,T)\right] \right) \\[0.1cm] ^{LF}L_{\sigma }\left[ T\right] =T(\varkappa ,0)+s^{-\sigma }\left( ^{LF}L_{\sigma}\left[ \psi (\varkappa ,\tau )\right] \right) -s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X}{\partial \varkappa ^{\sigma }}+N_{\sigma ,2}(X,T)+R_{\sigma ,2}(X,T)\right] \right) \end{array} \right. . \end{array}$$(28)

By taking the inverse transformation on both sides of each system equation (28), there follows:

$\begin{matrix} \{\begin{matrix} X = X (ϰ, 0) +^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [φ (ϰ, τ)])) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} T}{\partial ϰ^{σ}} + N_{σ, 1} (X, T) + R_{σ, 1} (X, T)])) \\ T = T (ϰ, 0) +^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [ψ (ϰ, τ)])) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} X}{\partial ϰ^{σ}} + N_{σ, 2} (X, T) + R_{σ, 2} (X, T)])) \end{matrix} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X=X(\varkappa ,0)+^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma } \left[ \varphi (\varkappa ,\tau )\right] \right) \right) -^{LF}L_{\sigma}^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma}T}{\partial \varkappa ^{\sigma }}+N_{\sigma ,1}(X,T)+R_{\sigma ,1}(X,T)\right] \right) \right) \\[0.1cm] T=T(\varkappa ,0)+^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma } \left[ \psi (\varkappa ,\tau )\right] \right) \right) -^{LF}L_{\sigma}^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma}X}{\partial \varkappa ^{\sigma }}+N_{\sigma ,2}(X,T)+R_{\sigma ,2}(X,T)\right] \right) \right) \end{array} \right. . \end{array}$$(29)

Step 2

By using the Adomian decomposition method [1], we represent the two unknown functions X and T as infinite series:

$\begin{matrix} X (ϰ, τ) = \sum_{n = 0}^{\infty} X_{n} (ϰ, τ), \\ T (ϰ, τ) = \sum_{n = 0}^{\infty} T_{n} (ϰ, τ) . \end{matrix}$ $$\begin{array}{} X(\varkappa ,\tau )=\sum_{n=0}^{\infty }X_{n}(\varkappa ,\tau ), \\[0.2cm] T(\varkappa ,\tau )=\sum_{n=0}^{\infty }T_{n}(\varkappa ,\tau ). \end{array}$$(30)

moreover, the nonlinear terms can be decomposed as:

$\begin{matrix} N_{σ, 1} (X, T) = \sum_{n = 0}^{\infty} A_{n}, \\ N_{σ, 2} (X, T) = \sum_{n = 0}^{\infty} B_{n}, \end{matrix}$ $$\begin{array}{} N_{\sigma ,1}(X,T)=\sum_{n=0}^{\infty }A_{n}, \\[0.1cm] N_{\sigma ,2}(X,T)=\sum_{n=0}^{\infty }B_{n}, \end{array}$$(31)

where A_n and B_n are Adomian polynomials [22].

Substituting (30) and (31) in (29), we get:

$\begin{matrix} \{\begin{matrix} \sum_{n = 0}^{\infty} X_{n} (ϰ, τ) = X (ϰ, 0) +^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [φ (ϰ, τ)])) \\ -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ}}{\partial ϰ^{σ}} (\sum_{n = 0}^{\infty} T_{n}) + \sum_{n = 0}^{\infty} A_{n} + R_{1, σ} (\sum_{n = 0}^{\infty} X_{n}, \sum_{n = 0}^{\infty} T_{n})])), \\ \sum_{n = 0}^{\infty} T_{n} (ϰ, τ) = T (ϰ, 0) +^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [ψ (ϰ, τ)])) \\ -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ}}{\partial ϰ^{σ}} (\sum_{n = 0}^{\infty} X_{n}) + \sum_{n = 0}^{\infty} B_{n} + R_{2, σ} (\sum_{n = 0}^{\infty} X_{n}, \sum_{n = 0}^{\infty} T_{n})])) . \end{matrix} \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} \sum_{n=0}^{\infty }X_{n}(\varkappa ,\tau )=X(\varkappa,0)+^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \varphi(\varkappa ,\tau )\right] \right) \right) \\[0.2cm] -^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }}{\partial \varkappa ^{\sigma }}\left(\sum_{n=0}^{\infty }T_{n}\right) +\sum_{n=0}^{\infty}A_{n}+R_{1,\sigma }\left( \sum_{n=0}^{\infty}X_{n},\sum_{n=0}^{\infty }T_{n}\right) \right] \right) \right) , \\[0.2cm] \sum_{n=0}^{\infty }T_{n}(\varkappa ,\tau )=T(\varkappa,0)+^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \psi(\varkappa ,\tau )\right] \right) \right) \\[0.2cm] -^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }}{\partial \varkappa ^{\sigma }}\left(\sum_{n=0}^{\infty }X_{n}\right) +\sum_{n=0}^{\infty}B_{n}+R_{2,\sigma }\left( \sum_{n=0}^{\infty}X_{n},\sum_{n=0}^{\infty }T_{n}\right) \right] \right) \right). \end{array} \right. \end{array}$$(32a)

By comparing both sides of (32a), we have:

$\begin{matrix} [\begin{matrix} X_{0} (ϰ, τ) = X (ϰ, 0) + L F L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [φ (ϰ, τ)])), \\ X_{1} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} T_{0}}{\partial ϰ^{σ}} + A_{0} + R_{1, σ} (X_{0}, T_{0})])), \\ X_{2} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} T_{1}}{\partial ϰ^{σ}} + A_{1} + R_{1, σ} (X_{1}, T_{1})])), \\ X_{3} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} T_{2}}{\partial ϰ^{σ}} + A_{2} + R_{1, σ} (X_{2}, T_{2})])), \\ ⋮ \end{matrix}] \end{matrix}$ $$\begin{array}{} \displaystyle \left[ \begin{array}{c} X_{0}(\varkappa ,\tau )=X(\varkappa ,0)+\text{ }LFL_{\sigma }^{-1}\left(s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \varphi (\varkappa ,\tau )\right]\right) \right) , \\[0.1cm] X_{1}(\varkappa ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }T_{0}}{\partial \varkappa^{\sigma }}+A_{0}+R_{1,\sigma }\left( X_{0},T_{0}\right) \right] \right)\right) , \\[0.1cm] X_{2}(\varkappa ,\tau )=-\text{ }^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }T_{1}}{\partial \varkappa^{\sigma }}+A_{1}+R_{1,\sigma }\left( X_{1},T_{1}\right) \right] \right)\right) , \\[0.1cm] X_{3}(\varkappa ,\tau )=-\text{ }^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }T_{2}}{\partial \varkappa^{\sigma }}+A_{2}+R_{1,\sigma }\left( X_{2},T_{2}\right) \right] \right)\right) , \\ \vdots \end{array} \right] \end{array}$$(33)

and

$\begin{matrix} [\begin{matrix} T_{0} (ϰ, τ) = T (ϰ, 0) +^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [ψ (ϰ, τ)])), \\ T_{1} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} X_{0}}{\partial ϰ^{σ}} + B_{0} + R_{2, σ} (X_{0}, T_{0})])), \\ T_{2} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} X_{1}}{\partial ϰ^{σ}} + B_{1} + R_{2, σ} (X_{1}, T_{1})])), \\ T_{3} (ϰ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\frac{\partial^{σ} X_{2}}{\partial ϰ^{σ}} + B_{2} + R_{2, σ} (X_{2}, T_{2})])), \\ ⋮ \end{matrix}] \end{matrix}$ $$\begin{array}{} \left[ \begin{array}{c} T_{0}(\varkappa ,\tau )=T(\varkappa ,0)+^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left(^{LF}L_{\sigma }\left[ \psi (\varkappa ,\tau )\right] \right) \right) , \\[0.1cm] T_{1}(\varkappa ,\tau )=-\text{ }^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X_{0}}{\partial \varkappa ^{\sigma }}+B_{0}+R_{2,\sigma }\left( X_{0},T_{0}\right) \right] \right)\right) , \\[0.1cm] T_{2}(\varkappa ,\tau )=-\text{ }^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X_{1}}{\partial \varkappa^{\sigma }}+B_{1}+R_{2,\sigma }\left( X_{1},T_{1}\right) \right] \right)\right) , \\[0.1cm] T_{3}(\varkappa ,\tau )=-\text{ }^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \frac{\partial ^{\sigma }X_{2}}{\partial \varkappa^{\sigma }}+B_{2}+R_{2,\sigma }\left( X_{2},T_{2}\right) \right] \right)\right) , \\ \vdots \end{array} \right] \end{array}$$(34)

Step 3

At the last step, we get the solution by a limit of the analytical solution (X, T) of the system (26):

$\begin{matrix} \{\begin{matrix} X (ϰ, τ) = lim_{N \to \infty} \sum_{n = 0}^{N} X_{n} (ϰ, τ) \\ T (ϰ, τ) = lim_{N \to \infty} \sum_{n = 0}^{N} T_{n} (ϰ, τ) \end{matrix} . \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} X(\varkappa ,\tau )=\lim_{N\rightarrow \infty}\sum_{n=0}^{N}X_{n}(\varkappa ,\tau ) \\[0.1cm] T(\varkappa ,\tau )=\lim_{N\rightarrow \infty}\sum_{n=0}^{N}T_{n}(\varkappa ,\tau ) \end{array} \right. . \end{array}$$(35)

Applications

In this section, we will implement the proposed method based on local fractional Yang-Laplace decomposition method (LFLDM) [11] and Adomian decomposition for solving two nonlinear systems of local fractional partial differential equations.

Example 4.1

Consider the following coupled nonlinear system of local fractional Burger equations

$\begin{matrix} \{\begin{matrix} \frac{\partial^{σ} X}{\partial τ^{σ}} - \frac{\partial^{2 σ} X}{\partial ϰ^{2 σ}} - 2 X X_{ϰ}^{(σ)} + {(X T)}_{ϰ}^{(σ)} = 0 \\ \frac{\partial^{σ} T}{\partial τ^{σ}} - \frac{\partial^{2 σ} T}{\partial ϰ^{2 σ}} - 2 T T_{ϰ}^{(σ)} + {(X T)}_{ϰ}^{(σ)} = 0 \end{matrix}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} \frac{\partial ^{\sigma }X}{\partial \tau ^{\sigma }}\ -\frac{\partial^{2\sigma }X}{\partial \varkappa ^{2\sigma }}-2XX_{\varkappa }^{\left(\sigma \right) }+\left( XT\right) _{\varkappa }^{\left( \sigma \right) }=0 \\[0.1cm] \frac{\partial ^{\sigma }T}{\partial \tau ^{\sigma }}\ -\frac{\partial^{2\sigma }T}{\partial \varkappa ^{2\sigma }}-2TT_{\varkappa }^{\left( \sigma \right) }+\left( XT\right) _{\varkappa }^{\left( \sigma \right) }=0 \end{array} \right. ,\text{ }0 \lt \sigma \leqslant 1, \end{array}$$(36)

under the initial conditions:

$\begin{matrix} X (ϰ, 0) = \sin_{σ} (ϰ^{σ}), T (ϰ, 0) = \sin_{σ} (ϰ^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle X(\varkappa ,0)=\sin _{\sigma }(\varkappa ^{\sigma }),\text{ }T(\varkappa,0)=\sin _{\sigma }(\varkappa ^{\sigma }). \end{array}$$(37)

From the equation (29), we obtain:

$\begin{matrix} \{\begin{matrix} X (ϰ, τ) = \sin_{σ} (ϰ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [- \frac{\partial^{2 σ} T}{\partial ϰ^{2 σ}} - 2 X X_{ϰ}^{(σ)} + {(X T)}_{ϰ}^{(σ)}])) \\ T (ϰ, τ) = \sin_{σ} (ϰ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [- \frac{\partial^{2 σ} X}{\partial ϰ^{2 σ}} - 2 T T_{ϰ}^{(σ)} + {(X T)}_{ϰ}^{(σ)}])) \end{matrix} . \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} X(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })-^{LF}L_{\sigma}^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ -\frac{\partial^{2\sigma }T}{\partial \varkappa ^{2\sigma }}-2XX_{\varkappa }^{\left(\sigma \right) }+\left( XT\right) _{\varkappa }^{\left( \sigma \right) }\right] \right) \right) \\[0.1cm] T(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })-^{LF}L_{\sigma}^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ -\frac{\partial^{2\sigma }X}{\partial \varkappa ^{2\sigma }}-2TT_{\varkappa }^{\left(\sigma \right) }+\left( XT\right) _{\varkappa }^{\left( \sigma \right) }\right] \right) \right) \end{array} \right. . \end{array}$$(38)

So that by using the Adomian decomposition [1], each function of the solution (X, T) can be decomposed as infinite series:

$\begin{matrix} X (ϰ, τ) = \sum_{n = 0}^{\infty} X_{n} (ϰ, τ), \\ T (ϰ, τ) = \sum_{n = 0}^{\infty} T_{n} (ϰ, τ), \end{matrix}$ $$\begin{array}{} \displaystyle X(\varkappa ,\tau )=\sum_{n=0}^{\infty }X_{n}(\varkappa ,\tau ), \\ \displaystyle T(\varkappa ,\tau )=\sum_{n=0}^{\infty }T_{n}(\varkappa ,\tau ), \end{array}$$(39)

and the nonlinear terms can be decomposed as:

$\begin{matrix} X X_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} A_{n} (X), \end{matrix}$ $$\begin{array}{} \displaystyle XX_{\varkappa }^{(\sigma )}=\sum_{n=0}^{\infty }A_{n}(X), \end{array}$$(40)

$\begin{matrix} (X T)_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} B_{n} (X, T), \end{matrix}$ $$\begin{array}{} \displaystyle (XT)_{\varkappa }^{\left( \sigma \right) }=\sum_{n=0}^{\infty}B_{n}(X,T), \end{array}$$(41)

and

$\begin{matrix} T T_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} C_{n} (X) . \end{matrix}$ $$\begin{array}{} \displaystyle TT_{\varkappa }^{(\sigma )}=\sum_{n=0}^{\infty }C_{n}(X). \end{array}$$(42)

Substituting (39), (40), (41) and (42) in (38), we get:

$\begin{matrix} \{\begin{matrix} \sum_{n = 0}^{\infty} X_{n} (ϰ, τ) = \sin_{σ} (ϰ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\begin{matrix} - \frac{\partial^{2 σ}}{\partial ϰ^{2 σ}} (\sum_{n = 0}^{\infty} X_{n} (ϰ, τ)) \\ - 2 \sum_{n = 0}^{\infty} A_{n} (X) + \sum_{n = 0}^{\infty} B_{n} (X, T) \end{matrix}])) \\ \sum_{n = 0}^{\infty} T_{n} (ϰ, τ) = \sin_{σ} (ϰ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\begin{matrix} - \frac{\partial^{2 σ}}{\partial ϰ^{2 σ}} (\sum_{n = 0}^{\infty} T_{n} (ϰ, τ)) \\ - 2 \sum_{n = 0}^{\infty} C_{n} (X) + \sum_{n = 0}^{\infty} B_{n} (X, T) \end{matrix}])) \end{matrix} . \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} \sum_{n=0}^{\infty }X_{n}(\varkappa ,\tau )=\sin _{\sigma}(\varkappa ^{\sigma })-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ \begin{array}{c} -\frac{\partial ^{2\sigma }}{\partial \varkappa ^{2\sigma }}\left(\sum_{n=0}^{\infty }X_{n}(\varkappa ,\tau )\right) \\[0.1cm] -2\sum_{n=0}^{\infty }A_{n}(X)+\sum_{n=0}^{\infty }B_{n}(X,T) \end{array} \right] \right) \right) \\[0.1cm] \sum_{n=0}^{\infty }T_{n}(\varkappa ,\tau )=\sin _{\sigma}(\varkappa ^{\sigma })-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left(^{LF}L_{\sigma }\left[ \begin{array}{c} -\frac{\partial ^{2\sigma }}{\partial \varkappa ^{2\sigma }}\left(\sum_{n=0}^{\infty }T_{n}(\varkappa ,\tau )\right) \\[0.0cm] -2\sum_{n=0}^{\infty }C_{n}(X)+\sum_{n=0}^{\infty }B_{n}(X,T) \end{array} \right] \right) \right) \end{array} \right. . \end{array}$$(43)

By comparing both sides of (43), it is:

$\begin{matrix} X_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}), \\ X_{1} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{0} (T, Z) - A_{0}^{'} (T, Z) + X_{0} (ϰ, υ, τ)])), \\ X_{2} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{1} (T, Z) - A_{0}^{'} (T, Z) + X_{1} (ϰ, υ, τ)])), \\ X_{3} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{2} (T, Z) - A_{0}^{'} (T, Z) + X_{2} (ϰ, υ, τ)])), \\ ⋮ \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\qquad\qquad\qquad\quad X_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma }),\, \\[0.1cm] X_{1}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left(s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{0}(T,Z)-A_{0}^{\prime}(T,Z)+X_{0}(\varkappa ,\upsilon ,\tau )\right] \right) \right) , \\[0.1cm] X_{2}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left(s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{1}(T,Z)-A_{0}^{\prime}(T,Z)+X_{1}(\varkappa ,\upsilon ,\tau )\right] \right) \right) ,{\, } \\[0.1cm] X_{3}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{2}(T,Z)-A_{0}^{\prime }(T,Z)+X_{2}(\varkappa ,\upsilon ,\tau )\right] \right) \right) , \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \vdots \end{array}$$(44)

$\begin{matrix} T_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}), \\ T_{1} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{0} (X, Z) + B_{0}^{'} (X, Z) - T_{0} (ϰ, υ, τ)])), \\ T_{2} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{1} (X, Z) + B_{1}^{'} (X, Z) - T_{1} (ϰ, υ, τ)])), \\ T_{3} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{2} (X, Z) + B_{2}^{'} (X, Z) - T_{2} (ϰ, υ, τ)])), \\ ⋮ \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\qquad\qquad\qquad T_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma }),\, \\[0.1cm] T_{1}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{0}(X,Z)+B_{0}^{\prime }(X,Z)-T_{0}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\[0.1cm] T_{2}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{1}(X,Z)+B_{1}^{\prime }(X,Z)-T_{1}(\varkappa,\upsilon ,\tau )\right] \right) \right) ,{\, } \\[0.1cm] T_{3}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{2}(X,Z)+B_{2}^{\prime }(X,Z)-T_{2}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\vdots \end{array}$$(45)

and so on.

For example, The first few components of A_n(X), B_n(X, t) and C(T) polynomials [22], are given by:

$\begin{matrix} \begin{matrix} A_{0} (X) = X_{0} X_{0, ϰ}^{(σ)}, \\ A_{1} (X) = X_{0} X_{1, ϰ}^{(σ)} + X_{1} X_{0, ϰ}^{(σ)}, \\ A_{2} (X) = X_{0} X_{2, ϰ}^{(σ)} + X_{2} X_{0, ϰ}^{(σ)} + X_{1} X_{1, ϰ}^{(σ)}, ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} A_{0}(X)=X_{0}X_{0,\varkappa }^{(\sigma )}, \\[0.1cm] A_{1}(X)=X_{0}X_{1,\varkappa }^{(\sigma )}+X_{1}X_{0,\varkappa }^{(\sigma )}, \\[0.1cm] A_{2}(X)=X_{0}X_{2,\varkappa }^{(\sigma )}+X_{2}X_{0,\varkappa }^{(\sigma)}+X_{1}X_{1,\varkappa }^{(\sigma )}, \vdots \end{array} \end{array}$$(46)

$\begin{matrix} \begin{matrix} B_{0} (X, T) = (X_{0} T_{0})_{ϰ}^{(σ)}, \\ B_{1} (X, T) = (X_{0} T_{1} + X_{1} T_{0})_{ϰ}^{(σ)}, \\ B_{2} (X, T) = (X_{1} T_{1} + X_{0} T_{2} + X_{2} T_{0})_{ϰ}^{(σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} B_{0}(X,T)=(X_{0}T_{0})_{\varkappa }^{\left( \sigma \right) }, \\[0.1cm] B_{1}(X,T)=(X_{0}T_{1}+X_{1}T_{0})_{\varkappa }^{\left( \sigma \right) }, \\[0.1cm] B_{2}(X,T)=(X_{1}T_{1}+X_{0}T_{2}+X_{2}T_{0})_{\varkappa }^{\left( \sigma \right) }, \\ \vdots \end{array} \end{array}$$(47)

and

$\begin{matrix} \begin{matrix} C_{0} (T) = T_{0} T_{0, ϰ}^{(σ)}, \\ C_{1} (T) = T_{0} T_{1, ϰ}^{(σ)} + T_{1} T_{0, ϰ}^{(σ)}, \\ C_{2} (T) = T_{0} T_{2, ϰ}^{(σ)} + T_{2} T_{0, ϰ}^{(σ)} + T_{1} T_{1, ϰ}^{(σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} C_{0}(T)=T_{0}T_{0,\varkappa }^{(\sigma )}, \\[0.1cm] C_{1}(T)=T_{0}T_{1,\varkappa }^{(\sigma )}+T_{1}T_{0,\varkappa }^{(\sigma )}, \\[0.1cm] C_{2}(T)=T_{0}T_{2,\varkappa }^{(\sigma )}+T_{2}T_{0,\varkappa }^{(\sigma)}+T_{1}T_{1,\varkappa }^{(\sigma )}, \\ \vdots \end{array} \end{array}$$(48)

According to the equations (44)-(45) and formulas (46)-(48), the first terms of local fractional Yang-Laplace decomposition method of the system (36), is given by:

$\begin{matrix} \begin{matrix} X_{0} (ϰ, τ) = \sin_{σ} (ϰ^{σ}), \\ X_{1} (ϰ, τ) = - \sin_{σ} (ϰ^{σ}) \frac{τ^{σ}}{Γ (1 + σ)}, \\ X_{2} (ϰ, τ) = \sin_{σ} (ϰ^{σ}) \frac{τ^{2 σ}}{Γ (1 + 2 σ)}, \\ X_{3} (ϰ, τ) = - \sin_{σ} (ϰ^{σ}) \frac{τ^{3 σ}}{Γ (1 + 3 σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} X_{0}(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma }),\ \\[0.1cm] X_{1}(\varkappa ,\tau )=-\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{\sigma }}{\Gamma (1+\sigma )}, \\[0.1cm] X_{2}(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{2\sigma }}{\Gamma (1+2\sigma )},\text{\ } \\[0.1cm] X_{3}(\varkappa ,\tau )=-\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{3\sigma }}{\Gamma (1+3\sigma )}, \\ \vdots \end{array} \end{array}$$(49)

and

$\begin{matrix} \begin{matrix} T_{0} (ϰ, τ) = \sin_{σ} (ϰ^{σ}), \\ T_{1} (ϰ, τ) = - \sin_{σ} (ϰ^{σ}) \frac{τ^{σ}}{Γ (1 + σ)}, \\ T_{2} (ϰ, τ) = \sin_{σ} (ϰ^{σ}) \frac{τ^{2 σ}}{Γ (1 + 2 σ)}, \\ T_{3} (ϰ, τ) = - \sin_{σ} (ϰ^{σ}) \frac{τ^{3 σ}}{Γ (1 + 3 σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} T_{0}(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma }),\ \\[0.1cm] T_{1}(\varkappa ,\tau )=-\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{\sigma }}{\Gamma (1+\sigma )}, \\[0.1cm] T_{2}(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{2\sigma }}{\Gamma (1+2\sigma )}, \\[0.1cm] T_{3}(\varkappa ,\tau )=-\sin _{\sigma }(\varkappa ^{\sigma })\frac{\tau^{3\sigma }}{\Gamma (1+3\sigma )}, \\ \vdots \end{array} \end{array}$$(50)

So that, the local fractional series solution (X, T), is:

$\begin{matrix} \{\begin{matrix} X (ϰ, τ) = \sin_{σ} (ϰ^{σ}) (1 - \frac{τ^{σ}}{Γ (1 + σ)} + \frac{τ^{2 σ}}{Γ (1 + 2 σ)} - \frac{τ^{3 σ}}{Γ (1 + 3 σ)} + \dots), \\ T (ϰ, τ) = \sin_{σ} (ϰ^{σ}) (1 - \frac{τ^{σ}}{Γ (1 + σ)} + \frac{τ^{2 σ}}{Γ (1 + 2 σ)} - \frac{τ^{3 σ}}{Γ (1 + 3 σ)} + \dots), \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })\left( 1-\frac{\tau^{\sigma }}{\Gamma (1+\sigma )}+\frac{\tau ^{2\sigma }}{\Gamma (1+2\sigma )}-\frac{\tau ^{3\sigma }}{\Gamma (1+3\sigma )}+\cdots \right) , \\[0.1cm] T(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })\left( 1-\frac{\tau^{\sigma }}{\Gamma (1+\sigma )}+\frac{\tau ^{2\sigma }}{\Gamma (1+2\sigma )}-\frac{\tau ^{3\sigma }}{\Gamma (1+3\sigma )}+\cdots \right) , \end{array} \right. \end{array}$$(51)

and in a closed form, we obtain the non-differentiable solution (X, T):

$\begin{matrix} \{\begin{matrix} X (ϰ, τ) = \sin_{σ} (ϰ^{σ}) E_{σ} (- τ^{σ}), \\ T (ϰ, τ) = \sin_{σ} (ϰ^{σ}) E_{σ} (- τ^{σ}) . \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })E_{\sigma }(-\tau^{\sigma }), \\[0.1cm] T(\varkappa ,\tau )=\sin _{\sigma }(\varkappa ^{\sigma })E_{\sigma }(-\tau^{\sigma }).{\, } \end{array} \right. \end{array}$$(52)

By letting σ = 1 into (52), we have:

$\begin{matrix} \{\begin{matrix} X (ϰ, τ) = \sin (ϰ) e^{- τ}, \\ T (ϰ, τ) = \sin (ϰ) e^{- τ} . \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\tau )=\sin (\varkappa )e^{-\tau }, \\[0.3cm] T(\varkappa ,\tau )=\sin (\varkappa )e^{-\tau }. \end{array} \right. \end{array}$$(53)

It should be noticed that, solution (52) satisfies the initial conditions (37), and in the case σ = 1, we have the same solution obtained in [23] by homotopy perturbation method, and in [24] by the natural decomposition method.

As a second example, let us now consider the following:

Example 4.2

Let

$\begin{matrix} \{\begin{matrix} X_{τ}^{(σ)} + T_{ϰ}^{(σ)} Z_{υ}^{(σ)} - T_{υ}^{(σ)} Z_{ϰ}^{(σ)} = - X \\ T_{τ}^{(σ)} + X_{ϰ}^{(σ)} Z_{υ}^{(σ)} + X_{υ}^{(σ)} Z_{ϰ}^{(σ)} = T \\ Z_{τ}^{(σ)} + X_{ϰ}^{(σ)} T_{υ}^{(σ)} + X_{υ}^{(σ)} T_{ϰ}^{(σ)} = Z \end{matrix}, 0 < σ ⩽ 1, \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X_{\tau }^{\left( \sigma \right) }\, +T_{\varkappa }^{\left( \sigma \right)}Z_{\upsilon }^{\left( \sigma \right) }-T_{\upsilon }^{\left( \sigma \right) }Z_{\varkappa }^{\left( \sigma \right) }=-X \\[0.1cm] T_{\tau }^{\left( \sigma \right) }+X_{\varkappa }^{\left( \sigma \right)}Z_{\upsilon }^{\left( \sigma \right) }+X_{\upsilon }^{\left( \sigma \right)}Z_{\varkappa }^{\left( \sigma \right) }=T{ \, \, \, \, } \\[0.1cm] Z_{\tau }^{\left( \sigma \right) }+X_{\varkappa }^{\left( \sigma \right)}T_{\upsilon }^{\left( \sigma \right) }+X_{\upsilon }^{\left( \sigma \right) }T_{\varkappa }^{\left( \sigma \right) }=Z{ \, \, \, \, } \end{array} \right. ,\text{ }0 \lt \sigma \leqslant 1, \end{array}$$(54)

be a nonlinear system of local fractional partial differential equations given under the initial conditions:

$\begin{matrix} X (ϰ, υ, 0) = E_{σ} (ϰ^{σ} + υ^{σ}), T (ϰ, υ, 0) = E_{σ} (ϰ^{σ} - υ^{σ}), Z (ϰ, υ, 0) = E_{σ} (- ϰ^{σ} + υ^{σ}) . \end{matrix}$ $$\begin{array}{} \displaystyle X(\varkappa ,\upsilon ,0)=E_{\sigma }(\varkappa ^{\sigma }+\upsilon ^{\sigma}),\text{ }T(\varkappa ,\upsilon ,0)=E_{\sigma }(\varkappa ^{\sigma}-\upsilon ^{\sigma }),\text{ }Z(\varkappa ,\upsilon ,0)=E_{\sigma}(-\varkappa ^{\sigma }+\upsilon ^{\sigma }). \end{array}$$(55)

Also in this case we will search for solutions by first applying the local fractional Yang-Laplace transform on both sides of each equation of the system (54):

$\begin{matrix} \{\begin{matrix} ^{L F} S_{σ} [X (ϰ, υ, τ)] = X (ϰ, υ, 0) - s^{- σ} (^{L F} L_{σ} [T_{ϰ}^{(σ)} Z_{υ}^{(σ)} - T_{υ}^{(σ)} Z_{ϰ}^{(σ)} + X]) \\ ^{L F} S_{σ} [T (ϰ, υ, τ)] = T (ϰ, υ, 0) - s^{- σ} (^{L F} L_{σ} [X_{ϰ}^{(σ)} Z_{υ}^{(σ)} + X_{υ}^{(σ)} Z_{ϰ}^{(σ)} - T]) \\ ^{L F} S_{σ} [Z (ϰ, υ, τ)] = Z (ϰ, υ, 0) - s^{- σ} (^{L F} L_{σ} [X_{ϰ}^{(σ)} T_{υ}^{(σ)} + X_{υ}^{(σ)} T_{ϰ}^{(σ)} - Z]) \end{matrix} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} ^{LF}S_{\sigma }\left[ X(\varkappa ,\upsilon ,\tau )\right] =X(\varkappa,\upsilon ,0)-s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ T_{\varkappa }^{\left(\sigma \right) }Z_{\upsilon }^{\left( \sigma \right) }-T_{\upsilon }^{\left(\sigma \right) }Z_{\varkappa }^{\left( \sigma \right) }+X\right] \right) \\[0.1cm] ^{LF}S_{\sigma }\left[ T(\varkappa ,\upsilon ,\tau )\right] =T(\varkappa,\upsilon ,0)-s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ X_{\varkappa }^{\left(\sigma \right) }Z_{\upsilon }^{\left( \sigma \right) }+X_{\upsilon }^{\left(\sigma \right) }Z_{\varkappa }^{\left( \sigma \right) }-T\right] \right) \\[0.1cm] ^{LF}S_{\sigma }\left[ Z(\varkappa ,\upsilon ,\tau )\right] =Z(\varkappa,\upsilon ,0)-s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ X_{\varkappa }^{\left(\sigma \right) }T_{\upsilon }^{\left( \sigma \right) }+X_{\upsilon }^{\left(\sigma \right) }T_{\varkappa }^{\left( \sigma \right) }-Z\right] \right) \end{array} \right. . \end{array}$$(56)

From here, by taking the inverse local fractional Yang-Laplace transform on both sides of each equation of (56) and taking into account the initial conditions (55), we have:

$\begin{matrix} \{\begin{matrix} X (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [T_{ϰ}^{(σ)} Z_{υ}^{(σ)} - T_{υ}^{(σ)} Z_{ϰ}^{(σ)} + X])) \\ T (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [X_{ϰ}^{(σ)} Z_{υ}^{(σ)} + X_{υ}^{(σ)} Z_{ϰ}^{(σ)} - T])) \\ Z (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [X_{ϰ}^{(σ)} T_{υ}^{(σ)} + X_{υ}^{(σ)} T_{ϰ}^{(σ)} - Z])) \end{matrix} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[T_{\varkappa }^{\left( \sigma \right) }Z_{\upsilon }^{\left( \sigma \right)}-T_{\upsilon }^{\left( \sigma \right) }Z_{\varkappa }^{\left( \sigma\right) }+X\right] \right) \right) \\[0.1cm] T(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[X_{\varkappa }^{\left( \sigma \right) }Z_{\upsilon }^{\left( \sigma \right)}+X_{\upsilon }^{\left( \sigma \right) }Z_{\varkappa }^{\left( \sigma \right) }-T\right] \right) \right) \\[0.1cm] Z(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[X_{\varkappa }^{\left( \sigma \right) }T_{\upsilon }^{\left( \sigma \right)}+X_{\upsilon }^{\left( \sigma \right) }T_{\varkappa }^{\left( \sigma\right) }-Z\right] \right) \right) \end{array} \right. . \end{array}$$(57)

Then by using the Adomian decomposition method [1] each function of the solution (X, T, Z) can be decomposed as an infinite series:

$\begin{matrix} \begin{matrix} X (ϰ, υ, τ) = \sum_{n = 0}^{\infty} X_{n} (ϰ, υ, τ), \\ T (ϰ, υ, τ) = \sum_{n = 0}^{\infty} T_{n} (ϰ, υ, τ), \\ Z (ϰ, υ, τ) = \sum_{n = 0}^{\infty} Z_{n} (ϰ, υ, τ), \end{matrix} \end{matrix}$ $$\begin{array}{} \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=\sum_{n=0}^{\infty }X_{n}(\varkappa,\upsilon ,\tau ), \\[0.3cm] T(\varkappa ,\upsilon ,\tau )=\sum_{n=0}^{\infty }T_{n}(\varkappa,\upsilon ,\tau ), \\[0.3cm] Z(\varkappa ,\upsilon ,\tau )=\sum_{n=0}^{\infty }Z_{n}(\varkappa,\upsilon ,\tau ), \end{array} \end{array}$$(58)

and the nonlinear terms can be decomposed as:

$\begin{matrix} T_{ϰ}^{(σ)} Z_{υ}^{(σ)} = \sum_{n = 0}^{\infty} A_{n} (T, Z), T_{υ}^{(σ)} Z_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} A_{n}^{'} (T, Z), \end{matrix}$ $$\begin{array}{} \displaystyle T_{\varkappa }^{\left( \sigma \right) }Z_{\upsilon }^{\left( \sigma \right)}=\sum_{n=0}^{\infty }A_{n}(T,Z),{ \, \, }T_{\upsilon }^{\left(\sigma \right) }Z_{\varkappa }^{\left( \sigma \right)}=\sum_{n=0}^{\infty }A_{n}^{\prime }(T,Z), \end{array}$$(59)

$\begin{matrix} X_{ϰ}^{(σ)} Z_{υ}^{(σ)} = \sum_{n = 0}^{\infty} B_{n} (X, Z), X_{υ}^{(σ)} Z_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} B_{n}^{'} (X, Z), \end{matrix}$ $$\begin{array}{} \displaystyle X_{\varkappa }^{\left( \sigma \right) }Z_{\upsilon }^{\left( \sigma \right)}=\sum_{n=0}^{\infty }B_{n}(X,Z),{ \, \, }X_{\upsilon }^{\left(\sigma \right) }Z_{\varkappa }^{\left( \sigma \right)}=\sum_{n=0}^{\infty }B_{n}^{\prime }(X,Z), \end{array}$$(60)

and

$\begin{matrix} X_{ϰ}^{(σ)} T_{υ}^{(σ)} = \sum_{n = 0}^{\infty} C_{n} (X, T), X_{υ}^{(σ)} T_{ϰ}^{(σ)} = \sum_{n = 0}^{\infty} C_{n}^{'} (X, T) . \end{matrix}$ $$\begin{array}{} \displaystyle X_{\varkappa }^{\left( \sigma \right) }T_{\upsilon }^{\left( \sigma \right)}=\sum_{n=0}^{\infty }C_{n}(X,T),{ \, \, }X_{\upsilon }^{\left( \sigma \right) }T_{\varkappa }^{\left( \sigma \right) }=\sum_{n=0}^{\infty }C_{n}^{\prime }(X,T). \end{array}$$(61)

Substituting (58), (59), (60) and (67) into (57), we get:

$\begin{matrix} \{\begin{matrix} \sum_{n = 0}^{\infty} X_{n} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}) \\ -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\sum_{n = 0}^{\infty} A_{n} (T, Z) - \sum_{n = 0}^{\infty} A_{n}^{'} (T, Z) + \sum_{n = 0}^{\infty} X_{n} (ϰ, υ, τ)])), \\ \sum_{n = 0}^{\infty} T_{n} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) \\ -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\sum_{n = 0}^{\infty} B_{n} (X, Z) + \sum_{n = 0}^{\infty} B_{n}^{'} (X, Z) - \sum_{n = 0}^{\infty} T_{n} (ϰ, υ, τ)])), \\ \sum_{n = 0}^{\infty} Z_{n} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) \\ -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [\sum_{n = 0}^{\infty} C_{n} (X, T) + \sum_{n = 0}^{\infty} C_{n}^{'} (X, T) - \sum_{n = 0}^{\infty} Z_{n} (ϰ, υ, τ)])) . \end{matrix} . \end{matrix}$ $$\begin{array}{} \left\{ \begin{array}{c} \sum_{n=0}^{\infty }X_{n}(\varkappa ,\upsilon ,\tau )=E_{\sigma}(\varkappa ^{\sigma }+\upsilon ^{\sigma }) \\[0.3cm] -^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[\sum_{n=0}^{\infty }A_{n}(T,Z)-\sum_{n=0}^{\infty}A_{n}^{\prime }(T,Z)+\sum_{n=0}^{\infty }X_{n}(\varkappa ,\upsilon,\tau )\right] \right) \right) , \\[0.3cm] \sum_{n=0}^{\infty }T_{n}(\varkappa ,\upsilon ,\tau )=E_{\sigma}(\varkappa ^{\sigma }-\upsilon ^{\sigma }) \\[0.3cm] -^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[\sum_{n=0}^{\infty }B_{n}(X,Z)+\sum_{n=0}^{\infty}B_{n}^{\prime }(X,Z)-\sum_{n=0}^{\infty }T_{n}(\varkappa ,\upsilon,\tau )\right] \right) \right) , \\[0.3cm] \sum_{n=0}^{\infty }Z_{n}(\varkappa ,\upsilon ,\tau )=E_{\sigma}(-\varkappa ^{\sigma }+\upsilon ^{\sigma }) \\[0.3cm] -^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[\sum_{n=0}^{\infty }C_{n}(X,T)+\sum_{n=0}^{\infty}C_{n}^{\prime }(X,T)-\sum_{n=0}^{\infty }Z_{n}(\varkappa ,\upsilon,\tau )\right] \right) \right) . \end{array} \right. . \end{array}$$(62)

By comparing both sides of (62), we have:

$\begin{matrix} X_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}), \\ X_{1} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{0} (T, Z) - A_{0}^{'} (T, Z) + X_{0} (ϰ, υ, τ)])), \\ X_{2} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{1} (T, Z) - A_{0}^{'} (T, Z) + X_{1} (ϰ, υ, τ)])), \\ X_{3} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [A_{2} (T, Z) - A_{0}^{'} (T, Z) + X_{2} (ϰ, υ, τ)])), \\ ⋮ \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\qquad\qquad\quad~~~ X_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma }), \\[0.1cm] X_{1}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left(s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{0}(T,Z)-A_{0}^{\prime}(T,Z)+X_{0}(\varkappa ,\upsilon ,\tau )\right] \right) \right) , \\[0.1cm] X_{2}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left(s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{1}(T,Z)-A_{0}^{\prime}(T,Z)+X_{1}(\varkappa ,\upsilon ,\tau )\right] \right) \right) ,{\, } \\[0.1cm] X_{3}(\varkappa ,\upsilon ,\tau )=\text{ }-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma }\left( ^{LF}L_{\sigma }\left[ A_{2}(T,Z)-A_{0}^{\prime }(T,Z)+X_{2}(\varkappa ,\upsilon ,\tau )\right] \right) \right) , \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad~~\vdots \end{array}$$(63)

$\begin{matrix} T_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}), \\ T_{1} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{0} (X, Z) + B_{0}^{'} (X, Z) - T_{0} (ϰ, υ, τ)])), \\ T_{2} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{1} (X, Z) + B_{1}^{'} (X, Z) - T_{1} (ϰ, υ, τ)])), \\ T_{3} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [B_{2} (X, Z) + B_{2}^{'} (X, Z) - T_{2} (ϰ, υ, τ)])), \\ ⋮ \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\qquad\qquad\quad T_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma }),\, \\[0.1cm] T_{1}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{0}(X,Z)+B_{0}^{\prime }(X,Z)-T_{0}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\[0.1cm] T_{2}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{1}(X,Z)+B_{1}^{\prime }(X,Z)-T_{1}(\varkappa,\upsilon ,\tau )\right] \right) \right) ,{\, } \\[0.1cm] T_{3}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ B_{2}(X,Z)+B_{2}^{\prime }(X,Z)-T_{2}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\vdots \end{array}$$(64)

and

$\begin{matrix} Z_{0} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}), \\ Z_{1} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [C_{0} (X, T) + C_{0}^{'} (X, T) - Z_{0} (ϰ, υ, τ)])), \\ Z_{2} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [C_{1} (X, T) + C_{1}^{'} (X, T) - Z_{1} (ϰ, υ, τ)])), \\ Z_{3} (ϰ, υ, τ) = -^{L F} L_{σ}^{- 1} (s^{- σ} (^{L F} L_{σ} [C_{2} (X, T) + C_{2}^{'} (X, T) - Z_{2} (ϰ, υ, τ)])), \\ ⋮ \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\qquad\qquad\quad Z_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma }),\, \\[0.1cm] Z_{1}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ C_{0}(X,T)+C_{0}^{\prime }(X,T)-Z_{0}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\[0.1cm] Z_{2}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ C_{1}(X,T)+C_{1}^{\prime }(X,T)-Z_{1}(\varkappa,\upsilon ,\tau )\right] \right) \right) ,{ } \\[0.1cm] Z_{3}(\varkappa ,\upsilon ,\tau )=-^{LF}L_{\sigma }^{-1}\left( s^{-\sigma}\left( ^{LF}L_{\sigma }\left[ C_{2}(X,T)+C_{2}^{\prime }(X,T)-Z_{2}(\varkappa,\upsilon ,\tau )\right] \right) \right) , \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \vdots \end{array}$$(65)

and so on.

For example, the first few components of A_n(T, Z), B_n(X, Z) and C_n(X, T) polynomials [22], are:

$\begin{matrix} A_{0} (T, Z) = T_{0 ϰ}^{(σ)} Z_{0 υ}^{(σ)}, \\ A_{1} (T, Z) = T_{1 ϰ}^{(σ)} Z_{0 υ}^{(σ)} + T_{0 ϰ}^{(σ)} Z_{1 υ}^{(σ)}, \\ A_{2} (T, Z) = T_{0 ϰ}^{(σ)} Z_{2 υ}^{(σ)} + T_{2 ϰ}^{(σ)} Z_{0 υ}^{(σ)} + T_{1 ϰ}^{(σ)} Z_{1 υ}^{(σ)}, \\ ⋮ \end{matrix}$ $$\begin{array}{c} \displaystyle A_{0}(T,Z)=T_{0\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}, \\[0.2cm] A_{1}(T,Z)=T_{1\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}+T_{0\varkappa }^{\left( \sigma \right) }Z_{1\upsilon }^{(\sigma )}, \\[0.2cm] A_{2}(T,Z)=T_{0\varkappa }^{\left( \sigma \right) }Z_{2\upsilon }^{(\sigma)}+T_{2\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}+T_{1\varkappa }^{\left( \sigma \right) }Z_{1\upsilon }^{(\sigma )}, \\ \vdots \end{array}$$(66)

$\begin{matrix} \begin{matrix} B_{0} (X, Z) = X_{0 ϰ}^{(σ)} Z_{0 υ}^{(σ)}, \\ B_{1} (X, Z) = X_{1 ϰ}^{(σ)} Z_{0 υ}^{(σ)} + X_{0 ϰ}^{(σ)} Z_{1 υ}^{(σ)}, \\ B_{2} (X, Z) = X_{0 ϰ}^{(σ)} Z_{2 υ}^{(σ)} + X_{2 ϰ}^{(σ)} Z_{0 υ}^{(σ)} + X_{1 ϰ}^{(σ)} Z_{1 υ}^{(σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} B_{0}(X,Z)=X_{0\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}, \\[0.2cm] B_{1}(X,Z)=X_{1\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}+X_{0\varkappa }^{\left( \sigma \right) }Z_{1\upsilon }^{(\sigma )}, \\[0.2cm] B_{2}(X,Z)=X_{0\varkappa }^{\left( \sigma \right) }Z_{2\upsilon }^{(\sigma)}+X_{2\varkappa }^{\left( \sigma \right) }Z_{0\upsilon }^{(\sigma)}+X_{1\varkappa }^{\left( \sigma \right) }Z_{1\upsilon }^{(\sigma )}, \\ \vdots \end{array} \end{array}$$(67)

and

$\begin{matrix} \begin{matrix} C_{0} (X, T) = X_{0 ϰ}^{(σ)} T_{0 υ}^{(σ)}, \\ C_{1} (X, T) = X_{1 ϰ}^{(σ)} T_{0 υ}^{(σ)} + X_{0 ϰ}^{(σ)} T_{1 υ}^{(σ)}, \\ C_{2} (X, T) = X_{0 ϰ}^{(σ)} T_{2 υ}^{(σ)} + X_{2 ϰ}^{(σ)} T_{0 υ}^{(σ)} + X_{1 ϰ}^{(σ)} T_{1 υ}^{(σ)}, \\ ⋮ \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{c} C_{0}(X,T)=X_{0\varkappa }^{\left( \sigma \right) }T_{0\upsilon }^{(\sigma)}, \\[0.2cm] C_{1}(X,T)=X_{1\varkappa }^{\left( \sigma \right) }T_{0\upsilon }^{(\sigma)}+X_{0\varkappa }^{\left( \sigma \right) }T_{1\upsilon }^{(\sigma )}, \\[0.2cm] C_{2}(X,T)=X_{0\varkappa }^{\left( \sigma \right) }T_{2\upsilon }^{(\sigma)}+X_{2\varkappa }^{\left( \sigma \right) }T_{0\upsilon }^{(\sigma)}+X_{1\varkappa }^{\left( \sigma \right) }T_{1\upsilon }^{(\sigma )}, \\ \vdots \end{array} \end{array}$$(68)

The other polynomials $\begin{matrix} A_{n}^{'}, B_{n}^{'} and C_{n}^{'}, \end{matrix}$ $\begin{array}{} \displaystyle A_{n}^{\prime },~B_{n}^{\prime }~\text{ and}~ C_{n}^{\prime }, \end{array}$ can be computed in the same way.

From the equations (63)-(65) and formulas of the polynomial terms, the first terms of local fractional Yang-Laplace decomposition method of the system (54), is given by:

$\begin{matrix} X_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}), \\ X_{1} (ϰ, υ, τ) = - E_{σ} (ϰ^{σ} + υ^{σ}) \frac{τ^{σ}}{Γ (1 + σ)}, \\ X_{2} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}) \frac{τ^{2 σ}}{Γ (1 + 2 σ)}, \\ X_{3} (ϰ, υ, τ) = - E_{σ} (ϰ^{σ} + υ^{σ}) \frac{τ^{3 σ}}{Γ (1 + 3 σ)}, \\ ⋮ \end{matrix}$ $$\begin{array}{c} \displaystyle X_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma }), \\[0.1cm] X_{1}(\varkappa ,\upsilon ,\tau )=-E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }, \\[0.1cm] X_{2}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) }, \\[0.1cm] X_{3}(\varkappa ,\upsilon ,\tau )=-E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }, \\ \vdots \end{array}$$(69)

$\begin{matrix} T_{0} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}), \\ T_{1} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) \frac{τ^{σ}}{Γ (1 + σ)}, \\ T_{2} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) \frac{τ^{2 σ}}{Γ (1 + 2 σ)}, \\ T_{3} (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) \frac{τ^{3 σ}}{Γ (1 + 3 σ)}, \\ ⋮ \end{matrix}$ $$\begin{array}{c} \displaystyle T_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma }),\, \\[0.1cm] T_{1}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }, \\[0.1cm] T_{2}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) },{ } \\[0.1cm] T_{3}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }, \\ \vdots \end{array}$$(70)

and

$\begin{matrix} Z_{0} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}), \\ Z_{1} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) \frac{τ^{σ}}{Γ (1 + σ)}, \\ Z_{2} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) \frac{τ^{2 σ}}{Γ (1 + 2 σ)}, \\ Z_{3} (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) \frac{τ^{3 σ}}{Γ (1 + 3 σ)}, \\ ⋮ \end{matrix}$ $$\begin{array}{c} \displaystyle Z_{0}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon ^{\sigma }),\, \\[0.1cm] Z_{1}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }, \\[0.1cm] Z_{2}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) }, \\[0.1cm] Z_{3}(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }, \\ \vdots \end{array}$$(71)

Each function of the solution (X, T, Z) is defined by an approximate series as follows:

$\begin{matrix} \{\begin{matrix} X (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}) (1 - \frac{τ^{σ}}{Γ (1 + σ)} + \frac{τ^{2 σ}}{Γ (1 + 2 σ)} - \frac{τ^{3 σ}}{Γ (1 + 3 σ)} + \dots), \\ T (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) (1 + \frac{τ^{σ}}{Γ (1 + σ)} + \frac{τ^{2 σ}}{Γ (1 + 2 σ)} + \frac{τ^{3 σ}}{Γ (1 + 3 σ)} + \dots), \\ Z (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) (1 + \frac{τ^{σ}}{Γ (1 + σ)} + \frac{τ^{2 σ}}{Γ (1 + 2 σ)} + \frac{τ^{3 σ}}{Γ (1 + 3 σ)} + \dots) . \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })(1-\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }+\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) }-\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }+\cdots ), \\ T(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })(1+\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }+\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) }+\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }+\cdots ), \\ Z(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })(1+\frac{\tau ^{\sigma }}{\Gamma \left( 1+\sigma \right) }+\frac{\tau ^{2\sigma }}{\Gamma \left( 1+2\sigma \right) }+\frac{\tau ^{3\sigma }}{\Gamma \left( 1+3\sigma \right) }+\cdots ) \, . \end{array} \right. \end{array}$$(72)

So that in closed form, the non-differentiable solution (X, T, Z) can be written as:

$\begin{matrix} \{\begin{matrix} X (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ}) E_{σ} (- τ^{σ}), \\ T (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ}) E_{σ} (τ^{σ}), \\ Z (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ}) E_{σ} (τ^{σ}), \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma })E_{\sigma }\left( -\tau ^{\sigma }\right) , \\[0.2cm] T(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma })E_{\sigma }\left( \tau ^{\sigma }\right) , \\[0.2cm] Z(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma })E_{\sigma }\left( \tau ^{\sigma }\right) , \end{array} \right. \end{array}$$(73)

and, according to [17],

$\begin{matrix} \{\begin{matrix} X (ϰ, υ, τ) = E_{σ} (ϰ^{σ} + υ^{σ} - τ^{σ}), \\ T (ϰ, υ, τ) = E_{σ} (ϰ^{σ} - υ^{σ} + τ^{σ}), \\ Z (ϰ, υ, τ) = E_{σ} (- ϰ^{σ} + υ^{σ} + τ^{σ}) . \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }+\upsilon^{\sigma }-\tau ^{\sigma }), \\[0.2cm] T(\varkappa ,\upsilon ,\tau )=E_{\sigma }(\varkappa ^{\sigma }-\upsilon^{\sigma }+\tau ^{\sigma }), \\[0.2cm] Z(\varkappa ,\upsilon ,\tau )=E_{\sigma }(-\varkappa ^{\sigma }+\upsilon^{\sigma }+\tau ^{\sigma }). \end{array} \right. \end{array}$$(74)

In particular, by letting σ = 1, from (74), we get

$\begin{matrix} \{\begin{matrix} X (ϰ, υ, τ) = e^{ϰ + υ - τ}, \\ T (ϰ, υ, τ) = e^{ϰ - υ + τ}, \\ Z (ϰ, υ, τ) = e^{- ϰ + υ + τ} . \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{c} X(\varkappa ,\upsilon ,\tau )=e^{\varkappa +\upsilon -\tau }, \\[0.2cm] T(\varkappa ,\upsilon ,\tau )=e^{\varkappa -\upsilon +\tau }, \\[0.2cm] Z(\varkappa ,\upsilon ,\tau )=e^{-\varkappa +\upsilon +\tau }. \end{array} \right. \end{array}$$(75)

Let us notice that, our solution (74) fulfills the initial conditions (55), and in the case σ = 1, we re-obtain the same solution already obtained both in [26] by the projected differential transform method and Elzaki transform, and in [24] by the natural decomposition method. Thus showing that the method proposed in this paper is the more general approach to the solution of nonlinear fractional differential system.

Conclusions

The local fractional Yang-Laplace transform decomposition method (LFLDM) has been used to solve nonlinear systems of local fractional partial differential equations. It has been shown that combined with the Adomian decomposiition method, LFLDM enables us to establish an efficient algorithm. This algorithm provides the solution in a series form that converges rapidly to the exact solution, as shown by the results obtained through the two non-trivial examples given in this paper. From the obtained results, it can be concluded that this algorithm is powerful and effective and it can be used to explore some more complicated nonlinear systems with local fractional derivative.

eISSN:: 2444-8656
Sprache:: Englisch

Zeitrahmen der Veröffentlichung:: Volume Open
Fachgebiete der Zeitschrift:: Biologie, andere, Mathematik, Angewandte Mathematik, Allgemeines, Physik

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Yang-Laplace Decomposition Method for Nonlinear System of Local Fractional Partial Differential Equations

Online veröffentlicht: 24. Dez. 2019

Seitenbereich: 489 - 502

Eingereicht: 15. Apr. 2019

Akzeptiert: 02. Juni 2019

DOI: https://doi.org/10.2478/AMNS.2019.2.00046

SchlüsselwörterLocal fractional derivative operator, local fractional Yang-Laplace decomposition method, nonlinear systems of local fractional partial differential equations

© 2019 Djelloul Ziane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Schlüsselwörter
Local fractional derivative operator, local fractional Yang-Laplace decomposition method, nonlinear systems of local fractional partial differential equations