Double Local Fractional Yang–Laplace Transform for Local Fractional PDEs on Fractal Domains

Abstract

This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ($LF{L}_{\zeta }^2$) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the $LF{L}_{\zeta }^2$ transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries.

1. Introduction

Fractional calculus has become a pivotal mathematical framework for modeling a wide range of complex phenomena characterized by memory effects, anomalous diffusion, and heterogeneous or porous media. It provides a powerful and indispensable tool for the analysis and description of dynamic systems across various scientific disciplines [1,2,3]. In recent years, a variety of fractional derivative operators have been proposed, including the Caputo, Riemann–Liouville, Atangana–Baleanu–Caputo, and Caputo–Fabrizio formulations. These operators have demonstrated significant efficacy and precision in capturing the behavior of complex systems in science and engineering. However, it is well established that no single operator is universally sufficient to model all aspects of such intricate dynamics. As a result, extensive research has been devoted to employing and comparing these operators in the investigation of diverse physical, engineering, and biological processes [4,5,6,7,8].

The fractal dimension is a fundamental tool for characterizing complex physical systems whose structures deviate from classical Euclidean geometry. In various applied domains, such as anomalous diffusion, material science, and biological transport processes, phenomena often unfold in environments with irregular, self-similar structures where standard calculus fails to provide accurate descriptions. These systems exhibit scale-dependent behavior that is naturally captured by fractal geometry, with the fractal dimension offering a quantitative measure of irregularity and heterogeneity.

To rigorously model such behavior, local fractional calculus has emerged as a powerful analytical framework for handling functions defined on fractal sets. Unlike classical derivatives, local fractional derivatives are well suited to functions that are continuous but nowhere differentiable—properties commonly observed in fractal media. The foundational concept of local fractional calculus, also referred to as fractal calculus, was first introduced by Kolwankar and Gangal [9,10], and later expanded by Yang and Jumarie [11,12], who formalized a specialized notation for local fractional derivatives on Cantor-type sets.

Xiao-Jun Yang [13,14] has made substantial contributions to the advancement of this field, particularly through his work in local fractional functional analysis and its applications to fractal integral transforms and wave propagation in fractal media. His formulation of local partial functional analysis has enabled new approaches to mathematical modeling and the exploration of fractal geometry using operators defined via local fractional derivatives and partial integrals.

The research conducted by Yang and collaborators [15,16,17,18,19] extends these theoretical developments into practical applications, providing a robust mathematical foundation for various integral transforms, most notably the Yang-Fourier transform, Yang–Laplace transform, and local partial wavelet transform. These tools offer new and precise methods for the analysis of fractal-based systems across mathematics and applied sciences.

The Laplace transform is a fundamental analytical tool in mathematics, extensively applied across diverse scientific and engineering disciplines, particularly in physics and systems theory. It facilitates the conversion of functions from the time domain (real variable) to the complex frequency domain, thereby simplifying the analysis of dynamic systems by transforming differential equations into algebraic equations. Traditionally, the Laplace transform has been effectively employed to solve linear differential equations of integer order. In recent years, however, increasing attention has been directed toward extending its applicability to linear differential equations of fractional order. This extension has proven valuable in capturing memory effects and anomalous dynamics inherent in complex systems. For comprehensive studies and applications of the Laplace transform in fractional-order systems, readers can refer to [20,21,22], which offer significant contributions to the fields of mathematical physics, control theory, and engineering.

The concept of double integrals has recently gained prominence in the formulation of novel integral transforms, commonly referred to as double transforms. These transforms provide a flexible and powerful analytical framework for solving a broad class of mathematical problems, including partial differential equations and integral equations of both integer and fractional orders. Notably, recent studies such as [23,24,25] have focused on the development and analysis of the double Laplace transform and the double Sumudu transform, investigating their structural properties and operational advantages. The double Laplace transform, in particular, has been effectively applied to solve a variety of complex equations, including the singular coupled Burgers equations, Poisson-type partial differential equations, and the fractional Caputo heat equation. Furthermore, it has demonstrated substantial utility in addressing partial integro-differential equations, as evidenced in works such as [26,27,28,29,30], thereby underscoring its growing significance in the field of applied mathematics.

The literature reflects a broad and growing interest in the applications of the Sumudu transform, which has been extensively studied and applied to various classes of differential equations, as documented. Building upon this foundation, several authors have introduced and utilized the double Elzaki transform to tackle complex mathematical problems, including singular systems of hyperbolic equations, the telegraph equation, and partial integro-differential equations [31,32,33,34]. In parallel, the double Natural transform has proven to be an effective analytical tool for solving a range of problems, such as one-dimensional heat equations, general linear telegraph equations, wave equations, and partial integro-differential equations [35,36]. More recently, the double Shehu transform has been proposed and investigated in [37,38], with successful applications to several physical models. These include integral equations, classical partial differential equations, and fractional partial differential equations in the Caputo sense. Collectively, these double transform techniques represent significant advancements in the analytical treatment of complex mathematical and physical models, offering enhanced flexibility and accuracy in solving equations across various domains.

Traditional fractional transform methods, such as single-variable Laplace or Fourier transforms, often struggle to handle complex local fractional partial differential equations, especially on fractal domains. These methods typically face challenges in accurately incorporating initial or boundary conditions and ensuring convergence for nonlinear or multi-dimensional problems. In this paper, the proposed double local fractional Yang–Laplace Transform addresses these limitations by extending the transform framework to handle two-dimensional and highly nonlinear structures with greater stability. It naturally embeds initial conditions, enhances convergence, and avoids the need for discretization or linearization, making it a powerful tool for solving a broader class of local fractional PDEs on fractal geometries.

The novel integral transform, called the double local fractional Yang–Laplace transform, was formulated using the double integrals within fractal spaces. This new transform extends the framework of local fractional calculus and is specifically designed to operate on functions defined over fractal sets, such as Cantor-type domains. We establish the fundamental properties of the double local fractional Yang–Laplace transform, including its behavior under local fractional partial differentiation, the formulation of a double local fractional Yang–Laplace convolution theorem, and its action on selected special functions. To demonstrate the efficacy of the proposed transform, we apply it to several benchmark models of local fractional partial differential equations, including the local fractional Laplace equation and the linear local fractional Klein–Gordon equation, among others. These applications highlight the transform’s capability to effectively handle the complexity inherent in systems governed by fractal geometry and local fractional dynamics. The results presented herein offer a significant advancement in the analytical treatment of local fractional partial differential equations and serve as a valuable tool to the broader field of fractional calculus.

The structure of this paper is organized as follows:

• Section 2 introduces the fundamental concepts and mathematical tools necessary for this study, including key definitions and properties of local fractional calculus and the local fractional Yang–Laplace transform.
• Section 3 presents the principal theoretical developments concerning the double local fractional Yang–Laplace transform, along with its core properties.
• Section 4 demonstrates the applicability of the proposed transform by solving representative local fractional partial differential equations, such as the local fractional diffusion equation and the local fractional Klein–Gordon equation.
• Section 5 concludes this paper with a summary of the main results, highlighting the theoretical and practical implications of this paper.

2. Fundamental Concepts

In this section, we will cover the fundamental concepts of the local fractional calculus, including the local fractional derivative, local fractional integral, significant results, and the local fractional Yang–Laplace transform.

2.1. Local Fractional Derivative

Definition 1.

The local fractional derivative of $\varphi (\varkappa )$ of order ζ at $\varkappa ={\varkappa }_0$ is defined as follows [14]:

$${\varphi }^{(\zeta )}(\varkappa )={\left.{\displaystyle \frac{d^{\zeta }\varphi }{d{\varkappa }^{\zeta }}}\right|}_{\varkappa ={\varkappa }_0}\underset{\varkappa \to {\varkappa }_0}{lim}{\displaystyle \frac{\Gamma (1+\zeta )\left[(\varphi (\varkappa )-\varphi ({\varkappa }_0))\right]}{{(\varkappa -{\varkappa }_0)}^{\zeta }}},$$

(1)

where $\varphi (\varkappa )\in C_{\zeta }(a,b),\zeta \in ]0,1]$, with the notation ${\varphi }^{(\zeta )}(\varkappa )=D_{\varkappa }^{\zeta }\varphi (\varkappa ).$

Let $0<\zeta \le 1$ be the local fractional order, and let $[a,b]\subset \mathbb{R}$ be a closed interval. The space $C_{\zeta }(a,b)$ is defined as the set of all functions $\varphi :[a,b]\to \mathbb{R}$ that are $\zeta$-Hölder continuous on the interval $[a,b]$. That is,

$$C_{\zeta }(a,b)=\left\{\varphi :[a,b]\to \mathbb{R};\exists L>0\mathrm{such}\mathrm{that}|\varphi (\varkappa )-\varphi (y)|\le {L|\varkappa -y|}^{\zeta },\forall \varkappa ,y\in [a,b]\right\}.$$

The local fractional derivative of a high order is expressed in the following manner:

$${\varphi }^{(k\zeta )}(\varkappa )=\stackrel{ktimes}{\overbrace{D_{\varkappa }^{(\zeta )}\cdots D_{\varkappa }^{(\zeta )}\varphi (\varkappa )},}$$

(2)

and the local fractional partial derivative of higher order is expressed as

$${\displaystyle \frac{{\partial }^{k\zeta }\varphi (\varkappa ,\tau )}{\partial {\varkappa }^{k\zeta }}}=\stackrel{ktimes}{\overbrace{{\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\cdots {\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\varphi (\varkappa ,\tau )}.}$$

(3)

Properties of Some Special Functions

The properties of the local fractional derivative of some non-differentiable functions are presented as follows:

$${\displaystyle \frac{d^{\zeta }{\varkappa }^{k\zeta }}{d{\varkappa }^{\zeta }}}={\displaystyle \frac{\Gamma (1+k\zeta ){\varkappa }^{(k-1)\zeta }}{\Gamma (1+(k-1)\zeta )}}.$$

(4)

Consider a particular case of $k=2;$ then, we would have

$${\displaystyle \frac{d^{\zeta }{\varkappa }^{2\zeta }}{d{\varkappa }^{\zeta }}}={\displaystyle \frac{\Gamma (1+2\zeta ){\varkappa }^{\zeta }}{\Gamma (1+\zeta )}},$$

(5)

and

$${\displaystyle \frac{d^{\zeta }}{d{\varkappa }^{\zeta }}}{E}_{\zeta }({\varkappa }^{\zeta })=E_{\zeta }({\varkappa }^{\zeta }),$$

(6)

$${\displaystyle \frac{d^{\zeta }}{d{\varkappa }^{\zeta }}}{sin}_{\zeta }({\varkappa }^{\zeta })={cos}_{\zeta }({\varkappa }^{\zeta }),$$

(7)

$${\displaystyle \frac{d^{\zeta }}{d{\varkappa }^{\zeta }}}{cos}_{\zeta }({\varkappa }^{\zeta })=-{sin}_{\zeta }({\varkappa }^{\zeta }),$$

(8)

where $E_{\zeta }$ is the Mittag–Leffler function, defined on a fractal set of fractal dimension by

$$E_{\zeta }({\varkappa }^{\zeta })=\sum _{k=0}^{\infty }{\displaystyle \frac{{\varkappa }^{k\zeta }}{\Gamma (1+k\zeta )}}.$$

(9)

2.2. Local Fractional Integral

Definition 2.

Let $\varphi (\varkappa )$ be a function belonging to the space $C_{\zeta }(a,b)$; then the local fractional integral of $\varphi (\varkappa )$ of order ζ is defined in the following manner [14]:

$$\begin{array}{ccc}\hfill {}_{a}I_b^{(\zeta )}\varphi (\varkappa )& =& {\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_a^b\varphi (\varkappa ){(dx)}^{\zeta }\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (1+\zeta )}}\underset{\Delta \tau ⟶0}{lim}\sum _{j=0}^{N-1}\varphi ({\varkappa }_j){(\Delta {\varkappa }_j)}^{\zeta },\hfill \end{array}$$

(10)

where $\Delta {\varkappa }_j={\varkappa }_{j+1}-{\varkappa }_j,$ $\Delta \varkappa =max\left\{\Delta {\varkappa }_0,\Delta {\varkappa }_1,\Delta {\varkappa }_2,\cdots \right\}$ and $\left[{\varkappa }_j,{\varkappa }_{j+1}\right]$ is a (subdivision) of the interval $[a,b]$, where ${\varkappa }_0=a,$ ${\varkappa }_N=b.$

Example 1.

We have

$${}_{0}I_{\varkappa }^{(\zeta )}{\displaystyle \frac{{\varkappa }^{k\zeta }}{\Gamma (1+k\zeta )}}={\displaystyle \frac{{\varkappa }^{(k+1)\zeta }}{\Gamma (1+(k+1)\zeta )}}.$$

(11)

Consider a particular case of k = 2; then, we have

$${}_{0}I_{\varkappa }^{(\zeta )}{\displaystyle \frac{{\varkappa }^{2\zeta }}{\Gamma (1+2\zeta )}}={\displaystyle \frac{{\varkappa }^{3\zeta }}{\Gamma (1+3\zeta )}}.$$

(12)

In the following definition, we will highlight some key past results.

Definition 3.

In the fractal space, the Mittag–Leffler function, sine, and cosine are defined as referenced in [14,17]:

$$E_{\zeta }({\varkappa }^{\zeta })=\sum _{k=0}^{+\infty }{\displaystyle \frac{{\varkappa }^{k\zeta }}{\Gamma (1+k\zeta )}},0<\zeta ⩽1,$$

(13)

$$E_{\zeta }({\varkappa }^{\zeta })E_{\zeta }({\tau }^{\zeta })=E_{\zeta }{(\varkappa +\tau )}^{\zeta },0<\zeta ⩽1,$$

(14)

$${sin}_{\zeta }({\varkappa }^{\zeta })=\sum _{k=0}^{+\infty }{(-1)}^k{\displaystyle \frac{{\varkappa }^{(2k+1)\zeta }}{\Gamma (1+(2k+1)\zeta )}},0<\zeta ⩽1,$$

(15)

$${cos}_{\zeta }({\varkappa }^{\zeta })=\sum _{k=0}^{+\infty }{(-1)}^k{\displaystyle \frac{{\varkappa }^{2k\zeta }}{\Gamma (1+2k\zeta )}},0<\zeta ⩽1.$$

(16)

The integral with respect to ${\left(d\varkappa \right)}^{\zeta }$ is defined as the solution to the fractional differential equation

$$dy=\varphi \left(\varkappa \right){\left(d\varkappa \right)}^{\zeta },\varkappa ⩾0\mathrm{and}y(0)=0,0<\zeta ⩽1.$$

(17)

which is provided by the following lemma:

Lemma 1.

Let $\varphi (\varkappa )$ be a continuous function, then the solution of (17) is defined by

$$\begin{array}{ccc}\hfill y(\varkappa )& =& {\int }_0^{\varkappa }\varphi \left(\tau \right){\left(d\tau \right)}^{\zeta },y(0)=0\hfill \\ & =& \zeta {\int }_0^{\varkappa }\left(\varkappa -\tau \right)\varphi \left(\tau \right)\left(d\tau \right),0<\zeta ⩽1.\hfill \end{array}$$

2.3. Local Fractional Yang–Laplace Transform

Here, we present the definition of the local fractional Yang–Laplace transform, denoted by $LF{L}_{\zeta }$, along with some properties related to this transformation [15,16,18].

Definition 4.

Let $\varphi \in L_{1,\zeta }[{\mathbb{R}}_{+}]$ and ${∥\varphi ∥}_{1,\zeta }<\infty$, then the Yang–Laplace transforms of $w(\varkappa ,\tau )$ are defined by

$$LF{L}_{\zeta }\left\{w(\varkappa ,\tau )\right\}=W(\varkappa ,\omega )={\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_0^{\infty }{E}_{\zeta }(-{\omega }^{\zeta }{\tau }^{\zeta })w(\varkappa ,\tau ){(d\tau )}^{\zeta },0<\zeta ⩽1.$$

(18)

The last integral converges for any real number ω.

Definition 5.

The inverse Yang–Laplace transforms of $W(\varkappa ,\nu )$ is defined by

$$LF{L}_{\zeta }^{-1}\left\{W(\varkappa ,\omega )\right\}=w(\varkappa ,\tau )={\displaystyle \frac{1}{{(2\pi )}^{\zeta }}}{\int }_{\beta -i\infty }^{\beta +i\infty }{E}_{\zeta }({\omega }^{\zeta }{\tau }^{\zeta })W(\varkappa ,\omega ){(d\omega )}^{\zeta },0<\zeta ⩽1.$$

(19)

A sufficient condition for convergence is presented as follows:

$${\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_0^{\infty }\left|w(\varkappa ,\tau )\right|{\left(d\tau \right)}^{\zeta }<m<\infty .$$

(20)

In the following, we present some properties of the Yang–Laplace transform:

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{a\varphi (\varkappa )+b\phi (\varkappa )\right\}& =& aLF{L}_{\zeta }(\varphi (\varkappa ))+bLF{L}_{\zeta }(\phi (\varkappa )),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{{\varphi }^{\left(n\zeta \right)}(\varkappa )\right\}& =& {\omega }^{n\zeta }LFL\left\{\varphi (\varkappa )\right\}-\sum _{k=1}^n{\omega }^{(k-1)\zeta }{\varphi }^{((n-k)\zeta )}(0),\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{{\varkappa }^{k\zeta }{E}_{\zeta }(a^{\zeta }{\varkappa }^{\zeta })\right\}& =& {\displaystyle \frac{\Gamma (1+k\zeta )}{{(\omega -a)}^{(k+1)\zeta }}},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{{sin}_{\zeta }(a^{\zeta }{\varkappa }^{\zeta })\right\}& =& {\displaystyle \frac{a^{\zeta }}{{\omega }^{2\zeta }+a^{2\zeta }}},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{{cos}_{\zeta }(a^{\zeta }{\varkappa }^{\zeta })\right\}& =& {\displaystyle \frac{{\omega }^{\zeta }}{{\omega }^{2\zeta }+a^{2\zeta }}},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left\{{\varkappa }^{k\zeta }\right\}& =& {\displaystyle \frac{\Gamma (1+k\zeta )}{{\omega }^{(k+1)\zeta }}}.\hfill \end{array}$$

(26)

3. Main Results

In this section, we introduce the double local fractional Yang–Laplace Transform and explore its fundamental properties and operational characteristics.

Definition 6.

Let

$${\int }_0^{\infty }{\int }_0^{\infty }\left|\varphi (\varkappa ,\tau )\right|{(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }<k<\infty .$$

(27)

The convergence condition can be clarified by using Formula (20) and [16].

The local fractional double Yang–Laplace transform of $\varphi (\varkappa ,\tau )$ is defined by

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}& =& F_{\zeta }^2(\omega ,\upsilon )\hfill \\ & =& {\displaystyle \frac{1}{{\Gamma }^2(1+\zeta )}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa +\upsilon \tau \right)}^{\zeta })\varphi (\varkappa ,\tau ){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta },\hfill \end{array}$$

(28)

where $\omega ,\upsilon \in \mathbb{C}$ and $E_{\zeta }(\varkappa )$ is the Mittag–Leffler function.

Corollary 1.

Using the properties of Mittag–Leffler functions, one can convert Formula (28) as follows:

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}& =& F_{\zeta }^2(\omega ,\upsilon )\hfill \\ & =& {\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-\left(\omega \varkappa \right))E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })\varphi (\varkappa ,\tau ){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }.\hfill \end{array}$$

(29)

In the following sections, we will demonstrate properties of the double local fractional Yang–Laplace transform method.

3.1. Linearity

Proposition 1.

Let $\varphi (\varkappa ,\tau )$ and $\phi (\varkappa ,\tau )$ be two functions of two variables. Then, we have

$$LF{L}_{\zeta }^2\left\{\lambda \varphi (\varkappa ,\tau )+\beta \phi (\varkappa ,\tau )\right\}=\lambda LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}+\beta LF{L}_{\zeta }^2\left\{\phi (\varkappa ,\tau )\right\},$$

(30)

where λ and β are constants.

Proof. 

To prove the above proposition, it is sufficient to use (28) as follows:

$$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{\lambda \varphi (\varkappa ,\tau )+\beta \phi (\varkappa ,\tau )\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa +\upsilon \tau \right)}^{\zeta })\left\{\lambda \varphi (\varkappa ,\tau )+\beta \phi (\varkappa ,\tau )\right\}{(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{\lambda }{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa +\upsilon \tau \right)}^{\zeta })\left\{\lambda \varphi (\varkappa ,\tau )\right\}{(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & +{\displaystyle \frac{\beta }{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa +\upsilon \tau \right)}^{\zeta })\left\{\beta \phi (\varkappa ,\tau )\right\}{(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & =\lambda LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}+\beta LF{L}_{\zeta }^2\left\{\phi (\varkappa ,\tau )\right\}.\hfill \end{array}$$

□

3.2. Derivative

Now, let us present the double local fractional Yang–Laplace transform of the local fractional partial derivative with respect to $\tau .$

Proposition 2.

If $LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}=F_{\zeta }^2(\omega ,\upsilon ),$ then we have

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}\right\}={\upsilon }^{\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-LF{L}_{\zeta }\left(\varphi (\varkappa ,0)\right),$$

(31)

and

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{n\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{n\zeta }}}\right\}={\upsilon }^{n\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^{n-1}{\upsilon }^{\left(n-k-1\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{k\zeta }}}\right).$$

(32)

Proof. 

For the proof of Formula (31), by utilizing Formula (29) and integrating by parts, we obtain

$$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-\left(\omega \varkappa \right))E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}{(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{E}_{\zeta }(-\left(\omega \varkappa \right))\left(\left[-\Gamma (1+\zeta )\varphi (\varkappa ,0)\right]\right.\hfill \\ & & +\left.{\upsilon }^{\zeta }\underset{t⟶\infty }{lim}{\int }_0^t{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })\varphi (\varkappa ,\tau ){\left(d\tau \right)}^{\zeta }\right){(dx)}^{\zeta }\hfill \\ & & =-{\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_0^{\infty }{E}_{\zeta }(-\left(\omega \varkappa \right))\varphi (\varkappa ,0){(dx)}^{\zeta }\hfill \\ & & +{\displaystyle \frac{{\upsilon }^{\zeta }}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-\left(\omega \varkappa \right))E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })\varphi (\varkappa ,\tau ){(dx)}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\upsilon }^{\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-LF{L}_{\zeta }\left(\varphi (\varkappa ,0)\right).\hfill \end{array}$$

To establish Formula (32), we will utilize the mathematical induction method.

• For $n=1$, applying Formula (32), we obtain

  $$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}\right\}={\upsilon }^{\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-LF{L}_{\zeta }\left(\varphi (\varkappa ,0)\right).$$
  Then, referring to Formula (31), we observe that the formula holds when $n=1$.
• Let us assume inductively that the formula holds for n, implying that

  $$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{n\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{n\zeta }}}\right\}={\upsilon }^{n\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^{n-1}{\upsilon }^{\left(n-k-1\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{k\zeta }}}\right).$$

  (33)
• It remains to show that (32) is true for $n+1$. Let ${\displaystyle \frac{{\partial }^{n\zeta }}{\partial {\tau }^{n\zeta }}}\varphi (\varkappa ,\tau )=\phi (\varkappa ,\tau )$ and according to (31) and (33), we have

  $$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{\left(n+1\right)\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{\left(n+1\right)\zeta }}}\right\}\hfill \\ & & =LF{L}_{\zeta }^2\left[{\displaystyle \frac{{\partial }^{\zeta }\phi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}\right]\hfill \\ & & ={\upsilon }^{\zeta }LF{L}_{\zeta }^2\left[\phi (\varkappa ,\tau )\right]-LF{L}_{\zeta }\left(\phi (\varkappa ,0)\right)\hfill \\ & & ={\upsilon }^{\zeta }\left[{\upsilon }^{n\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^{n-1}{\upsilon }^{\left(n-k-1\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }f(\varkappa ,0)}{\partial {\tau }^{k\zeta }}}\right)\right]-LF{L}_{\zeta }\left(\phi (\varkappa ,0)\right)\hfill \\ & & ={\upsilon }^{\left(n+1\right)\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^{n-1}{\upsilon }^{\left(n-k\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{k\zeta }}}\right)-LF{L}_{\zeta }\left(\phi (\varkappa ,0)\right)\hfill \\ & & ={\upsilon }^{\left(n+1\right)\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^n{\upsilon }^{\left(n-k\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{k\zeta }}}\right).\hfill \end{array}$$

Therefore, Formula (32) is true for $n+1$, which completes the proof. □

Example 2.

For $n=2$ and $n=3$, we conclude the following formulas:

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{2\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{2\zeta }}}\right\}={\upsilon }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\upsilon }^{\zeta }LF{L}_{\zeta }\left(\varphi (\varkappa ,0)\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{\zeta }}}\right).$$

(34)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{3\zeta }\varphi (\varkappa ,\tau )}{\partial {\tau }^{3\zeta }}}\right\}& =& {\upsilon }^{3\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\upsilon }^{2\zeta }LF{L}_{\zeta }\left(\varphi (\varkappa ,0)\right)\hfill \\ & & -{\upsilon }^{\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{\zeta }}}\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{2\zeta }\varphi (\varkappa ,0)}{\partial {\tau }^{2\zeta }}}\right).\hfill \end{array}$$

(35)

Proposition 3.

If

$$LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}=F_{\zeta }^2(\omega ,\upsilon ),$$

then we have

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{\zeta }\varphi (\varkappa ,\tau )}{\partial {\varkappa }^{\zeta }}}\right\}={\omega }^{\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-LF{L}_{\zeta }\left(\varphi (0,\tau )\right),$$

(36)

and

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{n\zeta }\varphi (\varkappa ,\tau )}{\partial {\varkappa }^{n\zeta }}}\right\}={\omega }^{n\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-\sum _{k=0}^{n-1}{\omega }^{\left(n-k-1\right)\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{k\zeta }\varphi (0,\tau )}{\partial {\varkappa }^{k\zeta }}}\right).$$

(37)

The proof follows a similar approach as that of the Proposition 2, making it straightforward to conclude.

Example 3.

For $n=2$ and $n=3$, we obtain the following formulas:

$$LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{2\zeta }\varphi (\varkappa ,\tau )}{\partial {\varkappa }^{2\zeta }}}\right\}={\omega }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\omega }^{\zeta }LF{L}_{\zeta }\left(\varphi (0,\tau )\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\varphi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right).$$

(38)

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\partial }^{3\zeta }\varphi (\varkappa ,\tau )}{\partial {\varkappa }^{3\zeta }}}\right\}& =& {\omega }^{3\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\omega }^{2\zeta }LF{L}_{\zeta }\left(\varphi (0,\tau )\right)\hfill \\ & & -{\omega }^{\zeta }LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\varphi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{2\zeta }\varphi (0,\tau )}{\partial {\varkappa }^{2\zeta }}}\right).\hfill \end{array}$$

(39)

3.3. Convolution

In this subsection, we present the double local fractional Yang–Laplace convolution.

Proposition 4.

If

$$LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}=F_{\zeta }^2(\omega ,\upsilon )\mathrm{and}LF{L}_{\zeta }^2\left\{\phi (\varkappa ,\tau )\right\}=G_{\zeta }^2(\omega ,\upsilon ).$$

Then, the local fractional double Yang–Laplace convolution can be defined as

$$LF{L}_{\zeta }^2\left\{{\left(\varphi (\varkappa ,\tau )\ast \ast \phi (\varkappa ,\tau )\right)}_{\zeta }\right\}=F_{\zeta }^2(\omega ,\upsilon )G_{\zeta }^2(\omega ,\upsilon ),$$

where

$${\left(\varphi (\varkappa ,\tau )\ast \ast \phi (\varkappa ,\tau )\right)}_{\zeta }={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\varkappa }{\int }_0^{\tau }\varphi (\varkappa -\eta )\phi (\tau -\theta ){(d\eta )}^{\zeta }{(d\theta )}^{\zeta }.$$

The proof can be concluded by using the local fractional double Yang–Laplace transform of $\varphi (\varkappa ,\tau )$ presented in (29) and by following the similar proof of the convolution in [39].

3.4. Transform of Special Functions

In the following, we will show the double local fractional Yang–Laplace transform of some special functions. In the following results, we will employ Formula (29) and its properties, along with those of some basic functions.

• If $\varphi (\varkappa ,\tau )=1,$ then we have

  $$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}& =& {\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta })E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & =& {\displaystyle \frac{1}{\Gamma {(1+\zeta )}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\int }_0^M{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & =& \underset{M⟶\infty }{lim}{\left[{\displaystyle \frac{-1}{{\omega }^{\zeta }}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta })\right]}_0^M\times \underset{M⟶\infty }{lim}{\left[{\displaystyle \frac{-1}{{\upsilon }^{\zeta }}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })\right]}_0^M\hfill \\ & =& {\displaystyle \frac{1}{{\omega }^{\zeta }{\upsilon }^{\zeta }}}.\hfill \end{array}$$
• If $\varphi (\varkappa ,\tau )={\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}$ $(0<\zeta ⩽1),$ then after the integration by parts, we get

  $$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta })E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{\Gamma {(1+\zeta )}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M{\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\int }_0^M{\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & & =\underset{M⟶\infty }{lim}\left({\left[{\displaystyle \frac{-1}{{\omega }^{\zeta }}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}\right]}_0^M+{\displaystyle \frac{1}{{\omega }^{\zeta }}}{\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_0^M{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){(d\varkappa )}^{\zeta }\right)\hfill \\ & & +\underset{M⟶\infty }{lim}\left({\left[{\displaystyle \frac{-1}{{\upsilon }^{\zeta }}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}\right]}_0^M+{\displaystyle \frac{1}{{\upsilon }^{\zeta }}}{\displaystyle \frac{1}{\Gamma (1+\zeta )}}{\int }_0^M{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\tau )}^{\zeta }\right).\hfill \end{array}$$
  Since

  $$\underset{M⟶\infty }{lim}{\left[{\displaystyle \frac{-1}{{\omega }^{\zeta }}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}\right]}_0^M=0\mathrm{and}\underset{M⟶\infty }{lim}{\left[{\displaystyle \frac{-1}{{\upsilon }^{\zeta }}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}\right]}_0^M=0.$$
  Consequently, we get

  $$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{{\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}\right\}& =& \underset{M⟶\infty }{lim}\left({\left[-{\displaystyle \frac{1}{{\omega }^{2\zeta }}}{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta })\right]}_0^M{\left[-{\displaystyle \frac{1}{{\upsilon }^{2\zeta }}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })\right]}_0^M\right)\hfill \\ & =& {\displaystyle \frac{1}{{\omega }^{2\zeta }{\upsilon }^{2\zeta }}}.\hfill \end{array}$$
• If $\varphi (\varkappa ,\tau )=E_{\zeta }({\left(a\varkappa +b\tau \right)}^{\zeta }),$ then, using Formula (29) and Mittag–Leffler properties, we obtain

  $$\begin{array}{ccc}\hfill LF{L}_{\zeta }^2\left\{\varphi (\varkappa ,\tau )\right\}& =& {\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta })E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })E_{\zeta }({\left(a\varkappa +b\tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & =& {\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-\left({\omega }^{\zeta }-a^{\zeta }\right){\varkappa }^{\zeta })E_{\zeta }(-\left({\upsilon }^{\zeta }-b^{\zeta }\right){\tau }^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & =& {\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M{E}_{\zeta }(-\left({\omega }^{\zeta }-a^{\zeta }\right){\varkappa }^{\zeta }){(d\varkappa )}^{\zeta }{\int }_0^M{E}_{\zeta }(-\left({\upsilon }^{\zeta }-b^{\zeta }\right){\tau }^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & =& \underset{M⟶\infty }{lim}\left({\left[{\displaystyle \frac{-1}{\left({\omega }^{\zeta }-a^{\zeta }\right)}}{E}_{\zeta }(-\left({\omega }^{\zeta }-a^{\zeta }\right){\varkappa }^{\zeta })\right]}_0^M\times {\left[{\displaystyle \frac{-1}{\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}{E}_{\zeta }(-\left({\upsilon }^{\zeta }-b^{\zeta }\right){\tau }^{\zeta })\right]}_0^M\right)\hfill \\ & =& {\displaystyle \frac{1}{\left({\omega }^{\zeta }-a^{\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}.\hfill \end{array}$$
• If $\varphi (\varkappa ,\tau )={sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$ $(0<\zeta ⩽1),$ then, from (29), we obtain

  $$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{{sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){\displaystyle \frac{E_{\zeta }\left(i^{\zeta }{\left(a\varkappa \right)}^{\zeta }\right)-E_{\zeta }\left(-i^{\zeta }{\left(a\varkappa \right)}^{\zeta }\right)}{2i^{\zeta }}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}{\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)-E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right]E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}{\displaystyle \frac{1}{\Gamma {(1+\zeta )}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right.\right.\hfill \\ & & \left.\left.-E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right]{(d\varkappa )}^{\zeta }{\int }_0^M{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}\underset{M⟶\infty }{lim}\left({\left[-{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }-a{i}^{\zeta }}}+{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }+a{i}^{\zeta }}}\right]}_0^M\left[-{\displaystyle \frac{E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })}{{\upsilon }^{\zeta }}}\right]{.}_0^M\right)\hfill \end{array}$$
  After straightforward computations, we get

  $$LF{L}_{\zeta }^2\left\{{sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })\right\}={\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right){\upsilon }^{\zeta }}}.$$
• If $\varphi (\varkappa ,\tau )={sinh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$ $(0<\zeta ⩽1),$ we have

  $$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{{sinh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){\displaystyle \frac{E_{\zeta }\left({\left(a\varkappa \right)}^{\zeta }\right)-E_{\zeta }\left(-{\left(a\varkappa \right)}^{\zeta }\right)}{2}}{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a\right){\varkappa }^{\zeta }\right)-E_{\zeta }\left(-\left({\omega }^{\zeta }+a\right){\varkappa }^{\zeta }\right)\right]E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{\Gamma {(1+\zeta )}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a\right){\varkappa }^{\zeta }\right)\right.\right.\hfill \\ & & \left.\left.-E_{\zeta }\left(-\left({\omega }^{\zeta }+a\right){\varkappa }^{\zeta }\right)\right]{(d\varkappa )}^{\zeta }{\int }_0^M{E}_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & & ={\displaystyle \frac{1}{2}}\underset{M⟶\infty }{lim}\left({\left[-{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }-a\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }-a}}+{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }+a\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }+a}}\right]}_0^M{\left[-{\displaystyle \frac{E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta })}{{\upsilon }^{\zeta }}}\right]}_0^M\right)\hfill \end{array}$$
  After performing the calculations, we obtain the following results:

  $$LF{L}_{\zeta }^2\left\{{sinh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })\right\}={\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }-a^{2\zeta }\right){\upsilon }^{\zeta }}}.$$
• If $\varphi (\varkappa ,\tau )={sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right),$ $(0<\zeta ⩽1),$ using Formula (29), we obtain

  $$\begin{array}{ccc}& & LF{L}_{\zeta }^2\left\{{sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })\right\}\hfill \\ & & ={\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }{E}_{\zeta }(-{\left(\omega \varkappa \right)}^{\zeta }){\displaystyle \frac{E_{\zeta }\left(i^{\zeta }{\left(a\varkappa \right)}^{\zeta }\right)-E_{\zeta }\left(-i^{\zeta }{\left(a\varkappa \right)}^{\zeta }\right)}{2i^{\zeta }}}{E}_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)E_{\zeta }(-{\left(\upsilon \tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}{\displaystyle \frac{1}{{\left(\Gamma (1+\zeta )\right)}^2}}{\int }_0^{\infty }{\int }_0^{\infty }\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right.\hfill \\ & & \left.-E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right]E_{\zeta }(-{\left(\left(\upsilon -b\right)\tau \right)}^{\zeta }){(d\varkappa )}^{\zeta }{\left(d\tau \right)}^{\zeta }\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}{\displaystyle \frac{1}{\Gamma {(1+\zeta )}^2}}\underset{M⟶+\infty }{lim}\left({\int }_0^M\left[E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)-E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)\right]{(d\varkappa )}^{\zeta }\right.\hfill \\ & & \left.\times {\int }_0^M{E}_{\zeta }(-{\left(\left(\upsilon -b\right)\tau \right)}^{\zeta }){\left(d\tau \right)}^{\zeta }\right)\hfill \\ & & ={\displaystyle \frac{1}{2i^{\zeta }}}\underset{M⟶\infty }{lim}\left({\left[-{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }-a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }-a{i}^{\zeta }}}+{\displaystyle \frac{E_{\zeta }\left(-\left({\omega }^{\zeta }+a{i}^{\zeta }\right){\varkappa }^{\zeta }\right)}{{\omega }^{\zeta }+a{i}^{\zeta }}}\right]}_0^M{\left[-{\displaystyle \frac{E_{\zeta }(-{\left(\left(\upsilon -b\right)\tau \right)}^{\zeta })}{{\left(\upsilon -b\right)}^{\zeta }}}\right]}_0^M\right)\hfill \end{array}$$
  Finally, we get

  $$LF{L}_{\zeta }^2\left\{{sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)\right\}={\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}.$$

In the following

Table 1. Double Local Fractional Yang-Laplace Transform of Special Functions.

$\mathit{\varphi }(\mathit{\varkappa },\mathit{\tau })$	${\mathit{LFL}}_{\mathit{\zeta }}^{\mathbf{2}}\left\{\mathit{\varphi }(\mathit{\varkappa },\mathit{\tau })\right\}$	Remarks
c	${\displaystyle \frac{c}{{\omega }^{\zeta }{\upsilon }^{\zeta }}}$	c is a constant
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}$	${\displaystyle \frac{1}{{\omega }^{2\zeta }{\upsilon }^{\zeta }}}$
${\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}$	${\displaystyle \frac{1}{{\omega }^{\zeta }{\upsilon }^{2\zeta }}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}$	${\displaystyle \frac{1}{{\omega }^{2\zeta }{\upsilon }^{2\zeta }}}$
$E_{\zeta }({\left(a\varkappa +b\tau \right)}^{\zeta })$	${\displaystyle \frac{1}{\left({\omega }^{\zeta }-a^{\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$	$E_{\zeta }\left({\left(a\varkappa \right)}^{\zeta }\right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{a^{k\zeta }{\varkappa }^{k\zeta }}{\Gamma (1+k\zeta )}}$
${\displaystyle \frac{{\tau }^{\zeta }}{\Gamma (1+\zeta )}}{E}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$	${\displaystyle \frac{1}{\left({\omega }^{\zeta }-a^{\zeta }\right){\upsilon }^{2\zeta }}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{E}_{\zeta }({\left(b\tau \right)}^{\zeta })$	${\displaystyle \frac{1}{{\omega }^{2\zeta }\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$
${sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right){\upsilon }^{\zeta }}}$	${sin}_{\zeta }({\varkappa }^{\zeta })={\sum }_{m=0}^{+\infty }{(-1)}^m{\displaystyle \frac{{\varkappa }^{(2m+1)\zeta }}{\Gamma (1+(2m+1)\zeta )}}$
${sin}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{{\omega }^{\zeta }\left({\upsilon }^{2\zeta }+a^{2\zeta }\right)}}$
${cos}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right){\upsilon }^{\zeta }}}$	${cos}_{\zeta }\left({\varkappa }^{\zeta }\right)={\sum }_{m=0}^{+\infty }{(-1)}^m{\displaystyle \frac{{\varkappa }^{2m\zeta }}{\Gamma (1+2m\zeta )}}$
${cos}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{{\omega }^{\zeta }\left({\upsilon }^{2\zeta }+a^{2\zeta }\right)}}$
${sinh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }-a^{2\zeta }\right){\upsilon }^{\zeta }}}$	${sinh}_{\zeta }(a{\varkappa }^{\zeta })={\sum }_{m=0}^{+\infty }{\displaystyle \frac{{\varkappa }^{(2m+1)\zeta }}{\Gamma (1+(2m+1)\zeta )}}$
${sinh}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{{\omega }^{\zeta }\left({\upsilon }^{2\zeta }-a^{2\zeta }\right)}}$
${cosh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{\left({\omega }^{2\zeta }-a^{2\zeta }\right){\upsilon }^{\zeta }}}$	${cosh}_{\zeta }\left(a{\varkappa }^{\zeta }\right)={\sum }_{m=0}^{+\infty }{\displaystyle \frac{{\varkappa }^{2m\zeta }}{\Gamma (1+2m\zeta )}}$
${cosh}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{{\omega }^{\zeta }\left({\upsilon }^{2\zeta }-a^{2\zeta }\right)}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{sin}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{{\omega }^{2\zeta }\left({\upsilon }^{2\zeta }+a^{2\zeta }\right)}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{cos}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{{\omega }^{2\zeta }\left({\upsilon }^{2\zeta }+a^{2\zeta }\right)}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{sinh}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{a^{\zeta }}{{\omega }^{2\zeta }\left({\omega }^{2\zeta }-a^{2\zeta }\right)}}$
${\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}{cosh}_{\zeta }({\left(a\tau \right)}^{\zeta })$	${\displaystyle \frac{{\omega }^{\zeta }}{{\omega }^{2\zeta }\left({\omega }^{2\zeta }-a^{2\zeta }\right)}}$
${sin}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)$	${\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$
${cos}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)$	${\displaystyle \frac{{\omega }^{\zeta }}{\left({\omega }^{2\zeta }+a^{2\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$
${sinh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)$	${\displaystyle \frac{a^{\zeta }}{\left({\omega }^{2\zeta }-a^{2\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$
${cosh}_{\zeta }({\left(a\varkappa \right)}^{\zeta })E_{\zeta }\left({\left(b\tau \right)}^{\zeta }\right)$	${\displaystyle \frac{{\omega }^{\zeta }}{\left({\omega }^{2\zeta }-a^{2\zeta }\right)\left({\upsilon }^{\zeta }-b^{\zeta }\right)}}$

, we will present the previous results along with some additional cases that can be proven using a similar approach as before.

4. Application of Double Local Fractional Yang–Laplace Transform

In this section, we will apply the double local fractional Yang–Laplace transform method $LF{L}_{\zeta }^2$ to solve several local fractional partial differential equations.

Problem 1.

We consider the following local fractional Laplace equation:

$${\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\tau }^{2\zeta }}}+{\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\varkappa }^{2\zeta }}}=0,0<\zeta ⩽1,$$

(40)

with the initial and boundary conditions

$$\Phi (\varkappa ,0)=0,{\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\varkappa }^{\zeta }}}=-E_{\zeta }({\varkappa }^{\zeta }),$$

(41)

$$\Phi (0,\tau )=-{sin}_{\zeta }({\tau }^{\zeta }),{\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}=-{sin}_{\zeta }({\tau }^{\zeta }).$$

(42)

By taking the double local fractional Yang–Laplace transform of Equation (40), we obtain

$$\begin{array}{c}{\upsilon }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\upsilon }^{\zeta }LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\tau }^{\zeta }}}\Phi (\varkappa ,0)\right)+{\omega }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )\\ -{\omega }^{\zeta }LF{L}_{\zeta }\left(\Phi (0,\tau )\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\Phi (0,\tau )\right)=0.\end{array}$$

(43)

By applying the single local fractional Yang–Laplace transform to Equations (41) and (42), we obtain

$$\begin{array}{ccc}\hfill LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)& =& 0,LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\varkappa }^{\zeta }}}\right)={\displaystyle \frac{-1}{{\omega }^{\zeta }-1}},\hfill \\ \hfill LF{L}_{\zeta }\left(\Phi (0,\tau )\right)& =& {\displaystyle \frac{-1}{{\upsilon }^{2\zeta }+1}},LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right)={\displaystyle \frac{-1}{{\upsilon }^{2\zeta }+1}}.\hfill \end{array}$$

(44)

Based on the conditions given in Equation (44), Equation (43) can be rewritten as

$$\left({\upsilon }^{2\zeta }+{\omega }^{2\zeta }\right)F_{\zeta }^2(\omega ,\upsilon )=-{\displaystyle \frac{1}{{\omega }^{\zeta }-1}}-{\displaystyle \frac{{\omega }^{\zeta }+1}{{\upsilon }^{2\zeta }+1}}.$$

Therefore, we have

$$F_{\zeta }^2(\omega ,\upsilon )=-{\displaystyle \frac{1}{\left({\omega }^{\zeta }-1\right)\left({\upsilon }^{2\zeta }+1\right)}}.$$

(45)

By applying the inverse local fractional double Yang–Laplace transform to Equation (45), we obtain the explicit solution

$$\Phi (\varkappa ,\tau )=-E_{\zeta }({\varkappa }^{\zeta }){sin}_{\zeta }({\tau }^{\zeta }),$$

(46)

which coincides with the one obtained in [19].

Problem 2.

Let us consider

$${\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,\tau )}{\partial {\tau }^{\zeta }}}-{\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\varkappa }^{2\zeta }}}=0,0<\zeta ⩽1,$$

(47)

with the initial and boundary conditions

$$\Phi (\varkappa ,0)={\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma \left(1+\zeta \right)}},$$

(48)

$$\Phi (0,\tau )=0,{\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}=1.$$

(49)

A double local fractional Yang–Laplace transform of Equation (47) implies

$$\begin{array}{c}{\upsilon }^{\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)+{\omega }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )\\ -{\omega }^{\zeta }LF{L}_{\zeta }\left(\Phi (0,\tau )\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right)=0.\end{array}$$

(50)

A single local fractional Yang–Laplace transform of Equations (48) and (49) leads to

$$LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)={\displaystyle \frac{1}{{\omega }^{2\zeta }}},LF{L}_{\zeta }\left(\Phi (0,\tau )\right)=0,LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\Phi (0,\tau )\right)={\displaystyle \frac{1}{{\upsilon }^{\zeta }}}.$$

(51)

Based on the conditions given in Equation (51), Equation (50) becomes

$$F_{\zeta }^2(\omega ,\upsilon )={\displaystyle \frac{1}{{\omega }^{2\zeta }{\upsilon }^{\zeta }}}.$$

(52)

By applying the inverse double local fractional Yang–Laplace transform to Equation (52), we obtain

$$\Phi (\varkappa ,\tau )={\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}.$$

(53)

This solution coincides with the one obtained in [40].

Problem 3.

Let us now solve the following problem:

$${\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\tau }^{2\zeta }}}-{\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\varkappa }^{2\zeta }}}+\Phi (\varkappa ,\tau )=0,0<\zeta ⩽1,$$

(54)

with the initial and boundary conditions

$$\Phi (\varkappa ,0)=0,{\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\tau }^{\zeta }}}={\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma \left(1+\zeta \right)}},$$

(55)

$$\Phi (0,\tau )=0,{\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}={sin}_{\zeta }\left({\tau }^{\zeta }\right).$$

(56)

By inserting the double local fractional Yang–Laplace transform of Equation (54), we obtain

$$\begin{array}{c}{\upsilon }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\upsilon }^{\zeta }LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\tau }^{\zeta }}}\Phi (\varkappa ,0)\right)-{\omega }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )\\ +{\omega }^{\zeta }LF{L}_{\zeta }\left(\Phi (0,\tau )\right)+LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\Phi (0,\tau )\right)+F_{\zeta }^2(\omega ,\upsilon )=0.\end{array}$$

(57)

By taking the single local fractional Yang–Laplace transform for each of the conditions (55) and (56), we have

$$\begin{array}{c}LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)=0,LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\varkappa }^{\zeta }}}\right)={\displaystyle \frac{1}{{\omega }^{2\zeta }}},\\ LF{L}_{\zeta }\left(\Phi (0,\tau )\right)=0,LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right)={\displaystyle \frac{1}{{\upsilon }^{2\zeta }+1}}.\end{array}$$

(58)

According to the conditions given in Equation (58), Equation (57) becomes

$$F_{\zeta }^2(\omega ,\upsilon )={\displaystyle \frac{1}{{\omega }^{2\zeta }({\upsilon }^{2\zeta }+1)}}.$$

(59)

By applying the inverse double local fractional Yang–Laplace transform to Equation (59), we obtain the explicit solution

$$\Phi (\varkappa ,\tau )={\displaystyle \frac{{\varkappa }^{\zeta }}{\Gamma (1+\zeta )}}sin({\tau }^{\zeta }).$$

(60)

This solution coincides with the one obtained for this equation in [41] when $\zeta =1$.

Problem 4.

Finally, we consider the linear local fractional Klein–Gordon equation

$${\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\tau }^{2\zeta }}}-{\displaystyle \frac{{\partial }^{2\zeta }\Phi (\varkappa ,\tau )}{\partial {\varkappa }^{2\zeta }}}+\Phi (\varkappa ,\tau )=2{sin}_{\zeta }\left({\varkappa }^{\zeta }\right),0<\zeta ⩽1,$$

(61)

with the initial and boundary conditions

$$\Phi (\varkappa ,0)={sin}_{\zeta }\left({\varkappa }^{\zeta }\right),{\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\tau }^{\zeta }}}=1,$$

(62)

$$\Phi (0,\tau )={sin}_{\zeta }\left({\tau }^{\zeta }\right),{\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}=1.$$

(63)

By applying the double local fractional Yang–Laplace transform to both sides of Equation (61), we obtain

$$\begin{array}{c}{\upsilon }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )-{\upsilon }^{\zeta }LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)-LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\tau }^{\zeta }}}\Phi (\varkappa ,0)\right)-{\omega }^{2\zeta }{F}_{\zeta }^2(\omega ,\upsilon )\\ +{\omega }^{\zeta }LF{L}_{\zeta }\left(\Phi (0,\tau )\right)+LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }}{\partial {\varkappa }^{\zeta }}}\Phi (0,\tau )\right)+F_{\zeta }^2(\omega ,\upsilon )=2{\displaystyle \frac{1}{{\upsilon }^{\zeta }\left({\omega }^{2\zeta }+1\right)}}.\end{array}$$

(64)

By taking the single local fractional Yang–Laplace transform for each condition given in Equations (62) and (63), we get

$$\begin{array}{c}LF{L}_{\zeta }\left(\Phi (\varkappa ,0)\right)={\displaystyle \frac{1}{{\omega }^{2\zeta }+1}},LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (\varkappa ,0)}{\partial {\tau }^{\zeta }}}\right)={\displaystyle \frac{1}{{\omega }^{\epsilon }}},\\ LF{L}_{\zeta }\left(\Phi (0,\tau )\right)={\displaystyle \frac{1}{{\upsilon }^{2\zeta }+1}},LF{L}_{\zeta }\left({\displaystyle \frac{{\partial }^{\zeta }\Phi (0,\tau )}{\partial {\varkappa }^{\zeta }}}\right)={\displaystyle \frac{1}{{\upsilon }^{\epsilon }}}.\end{array}$$

(65)

Based on the conditions given in Equation (65), Equation (64) becomes

$$F_{\zeta }^2(\omega ,\upsilon )={\displaystyle \frac{1}{{\upsilon }^{\zeta }\left({\omega }^{2\zeta }+1\right)}}+{\displaystyle \frac{1}{{\omega }^{\zeta }\left({\upsilon }^{2\zeta }+1\right)}}.$$

(66)

By applying the inverse local fractional double Yang–Laplace transform and utilizing the previously mentioned table, we obtain the solution to Equation (61):

$$\Phi (\varkappa ,\tau )={sin}_{\zeta }({\varkappa }^{\zeta })+{sin}_{\zeta }({\tau }^{\zeta }).$$

(67)

5. Conclusions

This paper introduces a novel analytical technique, the local fractional double Yang–Laplace transform, for solving local fractional partial differential equations. We establish several fundamental properties of the proposed transform, including linearity, behavior under differentiation, and a convolution theorem. The method is both conceptually accessible and computationally efficient, offering a systematic approach to obtaining exact solutions when they exist. To validate its effectiveness, we apply the local fractional double Yang–Laplace transform to solve the linear local fractional Klein–Gordon equation, demonstrating the method’s accuracy and practical utility.

The results underscore the potential of the local fractional double Yang–Laplace transform as a powerful tool for handling differential equations defined on fractal domains, particularly Cantor-type sets. Compared to conventional methods, this approach enables a more efficient derivation of exact solutions while maintaining mathematical rigor. The simplicity and robustness of the method are evident in the examples presented, and its applicability extends naturally to a wider class of problems involving local fractional models.