Analytical Solutions for the Nonlinear Partial Differential Equations Using the Conformable Triple Laplace Transform Decomposition Method

Department of Mathematics, LJIET, LJ University, Ahmedabad, Gujarat, India Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia Jabalia Camp, United Nations Relief and Works Agency (UNRWA) Palestinian Refugee Camp, Gaza Strip Jabalya, State of Palestine


Introduction
Fractional calculus has attracted many researchers in the last decades. e impact of this fractional calculus on both pure and applied branches of science and engineering has been increased. Many researchers started to approach with the discrete versions of this fractional of calculus which are summarized into two approaches: nonlocal (classical) and local. Most popular definitions in the area of nonlocal fractional calculus are the Riemann-Liouville, Caputo, and Grunwald-Letnikov definitions. e obtained fractional derivatives lack some basic properties such as chain rule and Leibniz rule for derivatives [1]. However, the semigroup properties of these fractional operators behave well in some cases. In [2], later on,  presented a new definition of a local fractional derivative, known as conformable derivative, which is well behaved and obeys the Leibniz rule and chain rule for derivatives. While conformable derivative has been criticized in [3,4], we believe that the new definition deserves to be explored further with its analysis and applications because many research studies have been conducted on this definition and its applications to various phenomena in physics and engineering. erefore, throughout this paper, we will call this definition as conformable derivative. It is defined as follows.
For a function f: (0, ∞) ⟶ R, the conformable derivative of order α ∈ (0, 1] of f at x > 0 is defined by For this derivative, Atanganana et al. (2018) presented new properties [5] which have been analysed for real valued multivariable functions [6] by Gozutok et al. (2018). In [7], conformable gradient vectors are defined, and a conformable sense Clairaut's theorem has also been proven. In [8][9][10][11][12][13][14], the researchers have worked on the linear ordinary and partial differential equations based on the conformable derivatives. Namely, two new results on homogeneous functions involving their conformable partial derivatives are introduced, specifically, homogeneity of the conformable partial derivatives of a homogeneous function and the conformable version of Euler's theorem. e conformable Laplace transform was studied and modified by Jarad et al. (2019) [15]. e conformable double Laplace transform was defined and applied in [16]. e conformable Laplace transform is not only useful to solve local conformable fractional dynamical systems but also it can be employed to solve systems within nonlocal conformable fractional derivatives that were defined and used in [17]. Finally, it is also a remarkable fact that there are a large number of studies in the theory and application of fractional differential equations based on this new definition of derivative, which have been developed in a short time. We refer to [4,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]] that many researchers have been worked on different analogues methods to solve partial fractional differential in conformable sense. Numerical and analytical techniques for solving conformable partial differential equations and conformable initial boundary value problems have been investigated in [36,37], respectively. In addition, some interesting problems have been studied in the sense of conformable derivative such as conformable gradient-based dynamical system for constrained optimization problem [38], conformable heat equation on radial symmetric plate [39], and optimal control problem for conformable heat conduction equation [40]. Several equations have been formulated with the help of conformable formulations to study their solutions such as the (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur equations [25] where the complex soliton solutions were investigated and the Date-Jimbo-Kashiwara-Miwa equation [26] where new travelling wave solutions were obtained. In addition, the conformable formulations of the coupled nonlinear Schrodinger equations [30] and Fokas-Lenells equation [35] were studied to obtain travelling wave solutions and optical solutions, respectively. e triple Laplace Adomian decomposition method and modified variational iteration Laplace transform method were studied in [27,28], respectively. Similarly, a combined method of both of the Laplace transform and resolvent kernel methods was introduced in [31]. Motivated by all these studies, we come up with the idea to study the nonlinear partial fractional differential equations by defining a function in 3-dimensional space. erefore, a conformable triple Laplace transform is defined and coupled with Adomian decomposition method to solve systematic nonlinear partial fractional differential equations. e triple Laplace transform has been rarely discussed in the literature which makes this topic as an open research topic. erefore, exploring new results concerning this interesting topic is always important. e main advantage of this method will give accurate solutions to nonlinear partial fractional derivatives in three-dimensional space.
is paper is divided into the following sections. In Section 2, some basic definitions on conformable partial derivatives are introduced. In Section 3, the main results and theorems on the conformable triple Laplace transform are investigated. In Section 4, a general nonlinear nonhomogeneous partial fractional differential equation is solved using the proposed method. In Section 5, numerical experiment is conducted using the proposed method to validate the obtained results. In Section 6, a conclusion of our research work is provided.
In the next preposition, we mention the conformable partial fractional derivative of some functions. By using eorem 1, it can be verified easily.

Definition 2.
Let the function u: (0, ∞) ⟶ R and 0 < α ≤ 1 be the piecewise continuous function. en, the conformable Laplace transform (CLT) of function u of exponential of order α is defined and denoted by Definition 3. Let u(x, y) be a piecewise continuous function on the domain D of R + × R + of exponential order α and β. en conformable double Laplace Transform (CDLT) of u(x, y) is defined and denoted by where x, y > 0, p, q ∈ ∁, α, β ∈ (0, 1]. Now, we define the conformable triple Laplace transform, for α, β, c ∈ (0, 1], and p, q, s ∈ ∁ are the Laplace variables. Definition 4. Let u(x, y, t) be a real valued piecewise continuous function of x, y, and t defined on the domain D of R + × R + × R + of exponential order α, β, and c, respectively. en, the conformable triple Laplace transform (CTLT) of u(x, y, t) is defined as follows: where p, q, s ∈ ∁ are Laplace variables and α, β, c ∈ (0, 1]. e conformable inverse triple Laplace transform, denoted by u(x, y, t), is defined by Definition 5. A unit step or Heaviside unit step function is defined as follows:

(d) e first shifting theorem for conformable triple
Laplace transform: Proof. From results (a)-(d) and (f ), it can be easily proved by using the definition of conformable triple Laplace transform (CTLT). Here only we see the proof of result (e).
So, by the definition of CTLT (equation (6)), we have By differentiating with respect to p, l-times, we get It reduces to Now, we again differentiate with respect to q and s, mand n-times, respectively, and we obtain the simplification as follows: which implies Now, we multiply (− 1) l+m+n , on both sides, and we get the required result: where H(x, y, t) is a Heaviside unit step function as defined in equation (8).
Proof. By applying the definition of CTLT, we have Now, using the definition of Heaviside unit step function H(x, y, t), we have □ Theorem 4. e conformable triple Laplace transform of the function (x α /α) l u(x, y, t) , (y β /β) m u(x, y, t), (t c /c) n u(x, y, t) and (z β+c /zy β zt c )(u(x, y, t)) is given by (t c /c)))) � (− 1) l (d l /dp l )(U α,β,c (p, q, s)) ((z β+c /zy β zt c ) (u((x α /α), (y β /β), (t c /c))))) Proof. By applying the definition of CTLT and eorem 2 (e), till equation (15) can be the required result (a). 6 International Journal of Differential Equations Remaining results (b)-(d) can be obtained via the same process. e CTLT (conformable triple Laplace transform) of the conformable partial fractional derivatives of order α, β, and c is given by Proof. Here, we go for proof of result (a), and the remaining results (b)-(g) can be proved. To obtain conformable triple Laplace transform of the fractional partial derivatives, we use integration by parts and eorem 1.
By applying the definition of CTLT, we have Since we have eorem 1, (z α (u)/zx α ) � x 1− α (zu/zx). We use this result in equation (23). erefore, equation (23) becomes e integral inside the bracket is given by By substituting equation (25) in equation (24), and simplifying, we get the required result (a), that is In general, the above results in eorem 5 can be extended as follows: International Journal of Differential Equations

Solving Nonlinear Partial Fractional Differential Equation Using the Conformable Triple Laplace Transform Decomposition Method
We consider a general nonlinear nonhomogeneous partial fractional differential equation: where m � 1, 2, 3, . . . and c ∈ (0, 1] with the initial conditions where R is the linear differential operator and N addresses the nonlinear partial fractional operator, and g � g(x α /α, y β /β, t c /c) is the source term. In order to solve equation (29), we follow the following steps: Step 1: applying the conformable triple Laplace transform to equation (29) on both sides, we have Using eorem 5 and equation (30), in equation (31), Step 2: divide by s m , and apply the conformable inverse triple Laplace transform to equation (32); it reduces to where G(x, y, t) represents the term coming from the source term and prescribed initial conditions.
Step 3: considering the conformable triple Laplace transform decomposition method, let the solution of equation (29) be an infinite series and the nonlinear term can be decomposed as where A n is called Adomian polynomials of u 1 , u 2 , u 3 , . . . , u n , and it can be calculated by the following formula: , where n � 0, 1, 2, 3, 4, . . . .
At the end, we approximate the analytical solution as follows:

Applications
In this section, a numerical experiment is done using the conformable triple Laplace decomposition method to solve nonlinear homogeneous and nonhomogeneous partial fractional differential equation in 3-dimensional space.

Solution 1. Rewrite equation (41) as
International Journal of Differential Equations Taking the conformable triple Laplace transform on both sides of equation (42), we have Recalling x, y, t))) � sU (p, q, s) − U(p, q, 0). So, equation (43) reduces to Since u(x, y, 0) � xy, we have (45) Now, by applying the conformable inverse triple Laplace transform to equation (44) and using initial condition equation (45), we obtain from equation (44) By applying the proposed method, in particular, equations (35)-(37), let u 0 (x, y, t) � xy, and the recursive relation is given by where A n is the Adomian polynomial to decompose the nonlinear terms by using the relation , where n � 0, 1, 2, 3, 4, . . .
For n � 0, We have For n � 1, (52) Using equation (51), u 0 , and u 1 and simplifying, equation (52) becomes We have Simplifying equation (54), we have For n � 2, Substituting u 0 , u 1 , and u 2 in equation (56) and simplifying, we obtain (57) erefore, we have Simplifying equation (58), we have Using initial condition (equation (63)) and taking the inverse triple Laplace transform on equation (67), we obtain By applying the proposed method, we have the following.
Let u 0 � yt − x, and the recursive relation is where A n is the Adomian polynomial to decompose the nonlinear terms by using the following relation: , where n � 0, 1, 2, 3, 4, . . . .  1  Figures 3 and 4 show the 3D graphical representations of equation (74) with various values of c and β.

Conclusion
In this work, the conformable triple Laplace transform has been investigated using all our obtained novel results and theorems.
e new conformable triple Laplace transform decomposition method is applied to find the solution of linear and nonlinear homogeneous and nonhomogeneous partial fractional differential equations. A numerical experiment has been conducted using this proposed method.
is proposed method can be applied for simultaneous two or more than two linear and nonlinear partial fractional differential equations. Note that, if we take α, β, c � 1, in Examples 1 and 2, we obtain an exact solution which was considered in [27]. Our results shed the light on the significance of exploring new generalized methods for solving partial differential equations, particularly nonlinear ones, due to the essential need to explore new analytical solutions to understand the dynamics of solutions for such important equations in physics and engineering.

Data Availability
No data were used to support this study.