# Analytical Solutions to Two-Dimensional Nonlinear Telegraph Equations Using the Conformable Triple Laplace Transform Iterative Method

Authors: Alemayehu Tamirie Deresse
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The conformable fractional triple Laplace transform approach, in conjunction with the new Iterative method, is used to examine the exact analytical solutions of the (2 + 1)-dimensional nonlinear conformable fractional Telegraph equation. All the fractional derivatives are in a conformable sense. Some basic properties and theorems for conformable triple Laplace transform are presented and proved. The linear part of the considered problem is solved using the conformable fractional triple Laplace transform method, while the noise terms of the nonlinear part of the equation are removed using the novel Iterative method’s consecutive iteration procedure, and a single iteration yields the exact solution. As a result, the proposed method has the benefit of giving an exact solution that can be applied analytically to the presented issues. To confirm the performance, correctness, and efficiency of the provided technique, two test modeling problems from mathematical physics, nonlinear conformable fractional Telegraph equations, are used. According to the findings, the proposed method is being used to solve additional forms of nonlinear fractional partial differential equation systems. Moreover, the conformable fractional triple Laplace transform iterative method has a small computational size as compared to other methods.


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## Body

## 1. Introduction

The fractional formulation of differential equations is a development of the fractional calculus, which was initially introduced in 1695 when L’Hôpital and Leibniz addressed the expansion of the integer-order derivative to the order 1/2 derivative. Both Euler and Lacroix researched the fractional-order derivative and defined it using the power function’s nth derivative formulation [1]. Fractional partial differential equations (FPDEs) have become increasingly important in recent years for modeling a wide range of applications in real-world sciences and engineering, including fluid dynamics, mathematical biology, electrical circuits, optics, and quantum mechanics [2]. As a result, many researchers have focused on solving FPDEs in recent decades [3, 4]. Since many physical and mechanical systems contain internal damping, which makes it impossible to derive equations describing the physical behavior of a non-conservative system using the traditional energy-based approach, fractional derivative formulations can be used to model them more accurately. In non-conservative systems, fractional derivative formulations can be constructed by minimizing specific functionals containing fractional derivative terms using techniques from the calculus of variations [5]. Many definitions of fractional derivatives and integrals have been published in the literature, including Riemann-Liouville fractional definitions [6], Caputo fractional definitions [7], Grünwald-Letnikov fractional derivatives [8], and Hadamard fractional integral [9]. All known fractional derivatives satisfy one of the well-known properties of classical derivatives, namely, the linear property. However, the other properties of classical derivatives, such as the derivatives of a constant are zero, the product rule, quotient rule, and the chain rule either do not hold or are too complicated for many fractional derivatives. For instance, Daα1=0 does not fulfill the Riemann-Liouville definition. In Caputo’s definition, fx is assumed to be differentiable; otherwise, one cannot use such a definition. Moreover, Liouville’s theorem in the fractional setting does not hold. Therefore, it is clear that all definitions of fractional derivatives seem deficient regarding certain mathematical properties, such as Rolle’s theorem and the mean value theorem [10].To resolve these issues, Khalil et al. [11] recently introduced a novel fractional derivative called conformable fractional derivative (CFD) in 2014. This definition is formulated as follows:Definition 1 (see [12–14]).
For the initial real valuea, the conformable fractional derivative Daαfx of a real functionf:a,∞⟶ℝ, α∈0,1 is defined as
(1)Daαfx=limh⟶0fx+hx−a1−α−fxh,forallx>a,α∈0,1.
The initial valuea can be zero, and if the limit exists,fx is said to be partially α−differentiable at t>0.The CFD’s Definition1 is very similar to the classical derivative. It depends upon the basic limit definition and consequently allows the easier extension of some typical theorems in calculus that the existing definitions of fractional derivatives did not allow, due to its simple nature. Along with the CFD’s Definition 1, various classical properties, such as the mean value theorem and the product, quotient, and chain rules, are fulfilled. Moreover, this definition is provided by the Leibniz rule, which other fractional derivatives cannot achieve (see [15]). Another study [16] conducted by Abdeljawad presented the left and right conformable fractional derivatives and fractional integrals of higher order concepts. In addition, the authors also defined the fractional chain rule, fractional integration by parts formulae, Gronwall inequality, fractional power series expansion, and fractional Laplace transform. Following this definition, a new approach for finding fractional operators was introduced by Antagan and Baleanu [17] with a nonsingular Mittag-Leffler kernel with a memory effect. Growing attention has been paid to exploring the conformable fractional derivative due to the enormous number of its meaningful applications in many fields of science. Recently, in [18], Rabha et al. introduced different vitalization of the growth of COVID-19 by using controller terms based on the concept of conformable calculus. Ghanbari et al. [19] studied the dynamic behavior of allelopathic stimulator phytoplankton species with Mittag-Leffler (ML) law by using the Atangana-Baleanu fractional derivative (ABC). The interested reader might consult the monograph [20–22] for more information.The conformable telegraph equations have a wide range of applications in science and engineering, with the most common application being in optimizing propagation-oriented and propagating electrical communication systems [23, 24]. Therefore, as one of the crucial equations in different fields of sciences, many scholars have recently focused their efforts on investigating the solutions of conformable fractional telegraph equations using various methodologies. Using a double conformable Sumudu matching transformation approach, [25] discovered accurate and convergent numerical solutions of linear space-time matching telegraph fractional equations in 2021. Using the cosine family of linear operators, Bouaouid et al. [26] established the existence, uniqueness, and stability of the integral solution of a nonlocal telegraph equation in the conformable time-fractional derivative (see [12, 27–30] for more related work on the solution of conformable telegraph equations).Because the Laplace transform method (LTM) [13, 14, 31] is an integral transform method for getting the approximate and precise solutions of FDEs, many authors are still working hard to develop and generalize this transform so that it can be used with the newly created fractional derivatives and integrals. For instance, the authors of the paper [32] present a fractional Laplace transform in terms of conformable fractional-order Bessel functions (CFBFs). They also established several important formulas of the fractional Laplace integral operator acting on the CFBFs and give the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of the fractional Laplace transform method. Ozan o¨zkan and Ali Kurt in 2018 proposed a new generalization of the double Laplace transform called the conformable double Laplace transform (CDLT), which they used to solve the conformable fractional partial heat equation and the conformable fractional partial Telegraph equation [33]. This method was later used by many authors to handle a variety of real-world challenges resulting from various occurrences such as conformable fractional partial differential equations, Singular conformable pseudoparabolic equations, and system of conformable fractional differential equations [34–36].Several researchers have recently extended the conformable double Laplace transform method to the conformable triple Laplace transform method (CTLTM) to obtain the exact/approximate solution of two-dimensional nonlinear CFDEs that occur in a variety of natural events. The conformable triple Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse triple conformable Laplace transform. In comparison to other known approaches, the conformable triple Laplace transform method provides rapid convergence of the exact solution without any restrictive assumptions about the answer. Unfortunately, some complex nonlinear partial differential equations that arise in mathematical physics, engineering, and other relevant branches of research that involve nonlinear phenomena are not solved by this technique. In such cases, the conformable triple Laplace transform method is frequently used with other efficient approaches, such as the Adomian decomposition method and homotopy analysis method to tackle a similar problem. For example, the conformable triple Laplace transform decomposition was employed by the authors in [37] to discover the solution of linear and nonlinear homogeneous and nonhomogeneous partial fractional differential equations. This method’s important result and theorems are also discussed. In 2022, [38] gives some key discoveries on conformable fractional partial derivatives and shows how to solve nonlinear partial differential equations in two dimensions using the conformable triple Laplace and Sumudu transform method in conjunction with the Adomian decomposition approach. In paper [39], the authors present the solution of the incompressible second-grade fluid models by using the generalized ρ-Laplace transform method in conjunction with the homotopy analysis method in the sense of the Liouville-Caputo fractional derivative.The main objective of this paper is to introduce the new method called the conformable triple Laplace transform iterative method (CTLTIM) to investigate an accurate solution to the two-dimensional nonlinear conformable telegraph equation under the given initial and boundary conditions. This method is the combination of the two powerful techniques, the conformable triple Laplace transform method (CTLTM) and the new iterative method (NIM) introduced by Daftardar-Gejji and Jafari [40]. In practical scientific areas, solving integer and fractional-order nonlinear differential equations with linear and nonlinear ordinary and partial differential equations utilizing the NIM is a fascinating problem [41]. The iterative strategy employed in this method produces a series that can be summed to obtain an analytical formula or utilized to construct an appropriate approximation with a faster convergent series solution [42, 43]. The approximation error can be reduced by properly truncating the series [44]. Recently, the NIM is combined with other known methods like the Sumudu transform method and Laplace transform method to obtain the approximate or exact solution of the nonlinear partial differential equation. The authors of the paper [45] successfully implemented the combined double Sumudu transform with the iterative method to get the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. Through this approach, the solution of the linear part was solved by the double Sumudu transform method, and the nonlinear part of the problem was solved by a successive iterative method. Deresse et al. [46] present the triple Laplace transform coupled with an iterative method to obtain the exact solution of the two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation vanished by a successive iterative method. As a result, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. However, the amalgamation of the conformable triple Laplace transform method and the new iterative method that is CTLTIM has not previously been studied to solve the two-dimensional nonlinear fractional telegraph equations; this is the main motivation of the current research work.The proposed CTLTIM has been utilized to solve the problems as follows. First, the source termfxβ/β,yγ/γ,tα/α of the considered problem (2) is decomposed into two functions namely f1xβ/β,yγ/γ,tα/α and f2xβ/β,yγ/γ,tα/α. The importance of this decomposition is that the part f1xβ/β,yγ/γ,tα/α with the terms in Equation (2) always leads to the simple algebraic expression while applying the conformable triple Laplace transform and the part f2xβ/β,yγ/γ,tα/α is combined with the nonlinear term of Equation (2) to avoid noise terms in the iteration process. Next, the conformable triple Laplace transform method is applied to the linear part of the problem. Finally, the successive iterative method is applied to the nonlinear part of the problem as it introduced in Section 4. While applying this iterative method, the noise terms in the iteration process are avoided, and a single iteration gives the exact solution. Therefore, using the described method one can obtain the exact solution to nonlinear partial fractional derivatives with less computational size. Moreover, the proposed approach allows the user with analytical approximation, and it is applied directly to the problems without requiring any discretization, linearization, or perturbation parameters like Adomian polynomials and SOS polynomials ([42, 47] see the references therein). This is the main advantage of the proposed method CFTLTIM over the other existing approaches in the literature.The following two-dimensional nonlinear conformable telegraph equation was the subject of the current study (forα=β=1 see [30]):
(2)∂2αu∂t2α+a∂αu∂tα+bNuxββ,yγγ,tαα=c∂2βu∂x2β+∂2γu∂y2γ+fxββ,yγγ,tαα,xββ,yγγ,tαα>0&α,β,γ∈0,1,depending on the starting conditions(3)uxββ,yγγ,0=φ1xββ,yγγ,∂α∂tαuxββ,yγγ,0=φ2xββ,yγγ,and boundary conditions (Cauchy type BCs)
(4)u0,yγγ,tαα=g1yγγ,tαα,∂β∂xβu0,yγγ,tαα=g2yγγ,tαα,(5)uxββ,0,tαα=g3xββ,tαα,∂γ∂yγuxββ,0,tαα=g4xββ,tαα,where a,b and c are known real constants; uxβ/β,yγ/γ,tα/α denotes either the voltage or current through the two-dimensional conductor at position x,y at the time t; N is the general continuous nonlinear resorting term; and fxβ/β,yγ/γ,tα/α is the source term which is assumed to be analytic in time t. Equation (2) reduces to the undamped telegraph equation in two space variables when α=0, and to the damped one when α>0.The rest of the paper is organized as follows: Section2 covers the definitions, properties, and theorems of conformable fractional derivatives. Section 3 contains some basic CFTLTM definitions, properties, and theorem proofs. In Section 4, the details of the new iterative method and its convergence are discussed. Section 5 displays the model’s description and how CTLTIM is used to obtain the exact analytical solutions to the specified conformable fractional telegraph equations. In Section 6, we demonstrate the proposed method’s reliability, convergence, and efficiency using two exemplary instances. Finally, Section 7 outlines concluding observations.


## 2. Conformable Fractional Derivative

This section introduces the essential definitions and features of conformable fractional partial derivatives, which are then applied to the current topic.Definition 2 (see [48]).
The fractional derivative of a suitable mappingf by Riemann-Liouville is given as
(6)Dxαfx=1Гn−αddxn∫0xx−tn−α−1ftdt,n−1<α≤n.
Caputo fractional derivative of a suitable mappingf is given as
(7)Dxαfx=1Гn−α∫0xx−tn−α−1fntdt,n−1<α≤n,where α is the order of fractional derivative and α∈Z+.
If this limit exists,f is said to be partially α−differentiable at t>0.Definition 3 (see [37, 38]).
Given a functionf:ℝ+×ℝ+×ℝ+, then the conformable partial fractional derivatives (CPFDs) of fxβ/β,yγ/γ,tα/α having order α,βandγ are defined by.
(8)∂xβf=∂β∂xβfxββ,yγγ,tαα=limh⟶0fxβ/β+hx1−β,yγ/γ,tα/α−fxβ/β,yγ/γ,tα/αh,∂yγf=∂γ∂yγfxββ,yγγ,tαα=limk⟶0fxβ/β,yγ/γ+ky1−γ,tα/α−fxβ/β,yγ/γ,tα/αk,∂tαf=∂α∂tαfxββ,yγγ,tαα=limε⟶0fxβ/β,yγ/γ,tα/α+εt1−α−fxβ/β,yγ/γ,tα/αε,where 0<α,β,γ≤1,xβ/β,yγ/γ,tα/α>0, and ∂xβf=∂β/∂xβfxβ/β,yγ/γ,tα/α,∂yγf=∂γ/∂yγfxβ/β,yγ/γ,tα/α, and ∂tαf=∂α/∂tαfxβ/β,yγ/γ,tα/α are called the fractional partial derivatives of orders α,βandγ, respectively.Theorem 4 (see [37, 38]).
Letα,β,γ∈0,1 and fxβ/β,yγ/γ,tα/α be differentiable at a point forxβ/β,yγ/γ,tα/α>0. Then,
(1)
∂xβfxβ/β,yγ/γ,tα/α=∂β/∂xβfxβ/β,yγ/γ,tα/α=x1−β∂/∂xfxβ/β,yγ/γ,tα/α.(2)
∂yγfxβ/β,yγ/γ,tα/α=∂γ/∂yγfxβ/β,yγ/γ,tα/α=y1−γ∂/∂yfxβ/β,yγ/γ,tα/α.(3)
∂tαfxβ/β,yγ/γ,tα/α=∂α/∂tαfxβ/β,yγ/γ,tα/α=t1−α∂/∂tfxβ/β,yγ/γ,tα/α.Proof (1).
With the help of CFPD definition, we have(9)∂xβf=∂β∂xβfxββ,yγγ,tαα=limh⟶0fxβ/β+hx1−β,yγ/γ,tα/α−fxβ/β,yγ/γ,tα/αh.
Usingλ=hx1−β in the above equation, we get
(10)∂xβf=∂β∂xβfxββ,yγγ,tαα=limλ⟶0fxβ/β+λ,yγ/γ,tα/α−fxβ/β,yγ/γ,tα/αλxβ−1,=x1−βlimλ⟶0fxβ/β+λ,yγ/γ,tα/α−fxβ/β,yγ/γ,tα/αλ=x1−βfxββ,yγγ,tαα.
We can prove the results of (2) and (3) in the same way.Proposition 5.
Letα,β,γ∈0,1 and a,b,c∈ℝ, l,m,n∈ℕ. Then, we have the following:
(i)
∂β/∂xβafxβ/β,yγ/γ,tα/α+bgxβ/β,yγ/γ,tα/α=a∂β/∂xβfxβ/β,yγ/γ,tα/α+b∂β/∂xβgxβ/β,yγ/γ,tα/α(ii)
∂β/∂xβfgxβ/β,yγ/γ,tα/α=gxβ/β,yγ/γ,tα/α∂β/∂xβfxβ/β,yγ/γ,tα/α+fxβ/β,yγ/γ,tα/α∂β/∂xβgxβ/β,yγ/γ,tα/α(iii)
∂β/∂xβf/gxβ/β,yγ/γ,tα/α=gxβ/β,yγ/γ,tα/α∂β/∂xβfxβ/β,yγ/γ,tα/α−fxβ/β,yγ/γ,tα/α∂β/∂xβgxβ/β,yγ/γ,tα/α/gxβ/β,yγ/γ,tα/α2
provided thatgxβ/β,yγ/γ,tα/α≠0.
(iv)
∂α/∂tαfxβ/β,yγ/γ,tα/α=0, if xβ/β,yγ/γ,tα/α was a function depending only on xβ/β&yγ/γ(v)
∂β+γ+α/∂xβ∂yγ∂tαxβ/βmyγ/γntα/αl=mnlxβ/βm−βyγ/γn−γtα/αl−α(vi)
∂β/∂xβxβ/βmyγ/γntα/αl=mxβ/βm−1yγ/γntα/αl,∂γ/∂yγxβ/βmyγ/γntα/αl=nxβ/βmyγ/γn−1tα/αland ∂α/∂tαxβ/βmyγ/γntα/αl=lxβ/βmyγ/γntα/αl−1(vii)
∂β/∂xβeaxβ/β+byγ/γ+ctα/α=aeaxβ/β+byγ/γ+ctα/α,∂γ/∂yγeaxβ/β+byγ/γ+ctα/α=beaxβ/β+byγ/γ+ctα/α and ∂α/∂tαeaxβ/β+byγ/γ+ctα/α=ceaxβ/β+byγ/γ+ctα/α(viii)
∂β/∂xβsinxβ/βcostα/α=cosxβ/βcostα/α and∂γ/∂yγsinxβ/βcosyγ/γcostα/α=−sinxβ/βsinyγ/γcostα/α.


## 3. Some Results and Theorems of the Conformable Triple Laplace Transform

In this section, we will go over the fundamental concepts of fractional conformable Laplace transforms as well as certain results that will be useful later. The conformable triple Laplace transform is also defined (see [36–39, 46] for more information).Definition 6.
Let the functionu:0,∞⟶ℝ and 0<α≤1. The conformable Laplace transform (CLT) of the exponential of order α function utα/α is thus defined and denoted by
(11)Ltαutαα=Uαs=∫0∞e−stααutααtα−1dt,t>0.Definition 7.
Letuxβ/β,yγ/γ be a piece-wise continuous function of exponential order on the domain D of ℝ+×ℝ+. After that, the conformable double Laplace transform (FCDLT) of uxβ/β,yγ/γ is defined and denoted by
(12)LxβLyγuxββ,yγγ=Uβ,γk,p=∫0∞∫0∞e−βxββ−γyγγuxββ,yγγxβ−1yγ−1dxdy,where xβ/β,yγ/γ>0,k,p∈ℂ,β,γ∈0,1.
Now, we define fractional conformable triple Laplace transform, forα,β,γ∈0,1 and k,p,s∈ℂ are the Laplace variables.Definition 8.
Letuxβ/β,yγ/γ,tα/α be a piece-wise continuous function of exponential order on the domain D of ℝ+×ℝ+×ℝ+. After that, the conformable triple Laplace transform (FCDLT) of uxβ/β,yγ/γ,tα/α is defined and denoted by
(13)LxβLyγLtαuxββ,yγγ,tαα=Uα,β,γk,p,s=∫0∞∫0∞∫0∞e−kxββ−pyγγ−stααuxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt,where k,p,s∈ℂ are Laplace variables of xβ/β,yγ/γandtα/α>0,, respectively, and α,β,γ∈0,1.The conformable inverse triple Laplace transform, abbreviated byuxβ/β,yγ/γ,tα/α, is defined as follows:
(14)uxββ,yγγ,tαα=Lk−1Lp−1Ls−1Uα,β,γk,p,s=12πi∫β−i∞β+i∞ekxββ12πi∫γ−i∞γ+i∞epyγγ12πi∫α−i∞α+i∞estααUα,β,γk,p,sdsdpdk.Definition 9.
The following is the definition of a unit step or Heaviside unit step function:(15)Hxββ−a,yγγ−b,tαα−c=1;xββ>a,yγγ>b,tαα>c0;xββ<a,yγγ<b,tαα<c.Theorem 10.
IfLxβLyγLtαuxβ/β,yγ/γ,tα/α=Uα,β,γk,p,s, LxβLyγLtαvxβ/β,yγ/γ,tα/α=Vα,β,γsk,p,s, and a,b andc are constants then the followings hold:
(1)
Linearity property:(16)LxβLyγLtαauxββ,yγγ,tαα+bvxββ,yγγ,tαα=aLxβLyγLtαuxββ,yγγ,tαα+bLxβLyγLtαvxββ,yγγ,tαα=aUα,β,γk,p,s+bVα,β,γk,p,s(2)
LxβLyγLtαc=c/kps, where c is the constant(3)
LxβLyγLtαxβ/βmyγ/γntα/αl=Γm+1Γn+1Γl+1/km+1pn+1sl+1, where Γ. is the gamma function. Note that Γn+1=n!, for n=0,1,2,3,⋯.(4)
LxβLyγLtαeaxβ/β+byγ/γ+ctα/α=LxLyLteax+by+ct=1/k−ap−bs−c and LxβLyγLtαeaxβ/β−byγ/γ−ctα/α=LxLyLteax−by−ct=1/k+ap+bs+c(5)
The conformable triple Laplace transform’s first shifting theorem:
IfLxβLyγLtαuxβ/β,yγ/γ,tα/α=Uα,β,γk,p,s, then LxβLyγLtαeaxβ/β+byγ/γ+ctα/αuxβ/β,yγ/γ,tα/α=Uα,β,γk−a,p−b,s−c.
(6)
LxβLyγLtαsinaxβ/βsinbyγ/γsinctα/α=abc/k2+a2p2+b2s2+c2,LxβLyγLtαcosaxβ/βcosbyγ/γcosctα/α=kps/k2+a2p2+b2s2+c2(7)
IfLxβLyγLtαuxβ/β,yγ/γ,tα/α=Uα,β,γk,p,s, then LxβLyγLtαxβ/βmyγ/γntα/αluxβ/β,yγ/γ,tα/α=−1m+n+ldm+n+l/dkmdpndslUα,β,γk,p,s.Proof (7).
Using the definition of the conformable triple Laplace transform method, the demonstration of outcomes 1–6 is simple. As a result, we will show how to prove result7 using the conformable triple Laplace transform definition.
(17)LxβLyγLtαuxββ,yγγ,tαα=Uα,β,γk,p,s=∫0∞∫0∞∫0∞e−kxββ−pyγγ−stααuxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
Differentiating with respect tok,m−times, we get
(18)dmdkmUα,β,γk,p,s=dmdkm∫0∞∫0∞∫0∞e−kxββ−pyγγ−stααuxββ,yγγ,tαα×xβ−1yγ−1tα−1dxdydt,=∫0∞∫0∞∫0∞dmdkme−kxββ−pyγγ−stαα×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
This suggests that(19)dmdkmUα,β,γk,p,s=∫0∞∫0∞∫0∞−xββme−kxββ−pyγγ−stαα×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
Differentiating (19) with respect to p,n−times, produces
(20)dndpnUα,β,γk,p,s=∫0∞∫0∞∫0∞−yγγne−kxββ−pyγγ−stαα×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
Again, differentiating (20) with respect to s,l−times, we obtain
(21)dldslUα,β,γk,p,s=∫0∞∫0∞∫0∞−tααle−kxββ−pyγγ−stαα×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
Using Equations (19), (20), and (21), we get
(22)dm+n+ldkmdpndslnUα,β,γk,p,s=∫0∞∫0∞∫0∞−xββm−yγγn−tααle−kxββ−pyγγ−stαα×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt,=−1m+n+l∫0∞∫0∞∫0∞e−kxββ−pyγγ−stαα×xββmyγγntααl×uxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
This suggests that(23)dm+n+ldkmdpndslUα,β,γk,p,s=−1m+n+lLxβLyγLtαxββmyγγntααluxββ,yγγ,tαα.
Multiplying both sides of the Equation (23) by −1m+n+l, we obtain
(24)LxβLyγLtαxββmyγγntααluxββ,yγγ,tαα=−1m+n+ldm+n+ldkmdpndslUα,β,γk,p,s.Theorem 11.
IfLxβLyγLtαuxβ/β,yγ/γ,tα/α=Uα,β,γk,p,s, then
(25)LxβLyγLtαuxββ−ξββ,yγγ−ηγγ,tαα−μαα×Hxββ−ξββ,yγγ−ηγγ,tαα−μαα=e−kξββ−pηγγ−sμααUα,β,γk,p,s,where the Heaviside unit step function Hx,y,tis defined as in Equation (15)
(see [37, 38] for the proof).Theorem 12.
Forα,β,γ∈0,1. Let uxβ/β,yγ/γ,tα/α be the real-valued piece-wise continuous function defined on the domain 0,∞×0,∞×0,∞. The CFTLT of the conformable partial fractional derivatives of order α,β,andγ is given bySS
(1)
LxβLyγLtα∂β/∂xβuxβ/β,yγ/γ,tα/α=kUk,p,s−U0,p,s(2)
LxβLyγLtα∂γ/∂yγuxβ/β,yγ/γ,tα/α=pUk,p,s−Uk,0,s(3)
LxβLyγLtα∂α/∂tαuxβ/β,yγ/γ,tα/α=sUk,p,s−Uk,p,0(4)
LxβLyγLtα∂2β/∂x2βuxβ/β,yγ/γ,tα/α=k2Uk,p,s−kU0,p,s−∂/∂xU0,p,s(5)
LxβLyγLtα∂2γ/∂y2γuxβ/β,yγ/γ,tα/α=p2Uk,p,s−pUk,0,s−∂/∂yUk,0,s(6)
LxβLyγLtα∂2α/∂t2αuxβ/β,yγ/γ,tα/α=s2Uk,p,s−sUk,p,0−∂/∂tUk,p,0(7)
LxβLyγLtα∂3β/∂x3βuxβ/β,yγ/γ,tα/α=k3Uk,p,s−k2U0,p,s−∂/∂xU0,p,s−∂2/∂x2U0,p,s(8)
LxβLyγLtα∂β+γ+α/∂xβ∂yγ∂αuxβ/β,yγ/γ,tα/α=kpsUk,p,s−kUk,0,0−pU0,p,0−sU0,0,s−kpUk,p,0−ksUk,0,s−psU0,p,s−U0,0,0Proof (1).
Using the CFTLT definition (6), we have
(26)LxβLyγLtα∂β∂xβuxββ,yγγ,tαα=∫0∞∫0∞∫0∞e−kxββ−pyγγ−stαα∂β∂xβuxββ,yγγ,tααxβ−1yγ−1tα−1dxdydt.
By using Theorem4, we have
(27)∂β∂xβuxββ,yγγ,tαα=x1−β∂∂xuxββ,yγγ,tαα.
Then, Equation (26) reduced to
(28)LxβLyγLtα∂β∂xβuxββ,yγγ,tαα=∫0∞∫0∞e−pyγγ−stαα∫0∞e−kxββ∂∂xuxββ,yγγ,tααdxyγ−1tα−1dydt.
Taking integration by parts and Theorem4 to the integrals inside the bracket produces
(29)∫0∞e−kxββ∂∂xuxββ,yγγ,tααdx=kUk,y,t−U0,y,t.
We get the needed outcome by substituting Equation (29) into Equation (28), then simplifying LxβLyγLtα∂β/∂xβuxβ/β,yγ/γ,tα/α=kUk,p,s−U0,p,s, and this result can be generalized to
(30)LxβLyγLtα∂mβ∂xmβuxββ,yγγ,tαα=kmUk,p,s−∑n=0m−1km−1−n∂n∂xU0,p,s.
The process outlined above can be used to receive verification of the remaining results.


## 4. Basic Idea of the New Iterative Method(NIM)

Consider the following general functional equation [40] for the main principle of the new iterative method:
(31)uxββ,yγγ,tαα=Nuxββ,yγγ,tαα+fxββ,yγγ,tαα,where N is a nonlinear operator in a Banach space such that N:B⟶B and f is a known function.We are looking for a solutionu of the Equation (31) having the series form:
(32)uxββ,yγγ,tαα=∑i=0∞uixββ,yγγ,tαα.The nonlinear operationN can then be decomposed as
(33)N∑i=0∞ui=Nu0+∑i=1∞uiN∑r=0iur−N∑r=0i−1ur.From Equations (33) and (32), Equation (31) is equivalent to
(34)∑i=0∞ui=f+Nu0+∑i=1∞uiN∑r=0iur−N∑r=0i−1ur.Equation (34) yields the following recurrence relation:
(35)G0=u0=f0,(36)G1=u1=Nu0,(37)Gm=um+1=Nu0+⋯+um−Nu0+⋯+um−1,m=1,2,⋯Then,(38)u1+⋯+um+1=Nu0+⋯+um,m=1,2,⋯,and hence,
(39)u=∑i=0∞ui=f+N∑i=1∞ui.As a result, them−term approximate solution of Equation (31) is defined as follows:
(40)u=u0+u1+u2+⋯+um−1,m>1.


### 4.1. Convergence of the NIM

The conditions for the series (32) convergence are presented in this subsection. And [41] is a good place to start for more information.Theorem 13.
IfN is a continuously differentiable functional in a neighborhood of u0 and Nnu0≤L, for each n and for some real L>0 and ui≤M<1/e,i=1,2,3,⋯, then the series ∑n=0∞Gn is absolutely convergent and moreover, Gn≤LMnen−1e−1,n=1,2,⋯.Theorem 14.
IfN is a continuously differentiable functional in a neighborhood of u0 and Nnu0≤M≤1/e for all n, then the series ∑n=0∞Gn is absolutely convergent.


## 4.1. Convergence of the NIM

The conditions for the series (32) convergence are presented in this subsection. And [41] is a good place to start for more information.Theorem 13.
IfN is a continuously differentiable functional in a neighborhood of u0 and Nnu0≤L, for each n and for some real L>0 and ui≤M<1/e,i=1,2,3,⋯, then the series ∑n=0∞Gn is absolutely convergent and moreover, Gn≤LMnen−1e−1,n=1,2,⋯.Theorem 14.
IfN is a continuously differentiable functional in a neighborhood of u0 and Nnu0≤M≤1/e for all n, then the series ∑n=0∞Gn is absolutely convergent.


## 5. Description of the Model

To solve the problem (2)-(5) by using the proposed method first, the source term fxβ/β,yγ/γ,tα/α must be decomposed into two functions namely f1xβ/β,yγ/γ,tα/α and f2xβ/β,yγ/γ,tα/α as explained in the introduction section. The part f1xβ/β,yγ/γ,tα/α with the terms in Equation (2) always leads to the simple algebraic expression while applying the fractional conformable triple Laplace transform. The part f2xβ/β,yγ/γ,tα/α is combined with the nonlinear term of Equation (2) to avoid noise terms in the iteration process.The following are the steps to determine the analytical solution of Equations (2)-(6) using the proposed techniques:Step 1.
On both sides of Equation (2), we use the properties of the conformable triple Laplace transform and Theorem (12) to obtain
(41)s2Uk,p,s−sUk,p,0−∂∂tUk,p,0+asUk,p,s−Uk,p,0=bLxβLyγLtαNuxββ,yγγ,tαα=ck2Uk,p,s−kU0,p,s−∂∂xU0,p,s+p2Uk,p,s−pUk,0,s−∂∂yUk,0,s+f¯1k,p,s+LxβLyγLtαf2xββ,yγγ,tαα.
Here,f¯1k,p,s is the conformable triple Laplace transform of f1xβ/β,yγ/γ,tα/α..Step 2.
We get the following result by applying the conformable double Laplace transform to the initial circumstance (2):
(42)Uk,p,0=φ1¯k,p,∂∂tUk,p,0=φ2¯k,p.Step 3.
The conformable double Laplace transform is applied to the boundary Conditions (4) and (5) to obtain
(43)U0,p,s=g1¯p,s,∂∂xU0,p,s=g2¯p,s,(44)Uk,0,s=g3¯k,s,∂∂yUk,0,s=g4¯k,s.Step 4.
Using Equations (42), (43), and (44) into the Equation (41), and simplifying, we obtain
(45)Uk,p,s=1s2+as−ck2−cp2s+aφ1¯k,p+φ2¯k,p−ckg1¯p,s+g2¯p,s+pg3¯Uk,s+g4¯Uk,s+f¯1k,p,s+LxβLyγLtαf2xββ,yγγ,tαα−bNuxββ,yγγ,tαα.Step 5.
Taking Equation (43) and applying the inverse conformable triple Laplace transform, we get
(46)ux,y,t=Lxyt−11s2+as−ck2−cp2s+aφ1¯k,p+φ2¯k,p−ckg1¯p,s+g2¯p,s+pg3¯Uk,s+g4¯Uk,s+f¯1k,p,s+LxβLyγLtαf2xββ,yγγ,tαα−bNuxββ,yγγ,tαα.Step 6.
Applying the iterative process to Equation (46), assume that the given problem (2) has the series solution of the form:
(47)uxββ,yγγ,tαα=∑i=0∞uixββ,yγγ,tαα.Step 7.
Plugging Equation (47) into Equation (46), we obtain
(48)∑i=0∞uixββ,yγγ,tαα=Lxyt−11s2+as−ck2−cp2s+aφ1¯k,p+φ2¯k,p+f¯1k,p,s−ckg1¯p,s+g2¯p,s+pg3¯Uk,s+g4¯Uk,s+LxβLyγLtαf2xββ,yγγ,tαα−bNuxββ,yγγ,tαα.Step 8.
Decomposing the nonlinear termNuxβ/β,yγ/γ,tα/α in Equation (48) as follows:
(49)N∑i=0∞uixββ,yγγ,tαα=Nu0xββ,yγγ,tαα+∑i=1∞uixββ,yγγ,tααN∑r=0iurxββ,yγγ,tαα−N∑r=0i−1urxββ,yγγ,tαα.Step 9.
Using the Equation (49) in Equation (48) produces
(50)∑i=0∞uixββ,yγγ,tαα=Lxyt−11s2+as−ck2−cp2s+aφ1¯k,p+φ2¯k,p+f¯1k,p,s−ckg1¯p,s+g2¯p,s+pg3¯Uk,s+g4¯Uk,s+LxβLyγLtαf2xββ,yγγ,tαα−bNu0+∑i=1∞uiN∑r=0iur−N∑r=0i−1ur.Step 10.
The following recurrence relations are defined from Equation (50) according to iteration ((35)-(37)):
(51)u0xββ,yγγ,tαα=Lxyt−11s2+as−ck2−cp2s+aφ1¯k,p+φ2¯k,p+f¯1k,p,s−ckg1¯p,s+g2¯p,s+pg3¯Uk,s+g4¯Uk,s,(52)u1xββ,yγγ,tαα=Lxyt−11s2+as−ck2−cp2LxβLyγLtαf2xββ,yγγ,tαα−bNu0xββ,yγγ,tαα,(53)um+1xββ,yγγ,tαα=Lxyt−11s2+as−ck2−cp2LxβLyγLtαf2xββ,yγγ,tαα−b∑i=1∞uiN∑r=0iur−N∑r=0i−1ur,n≥1.Step 11.
The series solution to the given problem (2)–(5) can be found by using (40) as follows:
(54)uxββ,yγγ,tαα=u0xββ,yγγ,tαα+u1xββ,yγγ,tαα+u2xββ,yγγ,tαα+⋯+umxββ,yγγ,tαα+.⋯


## 6. Application

Two test instances will be examined in this part to assess the performance of the suggested method.Example 1.
Consider the following two-dimensional nonlinear conformable telegraph equation in the regionΩ=0,22 as follows:
(55)∂2αu∂t2α+2∂αu∂tα=∂2βu∂x2β+∂2γu∂y2γ−u2+e2xβ/β+2yγ/γ−4tα/α−2exβ/β+yγ/γ−2tα/α,with preliminary conditions
(56)uxββ,yγγ,0=exβ/β+yγ/γ,∂α∂tαuxββ,yγγ,0=−2exβ/β+yγ/γ,and boundary conditions
(57)u0,yγγ,tαα=eyγ/γ−2tα/α,∂β∂xβu0,yγγ,tαα=eyγ/γ−2tα/α,(58)uxββ,0,tαα=exβ/β−2tα/α,∂γ∂yγuxββ,0,tαα=exβ/β−2tα/α.Solution: To both sides of Equation (55), we use the properties of the conformable triple Laplace transform and Theorem (12) to get
(59)s2Uk,p,s−sUk,p,0−∂∂tUk,p,0+2sUk,p,s−Uk,p,0=k2Uk,p,s−kU0,p,s−∂∂xU0,p,s+p2Uk,p,s−pUk,0,s−∂∂yUk,0,s−2k−1p−1s+2+LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−u2xββ,yγγ,tαα.We get the following results by applying the conformable fractional double Laplace transform to the initial condition (56).
(60)Uk,p,0=1k−1p−1,∂∂tUk,p,0=−2k−1p−1.The boundary conditions (57) and (58) are transformed using the conformable double Laplace transform as follows:
(61)U0,p,s=1p−1s+2,∂∂xU0,p,s=1p−1s+2,(62)Uk,0,s=1k−1s+2,∂∂yUk,0,s=1k−1s+2.Substituting Equations (60), (61), and (62) into Equation (59), and simplifying, we obtain
(63)Uk,p,s=1s2+2s−k2−p2s+2k−1p−1−k+1p−1s+2−p+1k−1s+2−2k−1p−1s+2+LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−u2xββ,yγγ,tαα.Again, performing some mathematical manipulation to Equation (63), it is reduced to
(64)Uk,p,s=2k−1p−1s+2+1s2+2s−k2−p2LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−u2xββ,yγγ,tαα.Taking the inverse conformable triple-Laplace transform to both sides of Equation (62), we get
(65)uxββ,yγγ,tαα=exβ/β+yγ/γ−2tα/α+Lxyt−11s2+2s−k2−p2LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−u2xββ,yγγ,tαα.Now, apply the new iterative method to Equation (65).Using (49) into (65) and using (51), (52), and (53), the components of the solution are obtained as follows:
(66)u0xββ,yγγ,tαα=exβ/β+yγ/γ−2tα/α,u1xββ,yγγ,tαα=Lxyt−11s2+2s−k2−p2LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−u02xββ,yγγ,tαα=Lxyt−11s2+2s−k2−p2LxβLyγLtαe2xβ/β+2yγ/γ−4tα/α−e2xβ/β+2yγ/γ−4tα/α=0,um+1xβ/β,yγ/γ,tα/α=Lxyt−11/s2+2s−k2−p2LxβLyγLtα∑r=0iur2xβ/β,yγ/γ,tα/α−∑r=0i−1ur2xβ/β,yγ/γ,tα/α,n≥1.As a result,
(67)u2xββ,yγγ,tαα=Lxyt−11s2+2s−k2−p2LxβLyγLtαu0xββ,yγγ,tαα+u1xββ,yγγ,tαα2−u0xββ,yγγ,tαα2=0,u3xββ,yγγ,tαα=Lxyt−11s2+2s−k2−p2LxβLyγLtαu0xββ,yγγ,tαα+u1xββ,yγγ,tαα+u2xββ,yγγ,tαα2−u0xββ,yγγ,tαα+u1xββ,yγγ,tαα2=0.Similarly, we getu4xβ/β,yγ/γ,tα/α=u5xβ/β,yγ/γ,tα/α=0 and so on.Therefore, using Equation (54), the solution to Example 1 is given by
(68)uxββ,yγγ,tαα=exβ/β+yγ/γ−2tα/α.Remark 15.
Forα=β=1, in two dimensions, the conformable fractional telegraph in Equation (55) reduces to the classical (or non-fractional) space-time telegraph equation, and its exact solution is ux,t=ex+y−2t, which is the same as the result obtained in [28, 30, 49].
Forβ=γ=1&α=0.5, the exact solution is ux,y,t=ex+ye2t+e2terfc2t−1.
Forβ=α=1&γ=0.5, the exact solution is ux,y,t=ex+2tey+ey−erfcy+1.
Forα=γ=1&β=0.5, the exact solution is ux,y,t=ey+2tex+ex−erfcx+1.
Whenevererft=2/π∫0te−z2dz and erfct=1−erft.The 3D graphical simulation of exact solution corresponding to Example1 for different fractional-order values α, β, and γ are depicted in Figure 1.Example 2.
Consider the time fractional-order nonlinear telegraph equation with the external source term as follows:(69)∂2αu∂t2α+5∂αu∂tα+u3=∂2βu∂x2β+∂2γu∂y2γ−2e−tα/αsinxββcosyγγ+e−3tα/αsin3xββcos3yγγ,
under the initial conditions(70)uxββ,yγγ,0=sinxββcosyγγ,∂α∂tαuxββ,yγγ,0=−sinxββcosyγγ,and boundary conditions
(71)u0,yγγ,tαα=0,∂β∂xβu0,yγγ,tαα=e−tα/αcosyγγ,(72)uxββ,0,tαα=e−tα/αsinxββ,∂γ∂yγuxββ,0,tαα=0.Figure 1
3D solution plots of Example1 at (a) α=0.5,β=0.8,γ=0.6,t=0.1, (b) α=0.6,β=0.9,γ=0.5,t=0.4, (c) α=0.8,β=0.5,γ=0.7,t=0.8, and (d) α=0.9,β=1,γ=0.2,t=1.
(a)(b)(c)(d)Solution. On both sides of Equation (69), we use the properties of the conformable triple Laplace transform and Theorem (12) to obtain
(73)s2Uk,p,s−sUk,p,0−∂∂tUk,p,0+5sUk,p,s−Uk,p,0=k2Uk,p,s−kU0,p,s−∂∂xU0,p,s+p2Uk,p,s−pUk,0,s−∂∂yUk,0,s−2pk2+1p2+1s+1+LxβLyγLtαe−3tα/αsin3xββcos3yγγ−u3xββ,yγγ,tαα.We get the following result by applying the conformable double Laplace transform to the initial condition (70):
(74)Uk,p,0=pk2+1p2+1,∂∂tUk,p,0=−pk2+1p2+1.The boundary conditions (70) and (71) are transformed using the conformable fractional double Laplace transform as follows:
(75)U0,p,s=0,∂∂xU0,p,s=pp2+1s+1,(76)Uk,0,s=1k2+1s+1,∂∂yUk,0,s=0.Using Equations (74), (75), and (76) into the Equation (73), and simplifying, we get
(77)Uk,p,s=1s2+5s−k2−p2ps+5−pk2+1p2+1−pp2+1s+1−pk2+1s+1−2pk2+1p2+1s+1+LxβLyγLtαe−3tα/αsin3xββcos3yγγ−u3xββ,yγγ,tαα.By simplifying Equation (77), we obtain
(78)Uk,p,s=pk2+1p2+1s+1+1s2+5s−k2−p2LxβLyγLtαe−3tα/αsin3xββcos3yγγ−u3xββ,yγγ,tαα.We get the following results by applying the inverse conformable triple-Laplace transform to both sides of Equation (78):
(79)uxββ,yγγ,tαα=e−tα/αsinxββcosyγγ+Lxyt−11s2+5s−k2−p2LxβLyγLtαe−3tα/αsin3xββcos3yγγ−u3xββ,yγγ,tαα.Now, apply the new iterative method to Equation (79).Substituting (49) into (79) and using (51), (52), and (53), we obtain the components of the solution as follows:
(80)u0xββ,yγγ,tαα=e−tα/αsinxββcosyγγ,u1xββ,yγγ,tαα=Lxyt−11s2+5s−k2−p2LxβLyγLtαe−3tα/αsin3xββcos3yγγ−u03xββ,yγγ,tαα=Lxyt−11s2+5s−k2−p2LxβLyγLtαe−3tα/αsin3xββcos3yγγ−e−3tα/αsin3xββcos3yγγ=0,um+1xβ/β,yγ/γ,tα/α=Lxyt−11/s2+5s−k2−p2LxβLyγLtα∑r=0iur3xβ/β,yγ/γ,tα/α−∑r=0i−1ur3xβ/β,yγ/γ,tα/α,n≥1.As a result,
(81)u2xββ,yγγ,tαα=Lxyt−11s2+5s−k2−p2LxβLyγLtαu0xββ,yγγ,tαα+u1xββ,yγγ,tαα3−u0xββ,yγγ,tαα3=0,u3xββ,yγγ,tαα=Lxyt−11s2+5s−k2−p2LxβLyγLtαu0xββ,yγγ,tαα+u1xββ,yγγ,tαα+u2xββ,yγγ,tαα3−u0xββ,yγγ,tαα+u1xββ,yγγ,tαα3=0.Similarly, we getu4xβ/β,yγ/γ,tα/α=u5xβ/β,yγ/γ,tα/α=0 and so on.Therefore, the solution to the Example2 using Equation (54) is
(82)uxββ,yγγ,tαα=e−tα/αsinxββcosyγγ.Remark 16.
Forα=β=1, in two dimensions, the conformable fractional telegraph equation (69) reduces to the classical (or non-fractional) space-time telegraph equation, and its exact solution is ux,t=e−tsinxcosy, which is the same as the result obtained by [29, 30].
Forβ=γ=1&α=0.5, the exact solution is ux,y,t=ex+yet+eterfct−1.
The 3D graphical simulation of exact solution corresponding to Example2 for different fractional-order values α, β, and γ are depicted in Figure 2.Figure 2
3D solution plots of Example1 at (a) α=0.4,β=0.5,γ=0.7,t=0.2, (b)α=0.5,β=0.6,γ=0.8,t=0.4, (c) α=0.7,β=0.9,γ=0.5,t=0.8, and (d) α=1,β=1,γ=0.9,t=1.
(a)(b)(c)(d)


## 7. Conclusion

The conformable triple Laplace transform has been examined in this study using all of our newly discovered results and theorems. The solution of the conformable fractional nonlinear Telegraph equation in two dimensions is found using the new conformable triple Laplace transform iterative approach. We give the basic definitions and properties of the conformable fractional derivative, conformable triple Laplace transform method, and the new iterative method. The proposed method CTLTIM has been put to the test in a numerical experiment, and the outcome is also supported by a 3D graphical representation for different values of fractional ordersα,β,andγ as shown in Figures 1 and 2. Note that in Examples 1 and 2, if we use α=β=γ, we obtain an exact solution which was considered in [28, 30, 49], and further, nontrivial problems that are solved using earlier methods become trivial in the sense that the decomposition,
(83)uxββ,yγγ,tαα=u0xββ,yγγ,tαα+u1xββ,yγγ,tαα+u2xββ,yγγ,tαα+⋯+umxββ,yγγ,tαα+⋯,consists of only one term, i.e.,
(84)uxββ,yγγ,tαα=u0xββ,yγγ,tαα.Due to the critical need to explore new analytical solutions to understand the dynamics of solutions for such important equations in physics and engineering, our findings highlight the importance of exploring new generalized methods for solving fractional partial differential equations, mainly the nonlinear ones.



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