Analytical Solutions to Two-Dimensional Nonlinear Telegraph Equations Using the Conformable Triple Laplace Transform Iterative Method
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.
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 . 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 . 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 . Many definitions of fractional derivatives and integrals have been published in the literature, including Riemann-Liouville fractional definitions , Caputo fractional definitions , Grünwald-Letnikov fractional derivatives , and Hadamard fractional integral . 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, does not fulfill the Riemann-Liouville definition. In Caputo’s definition, 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 .
To resolve these issues, Khalil et al.  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 value , the conformable fractional derivative of a real function, is defined as The initial value can be zero, and if the limit exists, is said to be partially differentiable at .
The CFD’s Definition 1 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 ). Another study  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  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 , 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.  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,  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.  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  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 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 . 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  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,  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 , 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 . 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 . 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 . 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  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.  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 term of the considered problem (2) is decomposed into two functions namely and . The importance of this decomposition is that the part with the terms in Equation (2) always leads to the simple algebraic expression while applying the conformable triple Laplace transform and the part 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 (for see ):
depending on the starting conditions and boundary conditions (Cauchy type BCs) where and are known real constants; denotes either the voltage or current through the two-dimensional conductor at position at the time ; is the general continuous nonlinear resorting term; and is the source term which is assumed to be analytic in time . Equation (2) reduces to the undamped telegraph equation in two space variables when and to the damped one when .
The rest of the paper is organized as follows: Section 2 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 ). The fractional derivative of a suitable mapping by Riemann-Liouville is given as
Caputo fractional derivative of a suitable mapping is given as
where is the order of fractional derivative and .
If this limit exists, is said to be partially differentiable at .
Definition 3 (see [37, 38]). Given a function , then the conformable partial fractional derivatives (CPFDs) of having order are defined by. where , and , and are called the fractional partial derivatives of orders , respectively.
Proposition 5. Let and , . Then, we have the following: (i)(ii)(iii)provided that ≠0. (iv), if was a function depending only on (v)(vi)and (vii) and (viii) and
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 function and . The conformable Laplace transform (CLT) of the exponential of order function is thus defined and denoted by
Definition 7. Let be a piece-wise continuous function of exponential order on the domain D of . After that, the conformable double Laplace transform (FCDLT) of is defined and denoted by
Now, we define fractional conformable triple Laplace transform, for and are the Laplace variables.
Definition 8. Let be a piece-wise continuous function of exponential order on the domain D of . After that, the conformable triple Laplace transform (FCDLT) of is defined and denoted by where are Laplace variables of , respectively, and
The conformable inverse triple Laplace transform, abbreviated by , is defined as follows:
Definition 9. The following is the definition of a unit step or Heaviside unit step function:
Theorem 10. If , and and are constants then the followings hold: (1)Linearity property: (2), where is the constant(3), where is the gamma function. Note that , for .(4) and (5)The conformable triple Laplace transform’s first shifting theorem:If then . (6) (7)If then
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 result using the conformable triple Laplace transform definition. Differentiating with respect to times, we get This suggests that Differentiating (19) with respect to times, produces Again, differentiating (20) with respect to times, we obtain Using Equations (19), (20), and (21), we get This suggests that Multiplying both sides of the Equation (23) by , we obtain
Theorem 12. For Let be the real-valued piece-wise continuous function defined on the domain The CFTLT of the conformable partial fractional derivatives of order is given bySS (1)(2)(3)(4)(5)(6)(7)(8)
Proof (1). Using the CFTLT definition (6), we have By using Theorem 4, we have Then, Equation (26) reduced to Taking integration by parts and Theorem 4 to the integrals inside the bracket produces We get the needed outcome by substituting Equation (29) into Equation (28), then simplifying and this result can be generalized to 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  for the main principle of the new iterative method: where is a nonlinear operator in a Banach space such that and is a known function.
We are looking for a solution of the Equation (31) having the series form:
The nonlinear operation can then be decomposed as
Equation (34) yields the following recurrence relation:
Then, and hence,
As a result, the term approximate solution of Equation (31) is defined as follows:
4.1. Convergence of the NIM
Theorem 13. If is a continuously differentiable functional in a neighborhood of and for each and for some real and then the series is absolutely convergent and moreover, .
Theorem 14. If is a continuously differentiable functional in a neighborhood of and for all , then the series is absolutely convergent.