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

Abstract

In this paper, a novel analytical method for solving nonlinear partial differential equations is studied. This method is known as triple Laplace transform decomposition method. This method is generalized in the sense of conformable derivative. Important results and theorems concerning this method are discussed. A new algorithm is proposed to solve linear and nonlinear partial differential equations in three dimensions. Moreover, some examples are provided to verify the performance of the proposed algorithm. This method presents a wide applicability to solve nonlinear partial differential equations in the sense of conformable derivative.

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

“…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][37][38][39]46] for more information). Definition 6.…”

“…where the Heaviside unit step function Hðx, y, tÞis defined as in Equation (15) (see [37,38] for the proof). Proof (1).…”

“…Definition 3 (see [37,38]). Given a function f : ℝ + × ℝ + × ℝ + , then the conformable partial fractional derivatives (CPFDs) of f ððx β /βÞ, ðy γ /γÞ, ðt α /αÞÞ having order α, β and γ are defined by.…”

“…Theorem 4 (see [37,38]). Let α, β, γ ∈ ð0, 1 and f ððx β /βÞ, ðy γ /γÞ, ðt α /αÞÞ be differentiable at a point forx β /β, y γ /γ, t α / α > 0.…”

“…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.…”