A Computational Quadruple Laplace Transform for the Solution of Partial Differential Equations

Abstract

In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homogeneous as well as non-homogeneous partial differential equations involving four variables. The performance of quadruple Laplace transform is shown to be very encouraging by concrete examples. An elementary table of quadruple Laplace transform is also provided.

Solutions of Three Dimensional Nonlinear Klein-Gordon Equations by Using Quadruple Laplace Transform

“…Provided the integral exists. Defnition 2 (see [23]). For a continuous function F(x, m, v, t) of four variables, then the quadruple Laplace transform is defned as follows:…”

“…where F(e, y, g, h) must be an analytic function for each e, y, g and h in the given region defned by the inequality Ree ≥ α, Rey ≥ β, Reg ≥ η, Reh ≥ ζ where α, β, η, and ζ are real constants to be chosen suitably based on [23]. To verify this we recall the Fourier integral formula given by ( 4) which expresses the representation of a function g(x) defned on…”

Exact Analytical Solutions of Linear Dissipative Wave Equations via Laplace Transform Method

“…Definition 2.1. [14] Let ψ(x, t) be a function of one dimensional variable x and a time variable t defined in positive quadrant of xt-plane. Then the double Lapalace transform of a function ψ(x, t) is given by…”

Triple Shehu transform and its properties with applications

“…With this, the associated differential equation can be directly reduced to either a differential equation of lower dimension or an algebraic equation in the new variable. There are several forms of integral transforms such as Laplace transform (Papoulis, 1957Debnath and Bhatta, 2014, Rehman et al, 2014and Dhunde et al, 2013, Sumudu transform (Kılıcman and Gadain, 2010;Mahdy et al, 2015 andMahdy et al, 2015a), Eltayeb and Kilicman, 2010, and Mechee and Naeemah, 2020. Aboodh transform (Aboodh, 2013, Elzaki transform (Elzaki, 2011), Variational homotopy perturbation method (Mahdy et al, 2015b), Alternative variational iteration method and one form may be obtained from the other by a transformation of the coordinates and the functions.…”