Approximate Solutions of Multidimensional Wave Problems Using an Effective Approach

Abstract

The main goal of this paper is to introduce a new scheme for the approximate solution of 1D, 2D, and 3D wave equations. The recurrence relation is very important to deal with the approximate solution of differential problems. We construct a scheme with the help of the Laplace-Carson integral transform ( ${\mathbb{L}}_c$ IT) and the homotopy perturbation method (HPM), called Laplace-Carson homotopy integral transform method ( ${\mathbb{L}}_c$ HITM). ${\mathbb{L}}_c$ IT produces the recurrence relation and destructs the restriction of variables whereas HPM gives the successive iteration of the relation using the initial conditions. The convergence analysis is provided to study the wave equation with multiple dimensions. Some numerical examples are considered to show the efficiency of this scheme. Graphical representation and plot distribution between the approximate and the exact solution predict the high rate of convergence of this approach.