A New Complex Integral Technique for solving The Differential Equations and Their Applications

Abstract

This paper introduces a new multi-parametric complex integral technique designed to generalize and extend the traditional Laplace operator. By incorporating four independent complex-valued parameter functions and accommodating arbitrary positive bases, the proposed technique provides a versatile mathematical framework that unifies several well-known single-parameter techniques, including the Laplace, Aboodh, and Elzaki transforms as special boundary cases. We establish the basic operational calculus of the transform, rigorously deriving differential theorems for the ordinary and the partial derivatives for an arbitrary order. Furthermore, we calculate the utility and algebraic flexibility of this approach by obtaining exact analytical solutions for key higher-order ordinary differential equations (ODEs) and partial differential equations (PDEs), such as wave, telegraph, and Klein–Gordon boundary value problems in mathematical physics and we used Mathematica for getting (3D) graph.