A method of directly defining the inverse mapping for solutions of cauchy reaction- diffusion problems

Abstract

In this paper, a semi-analytical approach called the Method of Directly Defining inverse Mapping (MDDiM) is applied to obtain an approximation series solution for the time-dependent reaction-diffusion problems. This novel approach is being applied to reaction-diffusion problems for the first time. Reaction- diffusion equations are used to describe, in modeling, nonlinear systems across various fields, including physics, chemistry, ecology, biology, and engineering. Traditionally, perturbation and asymptotic methods were widely used to obtain analytical approximations for nonlinear problems. However, these methods often failed under strong nonlinearity conditions. To overcome this obstacle, Shijun Liao introduced the Homotopy Analysis Method (HAM) in 1992, which is effective for highly nonlinear problems. However, determining the unknown functions requires computing the inverse of the linear operator. In scientific computing, calculating the inverse operator for a differential equation is a time-consuming process. To overcome this obstacle, in 2016 Liao introduced a new approach, namely the Method of Directly Defining inverse Mapping (MDDiM) with the freedom to choose the inverse linear mapping directly. The proposed method achieved a remarkable level of accuracy for solutions even with few iterations. Therefore, it can be concluded that MDDiM is not only easy to use but also accurate.