On the method of directly defining inverse mapping for nonlinear differential equations

TL;DR

This paper introduces the MDDiM, a novel approach based on homotopy analysis that directly defines inverse mappings to solve nonlinear differential equations without explicitly computing inverse operators.

Contribution

The paper proposes the MDDiM, a new method that simplifies solving nonlinear differential equations by directly defining inverse mappings, avoiding explicit inverse operator calculations.

Findings

MDDiM provides analytic approximations for nonlinear differential equations.

The method guarantees convergence to the true solution.

Demonstrated effectiveness on three nonlinear differential equations.

Abstract

In scientific computing, it is time-consuming to calculate an inverse operator ${\mathscr A}^{-1}$ of a differential equation ${\mathscr A}\varphi = f$, especially when ${\mathscr A}$ is a highly nonlinear operator. In this paper, based on the homotopy analysis method (HAM), a new approach, namely the method of directly defining inverse mapping (MDDiM), is proposed to gain analytic approximations of nonlinear differential equations. In other words, one can solve a nonlinear differential equation ${\mathscr A}\varphi = f$ by means of directly defining an inverse mapping $\mathscr J$, i.e. without calculating any inverse operators. Here, the inverse mapping $\mathscr J$ is even unnecessary to be explicitly expressed in a differential form, since "mapping" is a more general concept than "differential operator". To guide how to directly define an inverse mapping $\mathscr J$, some rules are provided. Besides, a convergence theorem is proved, which guarantees that a convergent series solution given by the MDDiM must be a solution of problems under consideration. In addition, three nonlinear differential equations are used to illustrate the validity and potential of the MDDiM, and especially the great freedom and large flexibility of directly defining inverse mappings for various types of nonlinear problems. The method of directly defining inverse mapping (MDDiM) might open a completely new, more general way to solve nonlinear problems in science and engineering, which is fundamentally different from traditional methods.