Numerical Algorithms

, Volume 72, Issue 4, pp 989–1020

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

Original Paper

Abstract

In scientific computing, it is time-consuming to calculate an inverse operator $$\mathcal {A}^{-1}$$ of a differential equation $$\mathcal {A}\phi = f$$, especially when $$\mathcal {A}$$ 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 $$\mathcal {A}\phi = f$$ by means of directly defining an inverse mapping $$\mathcal J$$, i.e. without calculating any inverse operators. Here, the inverse mapping $$\mathcal {J}$$ 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 $$\mathcal {J}$$, 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.

Keywords

Homotopy analysis method Analytical approximation Nonlinear differential equation Direct definition of inverse mapping

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