On the method of directly defining inverse mapping for nonlinear differential equations
For researchers in scientific computing and applied mathematics, this work introduces a new conceptual framework that may simplify solving nonlinear differential equations, but the results are preliminary and incremental relative to existing homotopy methods.
The paper proposes the method of directly defining inverse mapping (MDDiM) for solving nonlinear differential equations without explicitly computing inverse operators, and proves a convergence theorem. The method is validated on three nonlinear examples, demonstrating flexibility and potential for broad application.
In scientific computing, it is time-consuming to calculate an inverse operator ${\mathscr A}^{-1}$ of a differential equation ${\mathscr A}φ= f$, 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 ${\mathscr A}φ= f$ 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.