辅助函数法求解非线性偏微分方程精确解

Auxiliary Function Method for Exact Solution of Nonlinear Partial Differential Equation

在数学和物理学领域,将含有非线性项的偏微分方程称为非线性偏微分方程。非线性偏微分方程用于描述物理学中许多不同的物理模型,范围涉及从引力到流体动力学的众多领域,还在数学中用于验证庞加莱猜想和卡拉比猜想。在求解非线性偏微分方程的过程中,几乎没有通用的求解方法能够应用于所有的方程。通常,可依据模型方程的数学物理背景来先验地假设非线性偏微分方程解的形式,并根据解的特点给出辅助方程。非线性偏微分方程可通过行波变换转化为常微分方程,再借助辅助方程来求解常微分方程。为此,借助行波变换及辅助方程的求解思路对BBM方程和Burgers方程进行了研究,并获得了其双曲正切函数及三角函数形式的精确解。研究结果表明,所采用的方法可广泛应用于若干在数学物理中有典型应用背景的非线性偏微分方程的精确解求解中。

In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms, which can describe many different physical models ranging from gravitation to fluid dynamics, and have been adopted in mathematics to solve problems such as the Poincar- conjecture and the Calabi conjecture. There are almost no general solutions that can be applied for all equations. Nonlinear partial differential equation usually originates from mathematical and physical fields, such that the ansatz of the solutions has been given and an auxiliary function has been provided according to its mathematical and physical features. They can be transmitted to an ordinary differential equations via a traveling wave transformation. Through introduction of the auxiliary function into the ordinary dif- ferential equation a set of nonlinear algebra equations is acquired, which can supply solutions original partial differential equation in solving process. Therefore,BBM equation and Burgers equation can be solved with the auxiliary function. The exact solutions include tan- gent function and trigonometric functions. The research shows that the proposed auxiliary function method can be applied to solve some other nonlinear partial differential equations with mathematical and physical background.