利用特征线法求解方程u_t+b·Du+cu=(fx,t)的初值问题

Solving the Initial-Value Problem of Equation u_t +b·Du+cu=f(x,t) with the Method of Characteristics

本文研究具有初值条件u(x,0)=g(x)的方程u_t+b·Du+cu=f(x,t)的初值问题。方程u_t+b·Du+cu=f(x,t)是具有常系数的一阶非齐次线性偏微分方程,这类方程在变分法、质点力学和几何学中都出现过,因此研究这类方程的目的是更好地应用于这些学科。求解这类方程的最基本方法是特征线法。它是把偏微分方程转化为常微分方程或常微分方程组,通过求解这些常微分方程得到所要求的解。本文分别运用特征线法以及特征线法的特殊情况求解了该初值问题,两种方法所得到的解是一致的,都是u(x,t)=g(x-bt)e^(-ct)+e^(-ct)integral from n=0 to te^(cu)f(x+b(u-t),u)du。因此,有了通过特征线法所求得的该初值问题的解的公式,我们可以更好地研究相关的一些实际问题。

The paper studies initial-value problem of equation u, ＋b. Ut＋b·Du＋cu=f（x,t） with initial condition u （x, 0 ） =g（x）. Equation Ut＋b·Du＋cu=f（x,t） is of one order non homogeneous linear partial differential equation with constant coefficients, and this kind of equations appeared in the Variational method,particle mechanics and geometry, so the study of this kind of equation is intended to be better applied in these disciplines. The most basic method of solving this kind of equations is the method of characteristics. It converts the partial differential equation into ordinary differential equations, and the requested solution is get by solving the ordinary differential equations. The paper respectively makes use of the method of characteristics and a special case of the method of characteristics to solve the initial-value problem, and the solutions are consistent, being（x＋b （u-t）,u）du. Therefore, there is the formula of the solution of initial-value problem being obtained by the method of characteristics, and we can better study the related problem.