CAJ | 학술논문

本文研究具有初值条件u（x，0）＝g（x）的方程 ut＋b·Du＋cu＝f（x，t）的初值问题。方程ut＋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 t0 ecu 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。
본문연구구유초치조건u（x，0）＝g（x）적방정 ut＋b·Du＋cu＝f（x，t）적초치문제。방정ut＋b·Du＋cu＝f（x，t）시구유상계수적일계비제차선성편미분방정，저류방정재변분법、질점역학화궤하학중도출현과，인차연구저류방정적목적시경호지응용우저사학과。구해저류방정적최기본방법시특정선법。타시파편미분방정전화위상미분방정혹상미분방정조，통과구해저사상미분방정득도소요구적해。본문분별운용특정선법이급특정선법적특수정황구해료해초치문제，량충방법소득도적해시일치적，도시u（x，t）＝g（x-bt）e-ct＋e-ct t0 ecu f（x＋b（u-t），u）du。인차，유료통과특정선법소구득적해초치문제적해적공식，아문가이경호지연구상관적일사실제문제。
The paper studies initial-value problem of equation 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 u(x,t)=g (x-bt)e-ct+e-ct t0 ecu f (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.