【作者】 杜红；

【导师】 崔明根；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2007， 博士

【摘要】 许多自然现象都是借助于线性、非线性方程来描述的,而对这些现象的分析一般可归结为微分方程、积分方程和微分-积分方程的求解问题.因此,如何求解这些有实际意义的方程也就变得越来越重要了.本文运用再生核空间的技巧,给出了几类非线性积分和微分方程的求解方法.首先,在再生核空间中给出了第一类Fredholm积分方程的最小范数解的表达式.进一步地,如果方程的解不唯一,给出了它的所有解的表达形式.并给出了求解第一类Fredholm积分方程逼近解的稳定性分析.其次,讨论了非线性Volterra-Fredholm积分方程的求解.利用再生核表达式得到了一组标准正交基,将方程的未知函数在这组基上进行Fourier级数展开,构造了一种收敛的迭代序列,从而得到了方程解的一种级数表达形式.通过截断级数得到方程的逼近解.同时,将此迭代法推广到求解非线性Fredholm积分方程组.第三,在再生核空间中给出了带有积分边界条件的抛物型微分方程的求解方法.通过构造一组满足积分边界条件的标准正交基,利用再生核的技巧给出了方程解的表达式.第四,在再生核空间中给出了求解非线性非局部边界条件的二阶偏微分方程的方法.通过构造一组满足非线性非局部边界条件的标准正交基,给出了一个有界的并且收敛的迭代序列,从而得到了方程解的级数表达形式.更多还原

【Abstract】 Some natural phenomena may be described by linear or nonlinear equations.Analysis on these phenomena would generally come down to solve solutions ofdi?erential equations, integral equations or di?erential-integral equations. Thus,how to solve these equations with actual prospecting becomes more and moreimportant problems. In this paper, some methods of solving nonlinear di?erentialand integral equations are proposed using skills in reproducing kernel space(RKS).Firstly, a representation of minimal norm solutions on Fredholm integralequations of the first kind is obtained in RKS. Representations of the solutionsare further given if it has solutions. At the same time, a analysis on stability ofapproximate solutions of Fredholm integral equations of the first kind is provided.Secondly, a method of solving nonlinear Volterra-Fredholm integral equationsis proposed. An orthnormal basis is obtained using representation of reproducingkernel and unknown functions of nonlinear Volterra-Fredholm integral equationsare expressed as Fourier series using the basis. A convergent iterative sequence isconstructed. A series representation of solutions for the equations is obtained andapproximate solutions are given by truncating the series. The iterative methodis further generalized to solve systems of nonlinear Fredholm integral equation.Thirdly, a method of solving parabolic di?erential equations with integralboundary conditions is proposed in RKS. A representation of the solutions isprovided by constructing an orthnormal basis satisfying integral boundary con-ditions and applying skills of reproducing kernel.Finally, a method of solving partial di?erential equations with nonlinearnonlocal boundary conditions is proposed. A bounded and convergent iterativesequence is provided by constructing an orthnormal basis satisfying nonlinearnonlocal boundary conditions. Hence, a representation of solutions for the equa-tions is obtained.更多还原