Methods Of Solving Integral And Differential Equations In The Reproducing Kernel Space

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.