The Research Of Reproducing Kernel Algorithm For Solving Two Types Of Interface Problems

With the development of science and technology,interface problems are widely used in real life,such as crystal curing,composite materials,etc.If these applica-tions are modeled using partial or ordinary differential equations,the parameters that control the differential equations are often discontinuous at the interface that separates two materials or two states,and the source terms are usually singular.Due to these irregularities,the solutions of differential equations are usually dis-continuous.At this time,the numerical method explored for the problem with continuous solution is not particularly significant in solving the interface problems,and the required accuracy cannot be achieved around the interface,so exploring an efficient numerical method for solving interface problems has important theoretical significance and practical value.This paper mainly studies two types of interface problems:one-dimensional elliptic interface problem and one-dimensional parabolic interface problem.This paper studies one-dimensional elliptic interface problem numerically and proposes the reproducing kernel collocation method(RKCM).In this paper,a new reproducing kernel space(?)[a, b]is established,and a group of basis is construct-ed by using reproducing kernel functions and polynomials to span an approximate solution space and the approximate solution can be represented by a linear com-bination of the basis.Then,the collocation method is used to quickly solve the unknown coefficient of the approximate solution,so as to obtain the exact expres-sion of the approximate solution.Then the detailed numerical analysis is carried out for the approximate solution of RKCM.Finally,the efficiency of the algorithm can be shown by some numerical examples.This paper studies one-dimensional parabolic interface problem from both the-oretical and numerical aspects.Theoretically,the Rothe method is applied to prove the existence uniqueness of the weak solution of the parabolic interface problem.Nu-merically,the implicit Euler method and the reproducing kernel collocation method(RKCM)are combined to deal with the problem.First,the parabolic problem is transformed into an elliptic problem by using the implicit Euler method to dis-cretize time.Then the approximate solution is put into the reproducing kernel space(?)[a, b]and solved by using the reproducing kernel collocation method(RKCM).The approximate solution is analyzed in detail in the time and space directions.Finally,some numerical examples are given to verify the stability and operability of the numerical algorithm.