三区复合型连带Legendre方程边值问题的相似构造法

The Similar Construction Method of the Boundary Value Problem of the Three-Regions Composite Associated Legendre Equation

在关于三区复合型连带Legendre方程的某一类边值问题的探索中,对其解的表达式进行剖析,发现此边值问题的解可以由内区、中区和外区的引解函数、内外边界条件的系数和两组衔接性条件的系数来组装得到,其引解函数由连带Legendre方程的两个线性无关解组成,并且解的结构具有相似性.由此提出求解此类边值问题的新办法——相似构造法.该方法能够简化问题的求解过程,降低求解难度,使结果更加简洁美观,从而为相关软件的开发提供了新的思路.

In the exploration of a certain kind of boundary value problem of the three-regions composite associated Legendre equation,this paper analyzes the expression of its solution,and finds that the solution of this boundary value problem can be assembled by the functions of guide solution of inner,middle and outer regions,the coefficients of the inner and outer boundary conditions,and the coefficients of two sets of cohesive conditions,that the functions of guide solution are composed of two linear independent solutions of the associated Legendre equation,and that the structures of the solutions are similar.Therefore,a new method for solving such boundary value problems is proposed,which is the similar construction method.The similar construction method can simplify the process of solution,reduce the difficulties of solution,make the solutions more concise and beautiful,and provide a new idea for the development of related well-testing software.