复合Riccati-Bessel方程边值问题的相似构造法

The Similar Construction Method of the Composite Riccati-Bessel Equation Boundary Value Problem

摘要: 在微分方程边值问题解的结构式相似性的研究基础上，针对复合Riccati-Bessel方程的一类边值问题，通过对解式的分析和简化，发现求解该类定解问题可先利用左、右区Riccati-Bessel方程的两个线性无关的解构造引解函数，然后将右区引解函数与右边值条件的系数结合生成右相似核函数，将左区引解函数与交界处衔接条件的系数结合生成右相似核函数，最后将左边值条件的系数和左相似核函数组装，得到了其解具有连分式乘积形式的相似结构，并提出求解该类复合Riccati-Bessel方程边值问题的一种新方法——相似构造法。

Abstract: Based on the study of the structural similarity of solutions to boundary value problems of differ-ential equations, for a type of boundary value problem of the composite Riccati-Bessel equation, through the analysis and simplification of the solution, it is found that the solution to this type of definite solution problem can be solved by using two linearly independent solutions of the left and right Riccati-Bessel equations to construct the guide function. Then combine the right-area guide function with the coefficients of the right-side value condition to generate a right similar kernel function, and combine the left-area guide function with the coefficients of the junction condition to generate a right similar kernel function. Finally, the coefficients of the left-side value condition and the left-similar kernel function are assembled to obtain the similar structure whose solution has the form of continued fraction product. Hence, a new method, the similarity construction method, is proposed to solve the boundary value problem of the composite Riccati-Bessel equation.