本文先研究了一类变型Bessel方程的通解,由此进一步研究求解复合变型Bessel方程边值问题,发现这类微分方程边值问题的解在不同的边界条件下具有相似的结构,且其解是由边界条件的系数和相似核函数决定的,由此提出了相似构造法。该方法的主要步骤是首先求出方程的基础解系,构造引解函数;再利用引解函数和初边值条件、交界面条件的系数构造内区核函数和外区核函数;最后根据核函数和边值条件的系数得到微分方程组的相似结构解。利用相似构造法在求解微分方程的初边值问题时,能够极大地简化求解过程,便于试井软件的编写以及分析相应的参数。In this paper, the general solution of a class of variant Bessel equations is studied, and then the boundary value problem of the composite variant Bessel equation is further studied, and it is found that the solution of the boundary value problem of this kind of differential equation has a similar structure under different boundary conditions, and its solution is determined by the coefficient and similar kernel function of the boundary condition, so the similarity construction method is proposed. The main steps of this method are to first find the basic solution system of the equation and construct the induction function. Then, the coefficients of the induction function, the initial boundary value condition and the interface condition are used to construct the inner and outer kernel functions. Finally, according to the coefficients of the kernel function and the boundary value condition, the similar structural solutions of the differential equations are obtained. The similarity construction method can greatly simplify the solution process when solving the initial boundary value problem of differential equations, which is convenient for the compilation of well testing software and the analysis of corresponding parameters.
