一类二阶偏微分方程边值问题解的相似构造法

Similar Construction Method for Solution of a Class of Second-Order Partial Differential Equation Boundary Value Problems

摘要: 本文针对一类二阶偏微分方程边值问题，提出一种基于相似结构的求解方法。该方法通过定解方程的基础解系构造引解函数，再将其结合边界条件系数形成相似核函数，进而得到完整解式。所得解呈现稳定的连分式结构，不随外边界条件形式的变化而改变其整体框架。该方法避免了传统求解中繁琐的推导过程，使解析解构造更为简洁高效，并具有良好的统一性和适用性。

Abstract: This paper proposes a similarity-based method for solving a class of second-order partial differential equation boundary value problems. The method constructs guiding functions from the fundamental solution system of the governing equation and then combines them with boundary-condition coefficients to form a similarity kernel function, from which the complete analytical solution can be systematically derived. The resulting solutions exhibit a stable continued-fraction structure whose overall framework remains unchanged despite variations in the form of the external boundary conditions. This approach avoids the cumbersome derivations required by traditional solution methods, making the construction of analytical solutions more concise and efficient, while offering strong consistency and broad applicability.