三区间复合Legendre方程边值问题解的相似结构
Similar Structure of Solutions of Boundary Value Problem of Three-Region Composite Legendre Equation
摘要: 针对三区复合Legendre方程的一类边值问题，对其解的表达形式进行了分析，发现其解式具有相似的结构；首先通过方程的两个线性无关解构造内、中、外区引解函数；再借助外边界条件、衔接性条件的系数以及引解函数构造内、中、外区相似核函数；最后将引解函数、相似核函数以及定解条件的系数进行组装得到其解，进而获得了求解此类边值问题的新方法——相似构造法，该方法简化了求解该类边值问题的复杂性，并提高了求解的效率。
Abstract: For a class of boundary value problems of three-region composite Legendre equation, the expres-sion of the solutions is analyzed, and it is found that its solution has a similar structure. Firstly, we can construct the functions of guide solution of the inner, middle and outer regions through the two linear independent solutions of the definite solution equation. Secondly, through the outer boundary condition, the connection condition coefficients and the functions of guide solution to con-struct the similar kernel functions in the inner, middle and outer regions. Finally, the solution is ob-tained by assembling the functions of guide solution, the similar kernel function and the coefficients of the definite solution conditions. Thus a new method for solving such boundary value problems is obtained, which is called the similarity construction method. This method simplifies the complexity of solving this kind of boundary value problem and improves the efficiency of solving them.
文章引用：付雪倩, 李顺初, 邵东凤, 刘盼, 范林. 三区间复合Legendre方程边值问题解的相似结构[J]. 理论数学, 2023, 13(2): 172-181. https://doi.org/10.12677/PM.2023.132020

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