用函数联络理论求八阶边值问题的最小二乘解
Least-squares Solutions of Eighth-order Boundary Value Problems using the Theory of Functional Connections
摘要：
本文给出了如何求线性和非线性常微分方程八阶边值问题的高精度解。该方法基于功能联系理论，分两步求解。首先，函数连接理论将微分方程约束解析地嵌入到候选函数（称为$constrained \, expression$）中，该候选函数包含用户可以自由选择的函数。这个表达式总是满足约束条件，不管自由函数是什么。其次，将自由函数展开为系数未知的正交基函数的线性组合。然后将约束表达式（及其导数）代入八阶微分方程，将问题转化为无约束最优化问题，其中正交基函数的线性组合中的系数为优化参数。这些参数，然后找到线性/非线性最小二乘法。从这种方法得到的解决方案是一个高精度的解析近似的真实的解决方案。与文献中出现的替代方法的比较验证了所提出的方法。
英文摘要：
This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a $constrained \, expression$) that contains a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.
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