论文标题
使用功能连接理论的第八阶边界价值问题的最小二乘解决方案
Least-squares Solutions of Eighth-order Boundary Value Problems using the Theory of Functional Connections
论文作者
论文摘要
本文展示了如何获得线性和非线性普通微分方程的第八阶边界值问题的高度精确解。提出的方法基于功能连接理论，并以两个步骤解决。首先，功能连接理论分析将微分方程约束嵌入到候选函数（称为$约束\，表达式$）中，该函数包含用户可以自由选择的函数。无论自由功能是什么，此表达式总是满足约束。其次，自由函数作为具有未知系数的正交函数的线性组合扩展。然后将受约束的表达（及其衍生物）取代为八阶微分方程，将问题转换为不受约束的优化问题，其中正交基函数的线性组合中的系数是优化参数。然后，通过线性/非线性最小二乘形式找到这些参数。从该方法获得的解决方案是对真实溶液的高度准确的分析近似。与文献中出现的替代方法的比较验证了所提出的方法。
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.
