Johnston Hunter, Leake Carl, Mortari Daniele
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.
Mathematics (Basel). 2020 Mar;8(3):397. doi: 10.3390/math8030397. Epub 2020 Mar 11.
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) containing 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.
本文展示了如何获得线性和非线性常微分方程八阶边值问题的高精度解。所提出的方法基于泛函连接理论，分两步求解。首先，泛函连接理论将微分方程约束解析地嵌入到一个候选函数（称为约束表达式）中，该表达式包含一个用户可自由选择的函数。无论自由函数是什么，这个表达式始终满足约束条件。其次，将自由函数展开为具有未知系数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，将问题转化为一个无约束优化问题，其中正交基函数线性组合中的系数就是优化参数。然后通过线性/非线性最小二乘法找到这些参数。通过这种方法得到的解是真实解的高精度解析近似。与文献中出现的其他方法的比较验证了所提出的方法。
