Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections

Abstract

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.

Fast character modeling with sketch-based PDE surfaces

“…Through constructing the orthogonal basis from the obtained kernel functions where the constraint initial condition is satisfied, Arqub develop exact and numerical solutions to fuzzy Fredholm-Volterra integrodifferential equations with the reproducing kernel Hilbert space method [2]. By applying the theory of functional connections to embed the differential equation constraints into a constrained expression containing a free-function which is a linear combination of orthogonal basis functions with unknown coefficients and using linear/nonlinear least-squares to determine the unknown coefficients, Johnston et al obtain the solutions to eighthorder boundary-value problems [19].…”

Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

“…TFC has been developed for univariate and multivariate scenarios [2,7,8] to solve a variety of mathematical problems: a homotopy continuation algorithm for dynamics and control problems [14], domain mapping [15], data-driven parameters discovery applied to epidemiological compartmental models [16], transport theory problems such as radiative transfer [17] and rarefied-gas dynamics [18], nonlinear programming under equality constraints [19], Timoshenko-Ehrenfest beam [20], boundary-value problems in hybrid systems [21], eighth-order boundary value problems [22], and in Support Vector Machine [23]. TFC has been widely used for solving optimal control problems for space application, solved via indirect methods [24]: orbit transfer and propagation [25][26][27][28], energy-optimal in relative motion [29], energy-optimal and fuel-efficient landing on small and large planetary bodies [30,31], the minimum time-energy optimal intercept problem [32].…”

Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-Rectangular 2-Dimensional Domains

“…A mathematical proof [1] has shown that constrained expressions, like the one given in Equation (1), can be used to represent the whole set of functions satisfying the set of constraints they are derived for. The TFC has been mainly developed [1][2][3][4] to better solve constraint optimization problems, such as ODEs [5][6][7][8], PDEs [4,9], or programming [10,11], with effective applications in optimal control [12,13], as well as in machine learning [4,14,15]. In fact, a constrained expression restricts the whole space of functions to the subspace fully satisfying the constraints.…”

“…The TFC has been mainly developed [ 1 – 4 ] to better solve constraint optimization problems, such as ODEs [ 5 – 8 ], PDEs [ 4 , 9 ], or programming [ 10 , 11 ], with effective applications in optimal control [ 12 , 13 ], as well as in machine learning [ 4 , 14 , 15 ]. In fact, a constrained expression restricts the whole space of functions to the subspace fully satisfying the constraints.…”