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) 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.
Highlights
This paper has been motivated by several articles dedicated to estimating the solutions of high-order boundary-value problems (BVPs) including, fourth-order [1], sixth-order [2], eighth-order [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], 2m-order [20], and higher-order [21,22,23,24] BVPs
This paper focuses on eighth-order BVPs because of the volume of research done on them, which is covered in Refs. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
BVPs appear in the physics of specific hydrodynamic stability problems when instability sets in as overstability [25], and in orthotropic cylindrical shells under line load [26]
Summary
This paper has been motivated by several articles dedicated to estimating the solutions of high-order boundary-value problems (BVPs) including, fourth-order [1], sixth-order [2], eighth-order [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], 2m-order [20], and higher-order [21,22,23,24] BVPs. Once the expressions of the ηk coefficients are obtained, they are back substituted into Equation (1) to produce the constrained expression, a functional representing all possible functions satisfying the specified set of constraints The use of this constrained expression has already been applied to many areas of study, including solving low-order differential equations [29,30], hybrid systems [31], and optimal control problems, including energy-optimal landing, energy-optimal intercept [32], and fuel-optimal landing [33]. X0 = xi is the initial value and x (k = N ) = x N = x f is the final value, and k is defined in the description of Chebyshev-Gauss-Lobatto collocation points It follows that b( x) is composed of a linear combination of b( x ), its derivatives, and a potential forcing term f ( x ) for the discrete values of x.
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