This study shows how to obtain least-squares solutions to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by any existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a constrained expression, derived from Theory of Connections. In this expression, the differential equation constraints are embedded and are always satisfied. The resulting constrained expression is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10 −12. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals—called constrained expressions—and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second order differential equations. The proposed method has two steps. The first step consists of writing a constrained expression, introduced in Ref. [1], that has embedded the differential equation constraints. These expressions are given in term of a new unknown function, g(t), and they satisfy the constraints, no matter what g(t) is. The second step consists of expressing g(t) as a linear combination of m independent known basis functions, g(t) = ξ T h(t). Specifically, Chebyshev orthogonal polynomials of the first kind are adopted for the basis functions. This choice requires rewriting the differential equation and the constraints in term of a new independent variable, x ∈ [−1, +1]. The procedure leads to a set of linear equations in terms of the unknown coefficients vector, ξ, that is then computed by least-squares. Numerical examples are provided to quantify the solutions accuracy for initial and boundary values problems as well as for a control-type problem, where the state is defined in one point and the costate in another point.
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