Higher-Order Differential Correction Schemes for the Two-Point Boundary Value Problem

The conventional differential corrections approach for solving two-point boundary value problems (TPBVPs) using Newton’s method has several limitations, such as limited validity regions and high sensitivity to initial guesses. This paper proposes alternative third-and fourth-order iterative schemes in order to improve the robustness of the differential corrections process to the quality of the initial guess. The necessary higher-order sensitivities are obtained through a computationally tractable, derivative-free approach, where a least-squares process is adopted to calculate said sensitivities for the TPBVP in a prescribed domain of interest. A nonproduct quadrature scheme, the conjugate unscented transform, is used to compute the multidimensional integrals necessary for the least-squares procedure. Improved robustness, reduced computational cost, and simplicity of implementation of the method are demonstrated using three benchmark problems.