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Third (fourth) order accurate two-step super-stable cubic spline polynomial approximation for the second order non-linear initial-value problems

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Abstract

We report a new implicit cubic spline (CS) approximation for the initial-value problems w″ = f(t, w, w′), w(t0) = a0, w′ (t0) = a1, t0 < t < ∞ on a mesh not necessarily equidistant. We use only monotonically reducing mesh for computation. We apply the suggested CS approximation to a test equation w″ + 2b0w′ + c02w = f(t), b0 > c0 ≥ 0, and study the stability of the obtained linear scheme. It has been shown that the linear CS scheme is absolutely stable on a graded mesh and super stable on an unvarying mesh. Some benchmark test examples including boundary layer and singular problems are solved using the proposed CS approximation to demonstrate the viability of the suggested method. Computational results are specified to demonstrate the usefulness of the developed CS method.

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  1. Department of Applied Mathematics, South Asian University, Rajpur Road, Maidan Garhi, New Delhi, 110068, India

    R. K. Mohanty

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Mohanty, R.K. Third (fourth) order accurate two-step super-stable cubic spline polynomial approximation for the second order non-linear initial-value problems. J Math Chem 61, 2123–2145 (2023). https://doi.org/10.1007/s10910-023-01509-0

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