Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

• Published in: Algorithms Vol. 15; no. 9; p. 320
• Main Authors: Khalil, Omar, El-Sharkawy, Hany, Youssef, Maha, Baumann, Gerd
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2022
• Subjects: adaptive approximation; Algorithms; Analysis; Approximation; boundary value problems; Collocation methods; Convergence; Differential equations; Error analysis; Estimates; Exact solutions; Greedy algorithms; initial value problems; Interpolation; Lagrange interpolation; Mathematical analysis; Methods; Ordinary differential equations; Partial differential equations; Poly-Sinc interpolation; Polynomials; Sinc methods; Germany
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a15090320

We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.