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A fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity

A fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity In this paper, we use orthogonal spline collocation methods (OSCM) for the one dimensional Helmholtz equation with discontinuous coefficients. We discuss the existence uniqueness results and establish optimal error estimates. We use piecewise Hermite cubic basis functions to approximate the solution. Finally, we perform some numerical experiments and validate the theoretical results. Comparative to existing methods, we prove that the orthogonal spline collocation methods (OSCM) handles the discontinuous coefficients effectively and gives optimal order of convergence for $$\Vert y-y_h\Vert _{L^{\infty }}$$ ‖ y - y h ‖ L ∞ -norm and superconvergent result for $$\Vert y'-y'_h\Vert _{L^{\infty }}$$ ‖ y ′ - y h ′ ‖ L ∞ -norm at the grid points. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

A fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity

The Journal of Analysis , Volume 27 (2) – May 7, 2018

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Forum D'Analystes, Chennai
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-018-0082-9
Publisher site
See Article on Publisher Site

Abstract

In this paper, we use orthogonal spline collocation methods (OSCM) for the one dimensional Helmholtz equation with discontinuous coefficients. We discuss the existence uniqueness results and establish optimal error estimates. We use piecewise Hermite cubic basis functions to approximate the solution. Finally, we perform some numerical experiments and validate the theoretical results. Comparative to existing methods, we prove that the orthogonal spline collocation methods (OSCM) handles the discontinuous coefficients effectively and gives optimal order of convergence for $$\Vert y-y_h\Vert _{L^{\infty }}$$ ‖ y - y h ‖ L ∞ -norm and superconvergent result for $$\Vert y'-y'_h\Vert _{L^{\infty }}$$ ‖ y ′ - y h ′ ‖ L ∞ -norm at the grid points.

Journal

The Journal of AnalysisSpringer Journals

Published: May 7, 2018

References