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Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations

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Abstract

In this paper, we develop a numerical solution based on nonpolynomial B-spline (trigonometric B-spline) collocation method for solving time-dependent equations involving PHI-Four and Allen–Cahn equations. A three-time-level implicit algorithm has been derived. This algorithm combines the trigonometric B-spline interpolant and the \(\theta \)-weighted scheme for space and time discretization, respectively. Convergence analysis is discussed and the accuracy of the presented method is \(O( {\tau ^{2}+h^{2}} ).\) Applying von Neumann stability analysis, the proposed technique is shown to be unconditionally stable. Three test problems are demonstrated to reveal that our method is reliable, efficient and very encouraging.

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Authors and Affiliations

  1. Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Tanta, 31521, Egypt

    W. K. Zahra

  2. Department of Mathematics, Basic and Applied Sciences School, Egypt-Japan University of Science and Technology, New Borg El-Arab City, Alexandria, 21934, Egypt

    W. K. Zahra

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Zahra, W.K. Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations. Mediterr. J. Math. 14, 122 (2017). https://doi.org/10.1007/s00009-017-0916-8

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  • DOI: https://doi.org/10.1007/s00009-017-0916-8

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