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A meshfree technique for the numerical solutions of nonlinear Fornberg–Whitham and Degasperis–Procesi equations with their modified forms

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Abstract

An efficient and robust hybrid scheme based on radial basis functions (RBFs) and finite difference is implemented to solve nonlinear partial differential equations (PDEs). In the proposed method, first-order finite difference and the \(\theta \)-weighted scheme are coupled for temporal discretization, while RBFs are used for spatial discretization. The key feature of the scheme is to use quasilinearization with a collocation approach to reduce nonlinear PDEs to linear algebraic system of equations which are easy to solve. Stability analysis is carried out to examine the spectral radius of the amplification matrix versus the shape parameter. Furthermore, the scheme is applied to solve some nonlinear PDEs including Fornberg–Whitham, Degasperis–Procesi equations, and their modified forms. Efficiency of the proposed technique is demonstrated via different error norms and conservative quantities. Moreover, the computed results are compared with the existing results in the literature. Simulations reveal better accuracy for the considered problems.

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Authors and Affiliations

  1. Faculty of Engineering Sciences, GIK Institute, Topi, 23640, KP, Pakistan

    Sadaf Shaheen & Sirajul Haq

  2. Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, 26000, KP, Pakistan

    Abdul Ghafoor

Authors
  1. Sadaf Shaheen
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  2. Sirajul Haq
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  3. Abdul Ghafoor
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Correspondence to Sadaf Shaheen.

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Communicated by Justin Wan.

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Shaheen, S., Haq, S. & Ghafoor, A. A meshfree technique for the numerical solutions of nonlinear Fornberg–Whitham and Degasperis–Procesi equations with their modified forms. Comp. Appl. Math. 41, 183 (2022). https://doi.org/10.1007/s40314-022-01870-x

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  • DOI: https://doi.org/10.1007/s40314-022-01870-x

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