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A convergent hybrid numerical scheme for a class of nonlinear diffusion equations

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Abstract

An accurate and hybrid matrix technique is proposed for numerical treatments of a class of nonlinear parabolic PDEs occurring in the modeling of oil pollution in water. Using the Taylor series formula, the time variable is discretized, which converts the nonlinear model into a sequence of linearized problems continuous with respect to spatial variable. Then, a spectral collocation procedure based on novel shifted Morgan-Voyce (SMV) is applied to solve the resulting discretized equation at each time step yielding a linear algebraic system of equations. A rigorous error analysis shows that the proposed approach is uniformly convergent of order \(\mathcal {O}(\Delta \tau ^2+N^{-2})\), where \(\Delta \tau \) is the time step and N is the number of SMV basis. Three test examples including Allen–Cahn and Newell–Whitehead equations are provided to demonstrate the accuracy and efficiency of the presented hybrid collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

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Acknowledgements

The authors would like to thank the reviewers for their helpful and constructive comments that helped improving the manuscript. The support provided by Shahid Bahonar University of Kerman, Iran, and the German Jordanian University, Amman, Jordan, is greatly acknowledged by the authors.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Mohammad Izadi

  2. School of Basic Sciences and Humanities, German Jordanian University, Amman, Jordan

    Dia Zeidan

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  1. Mohammad Izadi
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  2. Dia Zeidan
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Correspondence to Mohammad Izadi.

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Communicated by Carla M.A. Pinto.

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Izadi, M., Zeidan, D. A convergent hybrid numerical scheme for a class of nonlinear diffusion equations. Comp. Appl. Math. 41, 318 (2022). https://doi.org/10.1007/s40314-022-02033-8

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  • DOI: https://doi.org/10.1007/s40314-022-02033-8

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