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A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative

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Abstract

This paper focuses on some mathematical and numerical aspects of reaction-diffusion problems pertaining to non-integer time derivatives using the well-known method of lower and upper solutions combined with the monotone iterative technique. First, we study the existence and uniqueness of weak solutions of the proposed models, then we prove some comparison results. Besides, linear finite element spaces on triangles are used to discretize the problem in space, whereas the generalized backward-Euler method is adopted to approximate the time non-integer derivative. Furthermore, the idea of this method is to construct two sequences of solutions of a linear initial value problem which are easier to compute and converge to the solution of the nonlinear problem. We show numerically through two examples that this convergence requires only few iterations. Some well-known examples with exact solutions and numerical results based on the finite element method in 2D are provided to validate the theoretical results. As a result, we confirm that the proposed method is efficient and easy to use to overcome the convergence and stability difficulties.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Dhar Elmahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas, 30000, Fez, Morocco

    Abdelouahed Alla Hamou & El Houssine Azroul

  2. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Zakia Hammouch

  3. Department of Medical Research, China Medical University Hospital, Taichung, Taiwan

    Zakia Hammouch

  4. Department of Sciences, École normale supérieure, Moulay Ismail University of Meknes, Meknes, Morocco

    Zakia Hammouch

  5. Department of Mathematics, Regional Center of Education and Professional Training, B.P. 49, 30000, Fez, Morocco

    Abdelilah Lamrani Alaoui

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  1. Abdelouahed Alla Hamou
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Correspondence to Zakia Hammouch.

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Hamou, A.A., Azroul, E., Hammouch, Z. et al. A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative. Engineering with Computers 39, 2515–2541 (2023). https://doi.org/10.1007/s00366-022-01635-4

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