A monotone iterative technique combined to finite element method for solving reaction-diffusion problems pertaining to non-integer derivative
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 soluti...
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01.08.2023
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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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| AbstractList | 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. |
| Author | Hammouch, Zakia Azroul, El Houssine Alaoui, Abdelilah Lamrani Hamou, Abdelouahed Alla |
| Author_xml | – sequence: 1 givenname: Abdelouahed Alla surname: Hamou fullname: Hamou, Abdelouahed Alla organization: Department of Mathematics, Faculty of Sciences Dhar Elmahraz, Sidi Mohamed Ben Abdellah University – sequence: 2 givenname: El Houssine surname: Azroul fullname: Azroul, El Houssine organization: Department of Mathematics, Faculty of Sciences Dhar Elmahraz, Sidi Mohamed Ben Abdellah University – sequence: 3 givenname: Zakia surname: Hammouch fullname: Hammouch, Zakia email: hammouch_zakia@tdmu.edu.vn organization: Division of Applied Mathematics, Thu Dau Mot University, Department of Medical Research, China Medical University Hospital, Department of Sciences, École normale supérieure, Moulay Ismail University of Meknes – sequence: 4 givenname: Abdelilah Lamrani surname: Alaoui fullname: Alaoui, Abdelilah Lamrani organization: Department of Mathematics, Regional Center of Education and Professional Training |
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| Keywords | Finite element method Non-integer derivative Monotone iterative technique Upper and lower solutions 35K91 35R11 34K37 Reaction-diffusion problems 35-XX 58J35 35K55 47J35 |
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