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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Bibliographic Details
Published in:Engineering with computers Vol. 39; no. 4; pp. 2515 - 2541
Main Authors: Hamou, Abdelouahed Alla, Azroul, El Houssine, Hammouch, Zakia, Alaoui, Abdelilah Lamrani
Format: Journal Article
Language:English
Published: London Springer London 01.08.2023
Springer Nature B.V
Subjects:
Boundary value problems
CAE) and Design
Calculus of Variations and Optimal Control; Optimization
Classical Mechanics
Computer Science
Computer-Aided Engineering (CAD
Control
Convergence
Exact solutions
Finite element analysis
Finite element method
Integers
Iterative methods
Math. Applications in Chemistry
Mathematical and Computational Engineering
Original Article
Sequences
Systems Theory
Finite element method
Non-integer derivative
Monotone iterative technique
Upper and lower solutions
35K91
35R11
34K37
Reaction-diffusion problems
35-XX
58J35
35K55
47J35
ISSN:0177-0667, 1435-5663
Online Access:Get full text
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Summary: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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ISSN:0177-0667
1435-5663
DOI:10.1007/s00366-022-01635-4