Mohamed Mbehou - Academia.edu (original) (raw)

Papers by Mohamed Mbehou

Research paper thumbnail of Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

Arab Journal of Mathematical Sciences

PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-o... more PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.Design/methodology/approachA rigorous error analysis is done, and optimal L2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.FindingsOptimal L2 error estimates and energy norm.Originality/valueThe goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal e...

Research paper thumbnail of A Behavior of Shallow-Water Wave under Mittag-Leffler Law

Chapman and Hall/CRC eBooks, Sep 14, 2022

Research paper thumbnail of Finite element method for nonlocal problems of Kirchhoff-type in domains with moving boundary

Research paper thumbnail of Finite Element Method for a Reaction-Diffusion Model for the Nonlocal Coupled System

A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonl... more A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonlinear coupled system of the reaction-diffusion problem is presented here. We establish the convergence and error bound for the fully discrete scheme. Some important results on exponential decay and vanishing of the solutions in finite time are also presented. Numerical experiments are performed using Matlab.

Research paper thumbnail of Coexistence and harvesting control policy in a food chain model with mutual defense of prey

Natural Resource Modeling, 2019

A model is proposed to understand the dynamics in a food chain (one predator-two prey). Unlike ma... more A model is proposed to understand the dynamics in a food chain (one predator-two prey). Unlike many approaches, we consider mutualism (for defense against predators) between the two groups of prey. We investigate the conditions for coexistence and exclusion. Unlike Elettreby's (2009) results, we show that prey can

Research paper thumbnail of Stokes and Navier-Stokes equations under power law slip boundary condition: Numerical analysis

Computers & Mathematics with Applications

In this work, we study theoretically and numerically the equations of Stokes and Navier-Stokes un... more In this work, we study theoretically and numerically the equations of Stokes and Navier-Stokes under power law slip boundary condition. We establish existence of a unique solution by using the monotone operators theory for the Stokes equations whereas for the Navier-Stokes equations, we construct the solution by means of Galerkin's approximation combined with some compactness results. Next, we formulate and analyze the finite element approximations associated to these problems. We derive optimal and sub-optimal a priori error estimate for both problems depending how the monotonicity is used. Iterative schemes for solving the nonlinear problems are formulated and convergence is studied. Numerical experiments presented confirm the theoretical findings.

Research paper thumbnail of Numerical Analysis of a Darcy-Forchheimer Model Coupled with the Heat Equation

Journal of Scientific Computing

This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlin... more This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlinear viscosity depending on the temperature coupled with the heat equation. We show unique solvability of the variational problem by using; Galerkin method, Brouwer's fixed point and some compactness properties. We propose and study in detail a finite element approximation. A priori error estimate is then derived and convergence is obtained. A solution technique is formulated to solve the nonlinear problem and numerical experiments that validate the theoretical findings are presented.

Research paper thumbnail of Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

Numerical Methods for Partial Differential Equations, 2019

In this work, we consider the heat equation coupled with Stokes equations under threshold type bo... more In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.

Research paper thumbnail of On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

Results in Applied Mathematics

Research paper thumbnail of Stability Analysis of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations Driven by Slip Boundary Conditions

Journal of Applied Mathematics, 2022

This paper is devoted to the study of numerical approximation for a class of two-dimensional Navi... more This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in L 2 -norm and H 1 -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H 1 -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in L 2 and H 1 -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Research paper thumbnail of Crank-Nicolson-Galerkin FEM for nonlocal p-Laplace problems depending on the W1,p 0 –norm

Research paper thumbnail of The Crank-Nicolson-Galerkin FEM for a nonlocal parabolic system

A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonl... more A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonlocal nonlinear coupled system of the reaction-diffusion problem is presented here. For sufficiently smooth solutions, the maximal error in the L-norm possesses the optimal rate of convergence O(δ + h) (where h is the mesh size and δ is the time step size with r ≥ 1) without any time step restriction. Some important results on the energy decay and vanishing of the solutions in finite time are also presented. To confirm our theoretical analysis, some numerical experiments are performed using Matlab.

Research paper thumbnail of Numerical Methods for a Nonlocal Parabolic Problem with Nonlinearity of Kirchhoff Type

Numerical Analysis and Applications, 2019

Research paper thumbnail of The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mathematical Methods in the Applied Sciences, 2017

Research paper thumbnail of Iterative methods for Stokes flow under nonlinear slip boundary condition coupled with the heat equation

Computers & Mathematics with Applications, 2018

Research paper thumbnail of A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics

Chaos, Solitons & Fractals, 2018

Abstract Recent discussions on the non validity of index law in fractional calculus have shown th... more Abstract Recent discussions on the non validity of index law in fractional calculus have shown the amazing filtering feature of Mittag–Leffler function foreseing Atangana–Baleanu derivative as one of reliable mathematical tools for describing some complex world problems, like problems of neuronal activities. In this paper, neuronal dynamics described by a three dimensional model of Hindmarsh–Rose nerve cells with external current are analyzed analytically and numerically. We make use of the Atangana–Baleanu fractional derivative in Caputo sense (ABC derivative) and asses its impact on the dynamic, especially the role played by its derivative order in combination with another control parameter, the intensity of the applied external current. Our analysis proves existence of equilibria whose some are unstable of type saddle point, paving the ways for possible bifurcations in the process. Numerical approximations of solutions reveal a system with initially regular bursts that evolve into period-adding chaotic bifurcations as the control parameters change, with precisely the Atangana–Baleanu fractional derivative’s order decreasing from 1 down to 0.5.

Research paper thumbnail of On the Numerical Solution of the Stationary Power-Law Stokes Equations: A Penalty Finite Element Approach

Journal of Scientific Computing, 2016

Research paper thumbnail of Numerical Analysis for a Nonlocal Parabolic Problem

East Asian Journal on Applied Mathematics, 2016

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear... more This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Research paper thumbnail of Analysis of a New MPFA Formulation for Flow Problems in Geologically Complex Media

Mathematical Modeling, Simulation, Visualization and e-Learning, 2008

This work analyzes some mathematical aspects of a new Multi-Point Flux Approximation (MPFA) formu... more This work analyzes some mathematical aspects of a new Multi-Point Flux Approximation (MPFA) formulation for flow problems. This MPFA formulation has been developed in [12, 13] for quadrilateral grids and [10] for unstructured grids. Our MPFA formulation displays capabilities for handling flow problems in geologically complex media modelled by spatially varying full permeability tensor. However in this work, we focus our attention on the case of anisotropic homogeneous porous media. In this framework, the proposed MPFA formulation leads to a well-posed discrete problem which is a linear system whose associated matrix is symmetric and positive definite, even if the permeability tensor governing the flow is only positive definite. Following the spirit of the finite element theory, we have introduced the concept of globally continuous and piecewise linear approximate solution. The convergence analysis of this solution is strongly based upon another concept: the weak approximate solution. Stability and convergence results for the weak approximate solution are proven for L 2-and L ∞-norm, and for a discrete energy norm as well. These results permit to prove some error estimates related to the globally continuous and piecewise linear approximate solution.

Research paper thumbnail of Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

Journal of Numerical Mathematics, 2015

In this work, we are concerned with the finite element approximation for the stationary power law... more In this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.

Research paper thumbnail of Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

Arab Journal of Mathematical Sciences

PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-o... more PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.Design/methodology/approachA rigorous error analysis is done, and optimal L2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.FindingsOptimal L2 error estimates and energy norm.Originality/valueThe goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal e...

Research paper thumbnail of A Behavior of Shallow-Water Wave under Mittag-Leffler Law

Chapman and Hall/CRC eBooks, Sep 14, 2022

Research paper thumbnail of Finite element method for nonlocal problems of Kirchhoff-type in domains with moving boundary

Research paper thumbnail of Finite Element Method for a Reaction-Diffusion Model for the Nonlocal Coupled System

A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonl... more A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonlinear coupled system of the reaction-diffusion problem is presented here. We establish the convergence and error bound for the fully discrete scheme. Some important results on exponential decay and vanishing of the solutions in finite time are also presented. Numerical experiments are performed using Matlab.

Research paper thumbnail of Coexistence and harvesting control policy in a food chain model with mutual defense of prey

Natural Resource Modeling, 2019

A model is proposed to understand the dynamics in a food chain (one predator-two prey). Unlike ma... more A model is proposed to understand the dynamics in a food chain (one predator-two prey). Unlike many approaches, we consider mutualism (for defense against predators) between the two groups of prey. We investigate the conditions for coexistence and exclusion. Unlike Elettreby's (2009) results, we show that prey can

Research paper thumbnail of Stokes and Navier-Stokes equations under power law slip boundary condition: Numerical analysis

Computers & Mathematics with Applications

In this work, we study theoretically and numerically the equations of Stokes and Navier-Stokes un... more In this work, we study theoretically and numerically the equations of Stokes and Navier-Stokes under power law slip boundary condition. We establish existence of a unique solution by using the monotone operators theory for the Stokes equations whereas for the Navier-Stokes equations, we construct the solution by means of Galerkin's approximation combined with some compactness results. Next, we formulate and analyze the finite element approximations associated to these problems. We derive optimal and sub-optimal a priori error estimate for both problems depending how the monotonicity is used. Iterative schemes for solving the nonlinear problems are formulated and convergence is studied. Numerical experiments presented confirm the theoretical findings.

Research paper thumbnail of Numerical Analysis of a Darcy-Forchheimer Model Coupled with the Heat Equation

Journal of Scientific Computing

This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlin... more This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlinear viscosity depending on the temperature coupled with the heat equation. We show unique solvability of the variational problem by using; Galerkin method, Brouwer's fixed point and some compactness properties. We propose and study in detail a finite element approximation. A priori error estimate is then derived and convergence is obtained. A solution technique is formulated to solve the nonlinear problem and numerical experiments that validate the theoretical findings are presented.

Research paper thumbnail of Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

Numerical Methods for Partial Differential Equations, 2019

In this work, we consider the heat equation coupled with Stokes equations under threshold type bo... more In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.

Research paper thumbnail of On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

Results in Applied Mathematics

Research paper thumbnail of Stability Analysis of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations Driven by Slip Boundary Conditions

Journal of Applied Mathematics, 2022

This paper is devoted to the study of numerical approximation for a class of two-dimensional Navi... more This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in L 2 -norm and H 1 -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H 1 -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in L 2 and H 1 -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Research paper thumbnail of Crank-Nicolson-Galerkin FEM for nonlocal p-Laplace problems depending on the W1,p 0 –norm

Research paper thumbnail of The Crank-Nicolson-Galerkin FEM for a nonlocal parabolic system

A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonl... more A theoretical analysis of a linearized Crank-Nicolson Galerkin finite element method for the nonlocal nonlinear coupled system of the reaction-diffusion problem is presented here. For sufficiently smooth solutions, the maximal error in the L-norm possesses the optimal rate of convergence O(δ + h) (where h is the mesh size and δ is the time step size with r ≥ 1) without any time step restriction. Some important results on the energy decay and vanishing of the solutions in finite time are also presented. To confirm our theoretical analysis, some numerical experiments are performed using Matlab.

Research paper thumbnail of Numerical Methods for a Nonlocal Parabolic Problem with Nonlinearity of Kirchhoff Type

Numerical Analysis and Applications, 2019

Research paper thumbnail of The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mathematical Methods in the Applied Sciences, 2017

Research paper thumbnail of Iterative methods for Stokes flow under nonlinear slip boundary condition coupled with the heat equation

Computers & Mathematics with Applications, 2018

Research paper thumbnail of A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics

Chaos, Solitons & Fractals, 2018

Abstract Recent discussions on the non validity of index law in fractional calculus have shown th... more Abstract Recent discussions on the non validity of index law in fractional calculus have shown the amazing filtering feature of Mittag–Leffler function foreseing Atangana–Baleanu derivative as one of reliable mathematical tools for describing some complex world problems, like problems of neuronal activities. In this paper, neuronal dynamics described by a three dimensional model of Hindmarsh–Rose nerve cells with external current are analyzed analytically and numerically. We make use of the Atangana–Baleanu fractional derivative in Caputo sense (ABC derivative) and asses its impact on the dynamic, especially the role played by its derivative order in combination with another control parameter, the intensity of the applied external current. Our analysis proves existence of equilibria whose some are unstable of type saddle point, paving the ways for possible bifurcations in the process. Numerical approximations of solutions reveal a system with initially regular bursts that evolve into period-adding chaotic bifurcations as the control parameters change, with precisely the Atangana–Baleanu fractional derivative’s order decreasing from 1 down to 0.5.

Research paper thumbnail of On the Numerical Solution of the Stationary Power-Law Stokes Equations: A Penalty Finite Element Approach

Journal of Scientific Computing, 2016

Research paper thumbnail of Numerical Analysis for a Nonlocal Parabolic Problem

East Asian Journal on Applied Mathematics, 2016

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear... more This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Research paper thumbnail of Analysis of a New MPFA Formulation for Flow Problems in Geologically Complex Media

Mathematical Modeling, Simulation, Visualization and e-Learning, 2008

This work analyzes some mathematical aspects of a new Multi-Point Flux Approximation (MPFA) formu... more This work analyzes some mathematical aspects of a new Multi-Point Flux Approximation (MPFA) formulation for flow problems. This MPFA formulation has been developed in [12, 13] for quadrilateral grids and [10] for unstructured grids. Our MPFA formulation displays capabilities for handling flow problems in geologically complex media modelled by spatially varying full permeability tensor. However in this work, we focus our attention on the case of anisotropic homogeneous porous media. In this framework, the proposed MPFA formulation leads to a well-posed discrete problem which is a linear system whose associated matrix is symmetric and positive definite, even if the permeability tensor governing the flow is only positive definite. Following the spirit of the finite element theory, we have introduced the concept of globally continuous and piecewise linear approximate solution. The convergence analysis of this solution is strongly based upon another concept: the weak approximate solution. Stability and convergence results for the weak approximate solution are proven for L 2-and L ∞-norm, and for a discrete energy norm as well. These results permit to prove some error estimates related to the globally continuous and piecewise linear approximate solution.

Research paper thumbnail of Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

Journal of Numerical Mathematics, 2015

In this work, we are concerned with the finite element approximation for the stationary power law... more In this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.