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

Papers by Mohamed Mbehou

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...

A Behavior of Shallow-Water Wave under Mittag-Leffler Law

Chapman and Hall/CRC eBooks, Sep 14, 2022

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

Scientific African

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.

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

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.

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.

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.

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

Results in Applied Mathematics

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.

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

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.

Numerical Analysis and Applications, 2019

The large neg a tive re sid ual Bouguer grav ity anom aly in north ern Po land called the Pomeran... more The large neg a tive re sid ual Bouguer grav ity anom aly in north ern Po land called the Pomerania Grav ity Low (PGL) was an alysed us ing Parker's ideal body the ory. A re sid ual grav ity anom aly along the pro file was in verted to find bounds on the den sity con trast, depth, and min i mum thick ness of its sources. As the ideal body reaches the sur face, the great est max i mum neg ative den sity con trast is -0.038 g/cm 3 , while the body it self has a thick ness of 52 km. If 8 km is taken as a depth to the source body top, the den sity con trast must cor re spond to at least -0.092 g/cm 3 , with a max i mum al low able thick ness of 18 km. The ideal body in ver sions show that the depth to the body top can not ex ceed 15 km. As sum ing a geo log i cally rea son able max imum den sity con trast as small as -0.2 g/cm 3 , the source body top can be no deeper than 11.5 km, and its thick ness greater than or equal to 6 km, as sum ing it ex tends up to the Earth sur face, or greater than or equal to 7 km, when its top is be low 8 km depth. It can be hy poth e sized that the main source of the neg a tive grav ity anom aly is re lated to a pre dom i nance of fel sic rocks in the Paleoproterozoic Dobrzyñ Do main of the East Eu ro pean Plat form base ment.

The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mathematical Methods in the Applied Sciences, 2017

Computers & Mathematics with Applications, 2018

We study two iterative schemes for the finite element approximation of the heat equation coupled ... more We study two iterative schemes for the finite element approximation of the heat equation coupled with Stokes flow under nonlinear slip boundary conditions of friction type. The iterative schemes are based on Uzawa's algorithm in which we decouple the computation of the velocity and pressure from that of the temperature by means of linearization. We derive some a priori estimates and prove convergence of these schemes. The theoretical results obtained are validated by means of numerical simulations.

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.

Journal of Scientific Computing, 2016

In this work, we study the penalty finite element approximation of the stationary power law Stoke... more In this work, we study the penalty finite element approximation of the stationary power law Stokes problem. We prove uniform convergence of the finite element solution with respect to the penalized parameter under classical assumptions on the weak solution. We formulate and analyze the convergence of a nonlinear saddle point problem by adopting a particular algorithm based on vanishing viscosity approach and long time behavior of an initial value problem. Finally, the predictions observed theoretically are validated by means of numerical experiments.

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.

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.

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.

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...

A Behavior of Shallow-Water Wave under Mittag-Leffler Law

Chapman and Hall/CRC eBooks, Sep 14, 2022

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

Scientific African

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.

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

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.

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.

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.

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

Results in Applied Mathematics

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.

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

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.

Numerical Analysis and Applications, 2019

The large neg a tive re sid ual Bouguer grav ity anom aly in north ern Po land called the Pomeran... more The large neg a tive re sid ual Bouguer grav ity anom aly in north ern Po land called the Pomerania Grav ity Low (PGL) was an alysed us ing Parker's ideal body the ory. A re sid ual grav ity anom aly along the pro file was in verted to find bounds on the den sity con trast, depth, and min i mum thick ness of its sources. As the ideal body reaches the sur face, the great est max i mum neg ative den sity con trast is -0.038 g/cm 3 , while the body it self has a thick ness of 52 km. If 8 km is taken as a depth to the source body top, the den sity con trast must cor re spond to at least -0.092 g/cm 3 , with a max i mum al low able thick ness of 18 km. The ideal body in ver sions show that the depth to the body top can not ex ceed 15 km. As sum ing a geo log i cally rea son able max imum den sity con trast as small as -0.2 g/cm 3 , the source body top can be no deeper than 11.5 km, and its thick ness greater than or equal to 6 km, as sum ing it ex tends up to the Earth sur face, or greater than or equal to 7 km, when its top is be low 8 km depth. It can be hy poth e sized that the main source of the neg a tive grav ity anom aly is re lated to a pre dom i nance of fel sic rocks in the Paleoproterozoic Dobrzyñ Do main of the East Eu ro pean Plat form base ment.

The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mathematical Methods in the Applied Sciences, 2017

Computers & Mathematics with Applications, 2018

We study two iterative schemes for the finite element approximation of the heat equation coupled ... more We study two iterative schemes for the finite element approximation of the heat equation coupled with Stokes flow under nonlinear slip boundary conditions of friction type. The iterative schemes are based on Uzawa's algorithm in which we decouple the computation of the velocity and pressure from that of the temperature by means of linearization. We derive some a priori estimates and prove convergence of these schemes. The theoretical results obtained are validated by means of numerical simulations.

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.

Journal of Scientific Computing, 2016

In this work, we study the penalty finite element approximation of the stationary power law Stoke... more In this work, we study the penalty finite element approximation of the stationary power law Stokes problem. We prove uniform convergence of the finite element solution with respect to the penalized parameter under classical assumptions on the weak solution. We formulate and analyze the convergence of a nonlinear saddle point problem by adopting a particular algorithm based on vanishing viscosity approach and long time behavior of an initial value problem. Finally, the predictions observed theoretically are validated by means of numerical experiments.

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.

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.

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.