Mohamed Khader

Benha University · Department of Mathematics

This study simulates the coronavirus infection (Covid-19) which is given as a mathematical model based on a set of ordinary differential equations. We provide this numerical treatment and simulation by using the finite element method (FEM). The proposed model is reduced to a set of algebraic equations using the FEM. The goal of this research is to...

This study aims to elucidates the effects of Ohmic dissipation and the magnetic field on the behavior of a Casson fluid flowing across a vertically stretched surface. The goal is to solve the problem by using numerical approaches. Furthermore, the fluid’s thermal conductivity is intended to vary proportionately with temperature. The effects of ther...

A research investigation is performed to examine the impact of non-uniform heat generation and viscous dissipation on the boundary layer flow of a power-law nanofluid over a non-linearly stretching sheet. Within the thermal domain, the analysis takes into account both thermal radiation and variable thermal conductivity. Through the utilization of s...

The objective of this article is to investigate how the properties of a non-Newtonian Williamson nanofluid flow, which occurs due to an exponential stretching sheet placed in a porous medium, are influenced by heat generation, viscous dissipation, and magnetic field. This study focuses on analyzing the heat transfer process by considering the impac...

The current research examines the rate of heat and mass transfer in MHD non-Newtonian Williamson nanofluid flow across an exponentially permeable stretched surface sensitive to heat generation/absorption and mass suction. The influences of Brownian motion and thermophoresis are included. In addition, the stretched surface is subjected to an angled...

Simulation and numerical study for the blood ethanol concentration system (BECS) and the Lotka-Volterra system, i.e., predator-prey equations (PPEs) (both of fractional order in the Caputo sense) by employing a development accurate variational iteration method are presented in this work. By assessing the absolute error, and the residual error funct...

Analysis of a steady flow of a viscous Casson fluid subject to Ohmic dissipation and an induced magnetic field is the main goal here. Through a stretched vertical sheet, the flow is managed. The energy equation is explained in a thermodynamical system along with Ohmic heating. By making the assumption that the viscosity and thermal conductivity are...

Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. A rough formula for the Caputo fractional derivative is...

This work aimed to present the influence of the magnetic field and Ohmic dissipation on the non-Newtonian Casson fluid on a vertical stretched sheet to numerically solve the problem. Here, the variable thermal conductivity is taken as a linear function of temperature. Electric fields, thermal slip, and viscous dissipation effects are taken into con...

The main goal of the research is to investigate the effects of temperature and concentration slip on the MHD Casson-Williamson nanofluid flow through a porous sheet, with a focus on viscous dissipation. This research has a novel approach, as it seeks to incorporate various factors that have not been previously studied in this context. Moreover, an...

The purpose of this research is to examine the effects of heat generation, viscous dissipation, and magnetic field on the characteristics of non-Newtonian Williamson nanofluid flow caused by an exponential stretching sheet implanted in a porous media. The process of heat transfer is examined while taking into account how the temperature affects the...

This study simulates the start of coronavirus infection using a mathematical model based on ordinary differential equations (Covid-19). Additionally, provide the numerical treatment and simulation of this model using the finite element approach (FEM). The goal of this research is to stop and slow the spread of a sickness that is ravaging the globe....

The present study is made to develop the fractional model of non-Newtonian Casson and Williamson boundary layer flow in the fluid flow taking into account the heat flux and the slip velocity. The temperature and the velocity fields, of the steady boundary layer flow, are generated by a stretched sheet with a non-uniform thickness. The governing non...

Heat transfer is a critical function in many technical, industrial, home, and commercial structures. As a result, the purpose of this study is to investigate the effects of slip velocity and variable fluid characteristics on Casson bionanofluid flow across a stretching sheet that has been saturated by gyrotactic microorganisms. The suggested system...

The fractional variable-order (VO) two-dimensional (2Dim) Cable equation is one of the most significant types of anomalous subdiffusion equations that emerge strongly in spiny neural dendrites and is solved by using an accurate numerical technique in this study. The non-standard weighted average finite difference approach is a simple proposed techn...

Theoretical investigation of magnetohydrodynamics (MHD) Casson and Williamson fluid flow and heat and mass transfer in laminar flow through a stretching sheet in the presence of heat generation is carried out in this study. The convective wall temperature and convective wall mass boundary condition are taken into account in this study. A study is a...

In the presented work, we present an accurate procedure, which is the spectral method, to find a solution to a certain class of the very important fractional (described by the Liouville–Caputo sense) models of the electrical RL, RC, and RLC circuits. This method is collocated using some important advantages of the generalized Legendre polynomials t...

The idea of the current investigation is to analyze the effect of thermal radiation and non-uniform heat source/sink on unsteady MHD micropolar fluid flow past a stretching/shirking sheet. The governing non-linear PDEs are transformed into a set of non-linear coupled ODEs which are then solved numerically by using the fourth order predictor–correct...

In this paper, the approximate solutions for systems of nonlinear algebraic equations by the power series method (PSM) are presented. Illustrative examples have been presented to demonstrate the efficiency of the proposed method. In addition, the obtained results are compared with those obtained from the standard Adomian decomposition method. It tu...

We implement an efficient computational scheme to study the effect of precursor consumption on chemical clock reactions. The proposed model is formulated as a system of FDEs with power kernel. This paper considers the fractional derivatives of Liouville–Caputo (LC). We use the spectral collocation method (SCM) with the help of the third-kind Chebys...

The purpose of this paper is to investigate the spectral collocation method with help of Chebyshev polynomials. We consider the space fractional Korteweg-de Vries and the space fractional Korteweg-de Vries-Burger's equations based on the Caputo-Fabrizio fractional derivative. The proposed method reduces the models under study to a set of ordinary d...

In the presented study, we are presenting the approximate solutions of two important equations, the Riccati and Logistic equations; the presented technique is based on the rational Legendre function. Since all the studied models are nonlinear, we convert these nonlinear equations to a sequence of linear ordinary differential equations (ODEs), then...

This article investigates a family of approximate solutions for the fractional model (in the Liouville-Caputo sense) of the Ebola virus via an accurate numerical procedure (Chebyshev spectral collocation method). We reduce the proposed epidemiological model to a system of algebraic equations with the help of the properties of the Chebyshev polynomi...

The purpose of this paper is to implement an approximate method for obtaining the solution of a physical model called the blood ethanol concentration system. This model can be expressed by a system of fractional differential equations (FDEs). Here, we will consider two forms of the fractional derivative namely, Caputo (with singular kernel) and Ata...

In this paper, we present an accurate numerical method to compute the approximate solutions of the Korteweg–de Vries, Korteweg–de Vries–Burger’s and Burger’s equations with Liouville–Caputo fractional space derivatives, respectively. We implement the spectral collocation method based on the shifted Chebyshev polynomials. The method reduces each mod...

This paper is devoted to introduce an efficient solver using a combination of the symbol of the operator and the windowed Fourier frames (WFFs) of the coupled system of second order ordinary differential equations. The given system has a basic importance in modeling various phenomena like, Cascades and Compartment Analysis, Pond Pollution, Home Hea...

The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equ...

In this article, we implement a spectral collocation method by using the properties of Legendre and Lagrange polynomials for solving the resulting nonlinear system of ODEs of the biochemical reaction model. This technique reduces the proposed model to a system of algebraic equations. We prove the uniqueness and present the local stability of the gi...

Here, we introduce a numerical solution by using the generalized Euler method for the (Caputo sense) fractional Susceptible-Infected-Recovered (SIR) model with a constant vaccination rate. We compare the obtained numerical solutions with those solutions by using the RK4. Hence, the obtained numerical results of the SIR model show the simplicity and...

We apply the operational matrices of fractional integration for Chebyshev wavelets for solving the fractional (Caputo form) Logistic differential equation (FLDE). We introduce a study of the convergence analysis and error estimation of the obtained approximation solution. The FLDE is reduced to a system of algebraic equations with the help of the p...

This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev...

This article is devoted to introduce a numerical treatment using Adams–Bashforth–Moulton method of the fractional model of HIV-1 infection of CD4\(^{+}\) T-cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. The fractional derivative is...

In this paper, we introduce a method based on replacement of the unknown function by truncated series of the well-known shifted Chebyshev (of third-kind) expansion of functions. We give an approximate formula for the integer derivative of this expansion. We state and prove some theorems on the convergence analysis. By means of collocation points th...

The proposed method is based on replacement of the unknown function by a truncated series of the shifted Legendre polynomial expansion. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error for the presented approximate formula. The introdu...

Herein, we study the numerical solution with the help of Chebyshev spectral collocation method for the ordinary differential equations which describe the flow of viscoelastic fluid over a stretching sheet embedded in a porous medium with viscous dissipation and slip velocity. The novel effects for the parameters which affect the flow and heat trans...

In this paper, Legendre spectral method is presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Caputo sense. The properties of the Legendre polynomials are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Mo...

In this paper, we implement the fractional complex transform method to convert the nonlinear fractional Klein-Gordon equation (FKGE) to an ordinary differential equation. We use the variational iteration method (VIM) to solve the resulting ODE. The fractional derivatives are presented in terms of the Caputo sense. Some numerical examples are presen...

The main aim of the present work is to present a new and simple algorithm for time fractional heat like physical models by using the new fractional homotopy analysis transform method (FHATM). The proposed method is an innovative adjustment in the Laplace transform algorithm (LTA) for fractional partial differential equations and makes the calculati...

In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement this technique to study the approximate solution of the hepatitis C model with different types of virus genome. The Hepatitis C virus (HCV) is a singlestranded RNA virus. The genomes of HCV display significant seque...

In this paper, we introduce a numerical treatment using fractional differential transform method (FDTM) of the fractional model of HIV-1 infection of CD4+ T-cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. The fractional derivative is...

In this paper, we implement the Adomian decomposition method (ADM) to approximate the deflection of polysilicon diaphragm with small flexural rigidity of Micro Electro Mechanical System (MEMS) capacitive microphone. We prove the existence and the uniqueness of the solution of considered problem with the help of the theory of semi-group. Special att...

In this article, a numerical study is introduced for solving the fractional wave equations by using an efficient class of finite difference methods. The proposed scheme is based on the Hermite formula. The stability and the convergence analysis of the proposed methods are given by a recently proposed procedure similar to the standard von Neumann st...

This paper is devoted to present an implementation of Legendre wavelets for solving fractional (Caputo) Logistic differential equation (FLDE). In the proposed method, the operational matrices of fractional integration for Legendre wavelets are used. We present a study to the convergence analysis. The properties of wavelets polynomials approximation...

This article is devoted to describe the boundary layer flow and heat transfer for non-Newtonian Powell–Eyring fluid over an exponentially stretching continuous impermeable surface with an exponential temperature distribution taking into account variable thermal conductivity. The fluid thermal conductivity is assumed to vary as a linear function of...

In this article, we introduce an efficient numerical method for solving the linear fractional Klein-Gordon equation (LFKGE). The proposed scheme is based on combined an efficient class of FDMs with Hermite formula. Special attention is given to study the stability analysis and the convergence of the proposed methods by a recently proposed procedure...

In this paper, Legendre spectral method is presented to study the approximate solution of the fractional hepatitis C model with different types of virus genome. The Hepatitis C virus (HCV) is a single-stranded RNA virus. The genomes of HCV display significant sequence heterogeneity and have been classified into types and subtypes. Types from 1 to 1...

In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for solving fractional SIRC model. The fractional derivative is described in Caputo sense. Special attention is given to present the local stability of the proposed model. We compare our numerical solutions with those numerical solutions using fourth-order Runge-...

This paper is devoted to introduce a numerical simulation with a theoretical study for flow of a Newtonian fluid over an impermeable stretching sheet which embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal condu...

In this paper, we implement an efficient numerical technique which we call fractional Chebyshev finite difference method (FChFDM). The fractional derivatives are presented in terms of Caputo sense. The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The proposed techn...

In this paper, we implement the shifted Jacobi operational matrix of derivative with spectral tau method and collocation method for numerical solution for the systems of linear and non-linear ordinary differential equations subject to initial or boundary conditions. By means of this approach, such problems are reduced for solving a system of algebr...

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by th...

The main aim of this article is to introduce the approximate solution for MHD flow of an electrically conducting Newtonian fluid over an impermeable stretching sheet with a power law surface velocity and variable thickness in the presence of thermal-radiation and internal heat generation/absorption. The flow is caused by the non-linear stretching o...

This paper is devoted to implementing the Legendre spectral collocation method to introduce numerical solutions of a certain class of fractional variational problems (FVPs). The properties of the Legendre polynomials and Rayleigh-Ritz method are used to reduce the FVPs to the solution of system of algebraic equations. Also, we study the convergence...

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed...

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear or...

In this article, an implementation of an efficient numerical method for solving the system of coupled nonlinear fractional diffusion equations (NFDEs) is introduced. The proposed system has many applications, such as porous media and plasma transport. The fractional derivative is described in the Caputo sense. The method is based upon a combination...

In this article, an implementation of an efficient numerical method for solving the linear fractional Klein–Gordon equation (LFKGE) is introduced. The fractional derivative is described in the Caputo sense. The method is based upon a combination between the properties of the Chebyshev approximations and finite difference method (FDM). The proposed...

In this article, we present a new numerical method to solve the integro-differential equations (IDEs). The proposed method uses the Legendre cardinal functions to express the approximate solution as a finite series. In our method the operational matrix of derivatives is used to reduce IDEs to a system of algebraic equations. To demonstrate the vali...

This paper is devoted to present an efficient approximate method for solving a certain class of fractional variational problems (FVPs). In the proposed method, we used the properties of Rayleigh-Ritz method and chain rule for fractional calculus to reduce FVPs to solve a system of algebraic equations which solved using a suitable numerical method....

In this paper, Legendre spectral method is presented to study the approximate solution of fractional SIRC model. The fractional derivative is described in the Caputo sense. The properties of the Legendre polynomials are used to reduce the proposed method to the solution of nonlinear system of algebraic equations using Newton iteration method. Moreo...

A numerical method is given for studying the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid towards a permeable stretching surface embedded in a porous medium with a second order slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using...

In this article, the homotopy analysis method (HAM) is implemented for obtaining semi-analytical solutions to the problem of the nonlinear vibrations of multiwalled carbon nanotubes embedded in an elastic medium. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the Van...

This paper is devoted to introduce a numerical simulation using the implicit finite difference method (FDM) with the theoretical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid towards a permeable stretching surface embedded in a porous medium with a second-order slip. The governing non-linear pa...

In this paper, A Chebyshev spectral method is presented to study the deals with the fractional SIRC model associated with the evolution of influenza A disease in human population. The properties of the Chebyshev polynomials are used to derive an approximate formula of the Caputo fractional derivative. This formula reduces the SIRC model to the solu...

In this paper, we present approximate analytical solution of the time-fractional biological population equation using the fractional iteration method (FIM). The fractional derivatives are described in the Caputo sense. The fractional complex transform (FCT) with help of the variational iteration method (VIM) is used to obtain the approximate soluti...

In this article, a new formula for Adomian's polynomials is introduced. It is applied to obtain the truncated series solutions for the fractional initial value problems with non-differentiable functions. This kind of equations contains a fractional single-term which is examined using Jumarie fractional derivatives and fractional Taylor series for n...

In this article, a numerical study for fractional reaction-subdiffusion equations is introduced using a class of finite difference methods. These methods are extensions of the weighted average methods for ordinary (non-fractional) reaction-subdiffusion equations. A stability analysis of the proposed methods is given by a recently proposed procedure...

In this paper, a matrix method for the approximate solution of high order fractional differential equations (FDEs) in terms of a truncated Legendre series is presented. The FDEs and its initial or boundary conditions are transformed to matrix equations, which correspond to a system of algebraic equations with unknown Legendre coefficients. The solu...

In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A...

In this paper, two efficient numerical methods for solving systems of fractional differential equations (SFDEs) are considered. The fractional derivative is described in the Caputo sense. The first method is based upon Chebyshev approximations, where the properties of Chebyshev polynomials are utilized to reduce SFDEs to system of algebraic equatio...

In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution...

In this paper, we are implemented the Chebyshev spectral method for solving the non-linear fractional Klein-Gordon equation (FKGE). The fractional derivative is considered in the Caputo sense. We presented an approximate formula of the fractional derivative. The properties of the Chebyshev polynomials are used to reduce FKGE to the solution of syst...

In this article, we introduce a modification of the power series method by using the Padé approximation with Laplace transform and apply it to solve systems of linear ordinary differential equations. This modification yields a series solution with accelerated convergence and obtains the solution in a closed form in most cases. Illustrative examples...

In this paper, we implement Chebyshev pseudo-spectral method for solving numerically system of linear and non-linear fractional integro-differential equations of Volterra type. The proposed technique is based on the new derived formula of the Caputo fractional derivative. The suggested method reduces this type of systems to the solution of system o...

In this article, an accurate numerical approach is introduced. In this approach we mixed between the fractional finite difference method and the restrictive Taylor approximation (RTA). The proposed method is implemented to solve numerically the perturbed fractional partial differential equations (FPDEs). Special attention is given to study the stab...

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a...

This paper presents an accurate numerical method for solving fractional SIRC model. In this work, we propose a method so called fractional Chebyshev finite difference method. In this technique, we approximate the proposed model with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomial...

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. Fractional advection-dispersion equation (FADE) is used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium and for modeling transpor...

This paper is devoted with numerical solution of the system fractional differential equations (FDEs) which are generated by optimization problem using the Chebyshev collocation method. The fractional derivatives are presented in terms of Caputo sense. The application of the proposed method to the generated system of FDEs leads to algebraic system w...

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet with a power law surface velocity, slip velocity and variable thickness. The flow is caused by a nonlinear stretching of a sheet. The governing partial differential equations are transformed into a nonlinear ordinary differential equati...

In this paper, a new approximate formula of the fractional derivative is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the er...

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The meth...