Mohamed Khader

Mohamed Khader
Benha University · Department of Mathematics

About

131
Publications
24,037
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3,679
Citations
Citations since 2016
34 Research Items
2317 Citations
20162017201820192020202120220100200300400
20162017201820192020202120220100200300400
20162017201820192020202120220100200300400
20162017201820192020202120220100200300400

Publications

Publications (131)
Article
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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...
Article
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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...
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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...
Article
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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...
Article
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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...
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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...
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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...
Article
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...
Article
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...
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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...
Article
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...
Article
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...
Article
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...
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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...
Article
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...
Article
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...
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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...
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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...
Article
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...
Article
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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...
Article
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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...
Article
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...
Article
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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...
Article
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...
Article
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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...
Article
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...
Article
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...
Article
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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...
Article
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...
Article
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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...
Article
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...
Article
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...
Article
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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...
Article
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...
Article
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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-...
Article
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...
Article
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...
Article
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...
Article
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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...
Article
Full-text available
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...
Article
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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...
Article
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...
Article
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...
Article
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...
Article
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...
Article
Full-text available
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...
Article
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....
Article
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...
Article
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...
Article
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...
Article
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...
Article
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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...
Article
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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...
Article
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...
Article
Full-text available
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...
Article
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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...
Article
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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...
Article
Full-text available
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...
Article
Full-text available
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...
Article
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...
Article
Full-text available
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...
Article
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...
Article
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...
Article
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...
Article
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...
Article
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...
Article
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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...
Article
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...
Article
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...
Article
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...
Article
Full-text available
This article is devoted to use the variational iteration method (VIM) established by J.H. He for solving linear and nonlinear delay differential equations (DDEs). This method is based on the use of Lagrange multiplier for identification of optimal value of a parameter in a functional. This procedure is a powerful tool for solving large amount of pr...
Article
A new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. Th...
Article
Full-text available
In this Letter, we introduced a modification of the Picard iteration method (PIM) using Padé approximation and the so called Picard–Padé technique. This technique is used to solve the chemical kinetics problem. This problem is formed by a system of nonlinear ordinary differential equations. Special attention is given to study the convergence analys...
Article
We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential—difference equations. The proposed method is based on the Laplace transform with the homotopy analysis method (HAM). This method is a powerful tool for solving a large amount of problems....
Article
An analysis is carried out to study the unsteady two-dimensional Powell-Eyring flow and heat transfer to a laminar liquid film from a horizontal stretching surface in the presence of internal heat generation. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of a similarity transform...
Article
Full-text available
This paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev...
Article
This article looks at the flow and heat transfer in the unsteady two-dimensional boundary layer of a non-Newtonian Maxwell fluid over a stretching sheet in the presence of variable fluid properties and internal heat generation. The governing differential equations are transformed into a set of coupled non-linear ordinary differential equations and...
Article
The Lie group method is applied to present an analysis of the magneto hydro-dynamics (MHD) steady laminar flow and the heat transfer from a warm laminar liquid flow to a melting moving surface in the presence of thermal radiation. By using the Lie group method, we have presented the transformation groups for the problem apart from the scaling group...
Article
In this paper, we introduce a modification of the Picard iteration method (PIM) using Padé approximation and so called Picard-Padé technique. Special attention is given to study the convergence analysis of the proposed method. Convergence analysis is reliable enough to estimate the maximum absolute error of the solution given by PIM. A basic enzyme...
Article
Full-text available
This paper is devoted to introduce a numerical simulation using finite difference method with the theoretical study for the problem of the flow and heat transfer over an unsteady stretching sheet embedded in a porous medium in the presence of a thermal radiation. The continuity, momentum and energy equations, which are coupled nonlinear partial dif...
Article
Full-text available
In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is base...
Article
Full-text available
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...
Article
Full-text available
A new approximate formula of the fractional derivatives 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 convergence anal...
Article
Full-text available
A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by mea...
Article
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional wave equation (FWE) is considered. The fractional derivative...
Article
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the linear fractional Klien-Gordon equation is considered. The fractional de...