Publications

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    M.M. Khader, N.H. Sweilam, A.M.S. Mahdy
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    ABSTRACT: 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 equations. Special attention is give to study the convergence and estimate the error of the presented method. The second method is fractional finite difference method (FDM), where we implement the Grünwald-Letnikov’s approach. We study the stability of the obtained numerical scheme. The numerical results show that the approaches are easy to implement and accurate when applied to SFDEs. The methods introduce promising tool for solving many systems of linear and non-linear fractional differential equations. Numerical examples are presented to illustrate the validity and the great potential of both proposed techniques.
    Arab Journal of Mathematical Sciences. 01/2014;
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    N.H. Sweilam, M.M. Khader, M. Adel
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    ABSTRACT: 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 simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.
    Journal of Advanced Research. 01/2014; 5(2):253–259.
  • A.M. Nagy, N.H. Sweilam
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    ABSTRACT: In this paper, we present an accurate numerical method for solving fractional Hodgkin–Huxley model. A non-standard finite difference method (NSFDM) is implemented to study the dynamic behaviors of the proposed model. The Grünwald–Letinkov definition is used to approximate the fractional derivatives. Numerical results are presented graphically reveal that NSFDM is easy to implement, effective and convenient for solving the proposed model.
    Physics Letters A. 01/2014;
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    M.M. Khader, N.H. Sweilam, W.Y. Kota
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    ABSTRACT: 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 validity and applicability of the proposed method, we present some numerical examples. We compare the obtained numerical results from the proposed method with some other methods. The results show that the proposed algorithm is of high accuracy, more simple and effective.
    Journal of the Egyptian Mathematical Society. 01/2014;
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    ABSTRACT: The understanding of the mechanism of DNA interactions and binding with metallic nanoparticles (NPs) and surfaces represents a great interest in today medicine applications due to diagnostic and treatment of oncology diseases. Recent experimental and simulation studies involve the DNA interaction with highly localized proton beams or metallic NPs (such as Ag, Au, etc.), aimed on targeted cancer therapy through the injection of metal micro- or nanoparticles into the tumor tissue with consequent local microwave or laser heating. The effects of mutational structure changes in DNA and protein structures could result in destroying of native chemical (hydrogen) bonds or, on the contrary, creating of new bonds that do not normally exist there. The cause of such changes might be the alteration of one or several nucleotides (in DNA) or the substitution of specific amino acid residues (in proteins), that can brought to the essential structural destabilization or unfolding. At the atomic or molecular level, the replacement of one nucleotide by another (in DNA double-helices) or replacement of one amino acid residue by another (in proteins) cause essential modifications of the molecular force fields of the environment that break locally important hydrogen bonds underlying the structural stability of the biological molecules. In this work, the molecular dynamics (MD) simulations were performed on four DNA models and the flexibilities of the purine and pyrimidine nucleotides during the interaction process with the metallic NPs and TiO2 surface were clarified.
    10/2013; , ISBN: 978-1-62808-052-0
  • Nasser Hassan Sweilam, Taghreed Abdul Rahman Assiri
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    ABSTRACT: The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered. The concept of variable order fractional derivative is considered in the sense of Caputo. The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presented.
    Journal of Applied Mathematics 06/2013; 2013. · 0.83 Impact Factor
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    ABSTRACT: A theoretical study is performed of the possible role of the methyl-directed mismatch repair system in the ultraviolet-induced mutagenesis of Escherichia coli bacterial cells. For this purpose, mathematical models of the SOS network, translesion synthesis and mismatch repair are developed. Within the proposed models, the key pathways of these repair systems were simulated on the basis of modern experimental data related to their mechanisms. Our model approach shows a possible mechanistic explanation of the hypothesis that the bacterial mismatch repair system is responsible for attenuation of mutation frequency during ultraviolet-induced SOS response via removal of the nucleotides misincorporated by DNA polymerase V (the UmuD'2C complex).
    Journal of Theoretical Biology 04/2013; · 2.35 Impact Factor
  • N H Sweilam, T A Assiri
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    ABSTRACT: In this paper, the explicit finite difference method (EFDM) for solving nonlinear fractional wave equation is considered. The stability and convergence of the method are discussed by means of Greschgorin theorem and using the stability matrix analysis. Numerical studies for a model is presented to confirm the accuracy and the effectiveness of the proposed method. The obtained results are compared with exact solutions. It is found that the proposed method is simple and efficient for such nonlinear problem.
    International Journal of Mathematics and Computer Applications Research (IJMCAR). 03/2013; 3(1):193-202.
  • N. H. Sweilam, M. M. Khader, W. Y. Kota
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    ABSTRACT: 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 means of collocation points to system of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshev coefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency and the accuracy of the present work.
    Mathematical Problems in Engineering 02/2013; 2013. · 1.38 Impact Factor
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    ABSTRACT: Molecular dynamics simulations were performed on ethanol–water–Pt system for studying the structural and diffusion behaviour of both ethanol and water molecules on the surface of Pt (111). This work is concerned with the differences between pure liquids and solutions in their diffusional behaviour. The self-diffusion coefficients and activation energies of diffusion of pure ethanol and water on Pt (111) surface were calculated and compared with the corresponding values of their mixtures. The results showed that the values of both the diffusion coefficients and activation energies are strongly affected by the purity of chemical species under investigation. A comparison between two different metal surfaces was also investigated and the results revealed that the nature of metal surface has a strong effect on the adsorption and diffusional behaviour of liquids based on their affinity towards a specific type of surfaces in addition to the hydrophobicity and hydrophilicity of the metal surface.
    European Chemical Bulletin. 01/2013; 2(5):147-254.
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  • M.M. Khader, N.H. Sweilam
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    ABSTRACT: 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 of linear or non-linear algebraic equations. We give the convergence analysis and derive an upper bound of the error for the derived formula. To demonstrate the validity and applicability of the suggested method, some test examples are given. Also, we present a comparison with the previous work using the homotopy perturbation method.
    Applied Mathematical Modelling 01/2013; 37(24):9819–9828. · 1.71 Impact Factor
  • N.H.Sweilam, M.M.Khader, M.Adel
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    ABSTRACT: 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 simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.
    Journal of Advanced Research. 01/2013;
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    ABSTRACT: The paper presents a mathematical model of the DNA mismatch repair system in Escherichia coli bacterial cells. The key pathways of this repair mechanism were simulated on the basis of modern experimental data. We have modelled in detail five main pathways of DNA misincorporation removal with different DNA exonucleases. Here we demonstrate an application of the model to problems of radiation-induced mutagenesis.
    Physics of Particles and Nuclei Letters 01/2013; 10(6).
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    N H Sweilam, M M Khader, A M S Mahdy
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    ABSTRACT: 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 derivative is described in the Caputo sense. The method is based on Legendre approximations. The properties of Legendre polynomials are utilized to reduce the proposed problem to a system of ordinary differential equations, which solved using the finite difference method. Numerical solutions are presented and the results are compared with the exact solution.
    International Journal of Mathematics and Computer Applications Research (IJMCAR). 12/2012; 2(4):1-10.
  • N H Sweilam, M M Khader, M Adel
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    ABSTRACT: In this article, a numerical study is introduced for the fractional reaction-subdiffusion equations by using an efficient class of finite difference methods (FDM). The proposed scheme is based on Hermite formula. The stability analysis and the convergence of the proposed methods are given by a recently proposed procedure similar to the standard John von Neumann stability analysis. Simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Finally, numerical examples are carried out to confirm the theoretical results.
    International Journal of Mathematics and Computer Applications Research (IJMCAR). 12/2012; 2(4):61-75.
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    ABSTRACT: An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and presence of the Pt (1 1 1) surface has been performed using DL_POLY version 2.19. The structure and diffusion properties of an ethanol–water system have been studied at various temperatures from 250 to 350 K. We have measured the self-diffusion coefficients of a 50:50% ethanol–water system; in the absence of a Pt surface our results have shown an excellent agreement with the experimental data (within an error of 7.4%). The enhancement of self-diffusion coefficients with the inclusion of the Pt (1 1 1) surface has been observed and estimated. Graphs of radial distribution functions (RDF) have been built; RDF correlations with the self-diffusion coefficients of both ethanol and water molecules are also illustrated.
    Chemical Physics 04/2012; 402:41–47. · 1.96 Impact Factor
  • NH Sweilam, HM Moharram, Sameh Ahmed
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    ABSTRACT: In this paper, a parallel iterative finite difference method (PIFD) for solving 2D Poisson's equation on a distributed system using Message Passing Interface (MPI) is investigated. This method is based on the domain decomposition method, where the 2D domain is divided into multiple sub-domains using horizontal and/or vertical axis depending on the available number of computer nodes. For interior points Poisson's equation is solved implicitly by four iterative schemes in combining with the boundary conditions. At the interface points of interior subdomains, Poisson's equation is solved by explicit iterative schemes. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication by restricting the exchange of data to the interface between the sub-domains. To examine the efficiency and accuracy of the iterative algorithm, several numerical experiments using different number of nodes of the Linux PC cluster are tested. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.
    Informatics and Systems (INFOS), 2012 8th International Conference on; 01/2012
  • N. H. Sweilam, M. M. Khader, A. M. S. Mahdy
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    ABSTRACT: A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.
    Journal of Applied Mathematics 01/2012; 2012. · 0.83 Impact Factor

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