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.
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    Nasser Hassan Sweilam, Tamer Mostafa Al-Ajami
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    ABSTRACT: In this paper, the Legendre spectral-collocation method is applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. Two different approches are presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian are approximated. In the second approach, the state equation is discretized first using the trapezoidal rule for the numerical integration followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Illustrative examples are included to demonstrate the validity and applicability of the proposed techniques.
    Journal of Advanced Research. 01/2014;
  • 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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    Amr Mahdy, Nasser Sweilam, Mohamed Khader
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    ABSTRACT: 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 solution of a system of algebraic equations which is solved using Newton iteration method. The convergence analysis and an upper bound of the error of the derived formula are given. We compared our numerical solutions with those numerical solutions using fourth-order Runge-Kutta method. The obtained results of the SIRC model show the simplicity and the efficiency of the proposed method. Also, illustration for propagation of influenza A virus and the relation between the four cases of it along the time at the fractional derivative are given.
    Applied Mathematics & Information Sciences An International Journal. 01/2014; 8(3):1-8.
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    ABSTRACT: In this paper, a parallel two-level algorithm for solving time-fractional parabolic equation is introduced, where the fractional derivative is in the sense of Caputo, and the domain of the problem contains a huge number of points. A parallel Crank-Nicholson finite difference method (P-CN-FDM) is used to obtain the approximate solution. The resultant large sparse linear system of equations is solved using a two-level Parallel Preconditioned Conjugate Gradient method (PPCG). The goal is to enhance the performance of our previous parallel PCG algorithm. The proposed algorithm is mainly based on the well-known Compressed Sparse Row (CSR) storage format, the parallel implementation of the Crank-Nicholson finite difference method, and a two-level parallelization model that distributes the workload on the cluster nodes in the first step and then split it on the processor’s cores in the second step. The implementation of the proposed algorithm is done on a Linux PC cluster. The obtained results are compared and show a great enhancement, in both memory utilization and execution time, compared to our previous algorithm that uses the matrix in its dense form without compression.
    Third International Conference on Mathematics and Information Sciences ICMIS 28-30 Dec. 2013, Luxor,Egypt.; 12/2013
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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. · 2.16 Impact Factor
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    Amr Mahdy, Nasser Sweilam, Mohamed khader
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    ABSTRACT: 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 which can be solved by the Newton iteration method. The method introduces a promising tool for solving many systems of non-linear FDEs. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed methods. Comparisons with the fractional finite difference method (FDM) and the fourth order Runge-Kutta (RK4) are given.
    Applied Mathematics & Information Sciences 01/2013; 7(5):2011-2018. · 1.23 Impact Factor
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    Nasser Hassan Sweilam, Tamer Mostafa Al-Ajami, Ronald H W Hoppe
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    ABSTRACT: We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm "optimize first, then discretize" and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.
    The Scientific World Journal 01/2013; 2013:306237. · 1.73 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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