Publications

  • N.H. Sweilam, A.M. Nagy, Adel A. El-Sayed
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    ABSTRACT: In this paper, an efficient numerical method for solving space fractional order diffusion equation is presented. The numerical approach is based on shifted Chebyshev polynomials of the second kind where the fractional derivatives are expressed in terms of Caputo type. Space fractional order diffusion equation is reduced to a system of ordinary differential equations using the properties of shifted Chebyshev polynomials of the second kind together with Chebyshev collocation method. The finite difference method is used to solve this system of equations. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.
    Chaos Solitons & Fractals 04/2015; 73. DOI:10.1016/j.chaos.2015.01.010 · 1.50 Impact Factor
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    ABSTRACT: This paper presents a numerical simulation technique for the fractional Cable equation in large scale domain. Special attention is given to the parallel execution of the fractional weighted average finite difference method (FWA-FDM) on distributed system with explicit message passing, where the fractional derivative is defined in Riemann–Liouville sense. The resultant huge system of equations is studied using precondition conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiments using different number of the Linux PC cluster nodes are studied. 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.
  • N. H. SWEILAM, A. M. NAGY, T. A. ASSIRI, N.Y.ALI
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    ABSTRACT: In this paper numerical studies for the variable-order fractional delay differential equations are presented. Adams-Bashforth-Moulton algorithm has been extended to study this problem, where the derivative is defined in the Caputo variable-order fractional sense. Special attention is given to prove the error estimate of the proposed method. Numerical test examples are presented to demonstrate utility of the method. Chaotic behaviors are observed in variable-order one dimensional delayed systems.
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    ABSTRACT: In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. 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.
    Journal of Numerical Mathematics 11/2014; 22(4):363–382. DOI:10.1515/jnma-2014-0016 · 0.63 Impact Factor
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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.
    10/2014; 22(3). DOI:10.1016/j.joems.2013.10.004
  • N. H. Sweilam, A. M. Nagy, T. F. Almajbri
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    ABSTRACT: In this paper, the space fractional diffusion equation (SFDE) is numerically studied, where the fractional derivative is defined in the sense of the right-shifted Grunwald. An explicit finite difference approximation (EFDA) for SFDE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.
  • N. H. Sweilam, A. M. Nagy, T. F. Almajbri
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    ABSTRACT: In this paper, the space fractional diffusion equation (SFDE) is numerically studied, where the fractional derivative is defined in the sense of the right-shifted Grunwald. An explicit finite difference approximation (EFDA) for SFDE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.
  • Applied Mathematics & Information Sciences 07/2014; 8(4):1675-1684. DOI:10.12785/amis/080423 · 1.23 Impact Factor
  • 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 06/2014; 378(30-31). DOI:10.1016/j.physleta.2014.06.012 · 1.63 Impact Factor
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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 05/2014; DOI:10.1016/j.jare.2014.05.004
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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 03/2014; 5(2):253–259. DOI:10.1016/j.jare.2013.03.006
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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.
    01/2014; 21(1). DOI:10.1016/j.ajmsc.2013.12.001
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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.
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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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    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 12/2013; 2013:306237. DOI:10.1155/2013/306237 · 1.73 Impact Factor
  • 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 12/2013; 37(24):9819–9828. DOI:10.1016/j.apm.2013.06.010 · 2.16 Impact Factor
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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 11/2013; 10(6). DOI:10.1134/S1547477113060046
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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.
    Models in Bioscience and Materials Research: Molecular Dynamics and Related Techniques, Edited by Kholmirzo T. Kholmurodov, 10/2013: chapter Molecular Dynamics Simulations of the DNA Interaction with Metallic Nanoparticles and TiO2 Surfaces.; NOVA publisher., ISBN: 978-1-62808-052-0
  • M. M. Khader, N. H. Sweilam
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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. 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 transport at the Earth surface. In this paper, an efficient numerical method for solving FADE 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 FADE to a system of ODEs, which is solved using the finite difference method. Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of FADE are presented and the results are compared with the exact solution.
    Computational & Applied Mathematics 10/2013; 33(3):739-750. DOI:10.1007/s40314-013-0091-x · 0.41 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 09/2013; 7(5):2011-2018. DOI:10.12785/amis/070541 · 1.23 Impact Factor

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