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  • Open Access

    ARTICLE

    A LARGE PARAMETER SPECTRAL PERTURBATION METHOD FOR NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS THAT MODELS BOUNDARY LAYER FLOW PROBLEMS

    T. M. Agbajea,b, S. S. Motsaa,* , S. Mondalc,† , P. Sibandaa

    Frontiers in Heat and Mass Transfer, Vol.9, pp. 1-13, 2017, DOI:10.5098/hmt.9.36

    Abstract In this work, we present a compliment of the spectral perturbation method (SPM) for solving nonlinear partial differential equations (PDEs) with applications in fluid flow problems. The (SPM) is a series expansion based approach that uses the Chebyshev spectral collocation method to solve the governing sequence of differential equation generated by the perturbation series approximation. Previously the SPM had the limitation of being used to solve problems with small parameters only. This current investigation seeks to improve the performance of the SPM by doing the series expansion about a large parameter. The new method namely… More >

  • Open Access

    ARTICLE

    Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets

    F. Bulut1,2, Ö. Oruç3, A. Esen3

    CMES-Computer Modeling in Engineering & Sciences, Vol.108, No.4, pp. 263-284, 2015, DOI:10.3970/cmes.2015.108.263

    Abstract In this paper, time fractional one dimensional coupled KdV and coupled modified KdV equations are solved numerically by Haar wavelet method. Proposed method is new in the sense that it doesn’t use fractional order Haar operational matrices. In the proposed method L1 discretization formula is used for time discretization where fractional derivatives are Caputo derivative and spatial discretization is made by Haar wavelets. L2 and L error norms for various initial and boundary conditions are used for testing accuracy of the proposed method when exact solutions are known. Numerical results which produced by the proposed method for More >

  • Open Access

    ARTICLE

    A Micromechanical Model for Estimating the Effective Stiffness of a Pair of Micro-cracked Interfaces in an Orthotropic Trimaterial under Inplane Deformations

    X. Wang1, W.T. Ang1,2, H. Fan1

    CMES-Computer Modeling in Engineering & Sciences, Vol.107, No.2, pp. 81-101, 2015, DOI:10.3970/cmes.2015.107.081

    Abstract A micromechanical model is proposed here for estimating the effective stiffness of a pair of parallel microscopically damaged interfaces in a trimaterial under inplane elastostatic deformations. The trimaterial is made of an orthotropic thin layer sandwiched between two orthotropic half-spaces. The microscopically damaged interfaces are modeled using periodically distributed interfacial micro-cracks. The micromechanical model is formulated and numerically solved in terms of hypersingular boundary integro-differential equations. The effects of the width of the thin layer, the micro-crack densities of the two interfaces and the material constants of the thin layer and the two half-spaces on More >

  • Open Access

    ARTICLE

    New Spectral Solutions of Multi-Term Fractional-Order Initial Value ProblemsWith Error Analysis

    W. M. Abd- Elhameed1,2, Y. H. Youssri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.5, pp. 375-398, 2015, DOI:10.3970/cmes.2015.105.375

    Abstract In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial value problems is established. The key idea for obtaining the suggested spectral numerical solutions for these equations is actually based on utilizing the ultraspherical wavelets along with applying the collocation method to reduce the fractional differential equation with its initial conditions into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested ultraspherical wavelets expansion are carefully discussed. For the sake of testing the proposed algorithm, some numerical examples are More >

  • Open Access

    ARTICLE

    Numerical Solution of Fractional Fredholm-Volterra Integro-Differential Equations by Means of Generalized Hat Functions Method

    Baofeng Li 1

    CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.2, pp. 105-122, 2014, DOI:10.3970/cmes.2014.099.105

    Abstract In this paper, operational matrix method based on the generalized hat functions is introduced for the approximate solutions of linear and nonlinear fractional integro-differential equations. The fractional order generalized hat functions operational matrix of integration is also introduced. The linear and nonlinear fractional integro-differential equations are transformed into a system of algebraic equations. In addition, the method is presented with error analysis. Numerical examples are included to demonstrate the validity and applicability of the approach. More >

  • Open Access

    ARTICLE

    Numerical Solution for a Class of Linear System of Fractional Differential Equations by the Haar Wavelet Method and the Convergence Analysis

    Yiming Chen1, Xiaoning Han1, Lechun Liu 1

    CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.5, pp. 391-405, 2014, DOI:10.3970/cmes.2014.097.391

    Abstract In this paper, a class of linear system of fractional differential equations is considered. It has been solved by operational matrix of Haar wavelet method which converts the problem into algebraic equations. Moreover the convergence of the method is studied, and three numerical examples are provided to demonstrate the accuracy and efficiency. More >

  • Open Access

    ARTICLE

    Bäcklund Transformations: a Link Between Diffusion Models and Hydrodynamic Equations

    J.R. Zabadal1, B. Bodmann1, V. G. Ribeiro2, A. Silveira2, S. Silveira2

    CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.4, pp. 215-227, 2014, DOI:10.3970/cmes.2014.103.215

    Abstract This work presents a new analytical method to transform exact solutions of linear diffusion equations into exact ones for nonlinear advection-diffusion models. The proposed formulation, based on Bäcklund transformations, is employed to obtain velocity fields for the unsteady two-dimensional Helmholtz equation, starting from analytical solutions of a heat conduction type model. More >

  • Open Access

    ARTICLE

    Space-time Discontinuous Galerkin Method Based on a New Generalized Flux Vector Splitting Method for Multi-dimensional Nonlinear Hyperbolic Systems

    P.A. Trapper1, P.Z. Bar-Yoseph2

    CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.1, pp. 19-47, 2014, DOI:10.3970/cmes.2014.103.019

    Abstract The space-time discontinuous Galerkin method for multi-dimensional nonlinear hyperbolic systems is enhanced with a generalized technique for splitting a flux vector that is not limited to the homogeneity property of the flux. This technique, based on the flux’s characteristic decomposition, extends the scope of the method’s applicability to a wider range of problems, including elastodynamics. The method is used for numerical solution of a number of representative problems based on models of vibrating string and vibrating rod that involve the propagation of a sharp front through the solution domain. More >

  • Open Access

    ARTICLE

    A Wavelet Method for Solving Bagley-Torvik Equation

    Xiaomin Wang1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.2, pp. 169-182, 2014, DOI:10.3970/cmes.2014.102.169

    Abstract In this paper, an efficient and robust wavelet Laplace inversion method of solving the fractional differential equations is proposed. Such an inverse function can be applied to any reasonable function categories and it is not necessary to know the properties of original function in advance. As an example, we have applied the proposed method to the solution of the Bagley–Torvik equations and Numerical examples are given to demonstrate the efficiency and accuracy of the proposed. More >

  • Open Access

    ARTICLE

    Operational Matrix Method for Solving Variable Order Fractional Integro-differential Equations

    Mingxu Yi1, Jun Huang1, Lifeng Wang1

    CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.5, pp. 361-377, 2013, DOI:10.3970/cmes.2013.096.361

    Abstract In this paper, operational matrix method based upon the Bernstein polynomials is proposed to solve the variable order fractional integro-differential equations in the Caputo derivative sense. We derive the Bernstein polynomials operational matrix of fractional order integration and introduce the product operational matrix of Bernstein polynomials. A truncated the Bernstein polynomials series together with the polynomials operational matrix are utilized to reduce the variable order fractional integro-differential equations to a system of algebraic equations. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Some examples are included to demonstrate the More >

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