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  • Open Access

    ARTICLE

    Legendre Polynomials Method for Solving a Class of Variable Order Fractional Differential Equation

    Lifeng Wang1, Yunpeng Ma1,2, Yongqiang Yang1

    CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.2, pp. 97-111, 2014, DOI:10.3970/cmes.2014.101.097

    Abstract In this paper, a numerical method based on the Legendre polynomials is presented for a class of variable order fractional differential equation. We adopt the Coimbra variable order fractional operator, which can be viewed as a Caputo-type definition. Three different kinds of operational matrixes with Legendre polynomials are derived. A truncated the Legendre polynomials series together with the products of several dependent matrixes are utilized to reduce the variable order fractional differential equation to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited n. An error More >

  • Open Access

    ARTICLE

    Investigation of Squeezing Unsteady Nanofluid Flow Using the Modified Decomposition Method

    Lei Lu1,2, Li-Hua Liu3,4, Xiao-Xiao Li1

    CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.1, pp. 1-15, 2014, DOI:10.3970/cmes.2014.101.001

    Abstract In this paper, we use the modified decomposition method (MDM) to solve the unsteady flow of a nanofluid squeezing between two parallel equations. Copper as nanoparticle with water as its base fluid has considered. The effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell- Garnetts (MG) and Brinkman models, respectively. The effects of the squeeze number, the nanofluid volume fraction, Eckert number, δ on Nusselt number and the Prandtl number are investigated. The figures and tables clearly show high accuracy of the method to solve the unsteady flow. More >

  • Open Access

    ARTICLE

    Solution of Two-Dimensional Viscous Flow in a Rectangular Domain by the Modified Decomposition Method

    Lei Lu1,2,3, Jun-Sheng Duan2, Long-Zhen Fan1

    CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.6, pp. 463-475, 2014, DOI:10.3970/cmes.2014.100.463

    Abstract In this paper, the modified decomposition method (MDM) for solving the nonlinear two-dimensional viscous flow equations is presented. This study investigates the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions. We first transform the original two-dimensional viscous flow problem into an equivalent fourth-order boundary value problem (BVP), then solve the problem by the MDM. The figures and tables clearly show high accuracy of the method to solve two-dimensional viscous flow. 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 >

  • Open Access

    ARTICLE

    Numerical Algorithm to Solve Fractional Integro-differential Equations Based on Operational Matrix of Generalized Block Pulse Functions

    Yunpeng Ma1, Lifeng Wang1, Zhijun Meng1

    CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.1, pp. 31-47, 2013, DOI:10.3970/cmes.2013.096.031

    Abstract In this paper, we propose a numerical algorithm for solving linear and nonlinear fractional integro-differential equations based on our constructed fractional order generalized block pulse functions operational matrix of integration. The linear and nonlinear fractional integro-differential equations are transformed into a system of algebraic equations by the matrix and these algebraic equations are solved through known computational methods. Further some numerical examples are shown to illustrate the accuracy and reliability of the proposed approach. Moreover, comparing the methodology with the known technique shows that our approach is more efficient and more convenient. More >

  • Open Access

    ARTICLE

    A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

    Xiaojing Liu1, Jizeng Wang1,2, Youhe Zhou1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.3, pp. 225-238, 2013, DOI:10.32604/cmes.2013.094.225

    Abstract A wavelet method is proposed for solving a class of nonlinear timedependent partial differential equations. Following this method, the nonlinear equations are first transformed into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers’ equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and More >

  • Open Access

    ARTICLE

    The Second Kind Chebyshev Wavelet Method for Fractional Differential Equations with Variable Coefficients

    Baofeng Li1

    CMES-Computer Modeling in Engineering & Sciences, Vol.93, No.3, pp. 187-202, 2013, DOI:10.3970/cmes.2013.093.187

    Abstract In this article, the second kind Chebyshev wavelet method is presented for solving a class of multi-order fractional differential equations (FDEs) with variable coefficients. We first construct the second kind Chebyshev wavelet, prove its convergence and then derive the operational matrix of fractional integration of the second kind Chebyshev wavelet. The operational matrix of fractional integration is utilized to reduce the fractional differential equations to a system of algebraic equations. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method. More >

  • Open Access

    ARTICLE

    Wavelet operational matrix method for solving fractional integral and differential equations of Bratu-type

    Lifeng Wang1, Yunpeng Ma1, Zhijun Meng1, Jun Huang1

    CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.4, pp. 353-368, 2013, DOI:10.3970/cmes.2013.092.353

    Abstract In this paper, a wavelet operational matrix method based on the second kind Chebyshev wavelet is proposed to solve the fractional integral and differential equations of Bratu-type. The second kind Chebyshev wavelet operational matrix of fractional order integration is derived. A truncated second kind Chebyshev wavelet series together with the wavelet operational matrix is utilized to reduce the fractional integral and differential equations of Bratu-type to a system of nonlinear algebraic equations. The convergence and the error analysis of the method are also given. Two examples are included to verify the validity and applicability of More >

  • Open Access

    ARTICLE

    Numerical solution of fractional partial differential equations using Haar wavelets

    Lifeng Wang1, Zhijun Meng1, Yunpeng Ma1, Zeyan Wu2

    CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.4, pp. 269-287, 2013, DOI:10.3970/cmes.2013.091.269

    Abstract In this paper, we present a computational method for solving a class of fractional partial differential equations which is based on Haar wavelets operational matrix of fractional order integration. We derive the Haar wavelets operational matrix of fractional order integration. Haar wavelets method is used because its computation is sample as it converts the original problem into Sylvester equation. Finally, some examples are included to show the implementation and accuracy of the approach. More >

  • Open Access

    ARTICLE

    Numerical solution of nonlinear fractional integral differential equations by using the second kind Chebyshev wavelets

    Yiming Chen1, Lu Sun1, Xuan Li1, Xiaohong Fu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.90, No.5, pp. 359-378, 2013, DOI:10.3970/cmes.2013.090.359

    Abstract By using the differential operator matrix and the product operation matrix of the second kind Chebyshev wavelets, a class of nonlinear fractional integral-differential equations is transformed into nonlinear algebraic equations, which makes the solution process and calculation more simple. At the same time, the maximum absolute error is obtained through error analysis. It also can be used under the condition that no exact solution exists. Numerical examples verify the validity of the proposed method. More >

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