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  • 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 >

  • Open Access

    ARTICLE

    Numerical Solving of a Boundary Value Problem for Fuzzy Differential Equations

    Afet Golayoğlu Fatullayev1, Canan Köroğlu2

    CMES-Computer Modeling in Engineering & Sciences, Vol.86, No.1, pp. 39-52, 2012, DOI:10.3970/cmes.2012.086.039

    Abstract In this work we solve numerically a boundary value problem for second order fuzzy differential equations under generalized differentiability in the form y''(t) = p(t)y'(t) + q(t)y(t) + F(t) y(0) = γ, y(l) = λ where t ∈T = [0,l], p(t)≥0, q(t)≥0 are continuous functions on [0,l] and [γ]α = [γ_αα], [λ]α = [λ_α¯α] are fuzzy numbers. There are four different solutions of the problem (0.1) when the fuzzy derivative is considered as generalization of the H-derivative. An algorithm is presented and the finite difference method is used for solving obtained problems. The applicability More >

  • Open Access

    ARTICLE

    Haar Wavelet Operational Matrix Method for Solving Fractional Partial Differential Equations

    Mingxu Yi1, Yiming Chen1

    CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.3, pp. 229-244, 2012, DOI:10.3970/cmes.2012.088.229

    Abstract In this paper, Haar wavelet operational matrix method is proposed to solve a class of fractional partial differential equations. We derive the Haar wavelet operational matrix of fractional order integration. Meanwhile, the Haar wavelet operational matrix of fractional order differentiation is obtained. The operational matrix of fractional order differentiation is utilized to reduce the initial equation to a Sylvester equation. Some numerical examples are included to demonstrate the validity and applicability of the approach. More >

  • Open Access

    ARTICLE

    A Lie-Group Adaptive Method to Identify the Radiative Coefficients in Parabolic Partial Differential Equations

    Chein-Shan Liu1, Chih-Wen Chang2

    CMC-Computers, Materials & Continua, Vol.25, No.2, pp. 107-134, 2011, DOI:10.3970/cmc.2011.025.107

    Abstract We consider two inverse problems for estimating radiative coefficients α(x) and α(x, y), respectively, in Tt(x, t) = Txx(x, t)-α(x)T(x, t), and Tt(x, y, t) = Txx(x, y, t) + Tyy(x, y, t)-α(x, y)T(x, y, t), where a are assumed to be continuous functions of space variables. A Lie-group adaptive method is developed, which can be used to find a at the spatially discretized points, where we only utilize the initial condition and boundary conditions, such as those for a typical direct problem. This point is quite different from other methods, which need the overspecified final time data. Three-fold advantages can be gained More >

  • Open Access

    ARTICLE

    An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F(x) = 0, Using the System of ODEs with an Optimum α in x· = λ[αF + (1−α)BTF]; Bij = ∂Fi/∂xj

    Chein-Shan Liu1, Satya N. Atluri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.73, No.4, pp. 395-432, 2011, DOI:10.3970/cmes.2011.073.395

    Abstract In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x· = λ[αF + (1−α)BTF], where λ and α are scalars and Bij = ∂Fi/∂xj. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian… More >

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