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  • 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 New Approach to Non-Homogeneous Fuzzy Initial Value Problem

    N.A. Gasilov1, I.F. Hashimoglu2, S.E. Amrahov3, A.G. Fatullayev1

    CMES-Computer Modeling in Engineering & Sciences, Vol.85, No.4, pp. 367-378, 2012, DOI:10.3970/cmes.2012.085.367

    Abstract In this paper, we consider a high-order linear differential equation with fuzzy forcing function and with fuzzy initial values. We assume the forcing function be in a special form, which we call triangular fuzzy function. We present solution as a fuzzy set of real functions such that each real function satisfies the initial value problem by some membership degree. We propose a method to find the fuzzy solution. We present an example to illustrate applicability of the proposed method. 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

    A Wavelet Numerical Method for Solving Nonlinear Fractional Vibration, Diffusion and Wave Equations

    Zhou YH1,2, Wang XM2, Wang JZ1,2 , Liu XJ2

    CMES-Computer Modeling in Engineering & Sciences, Vol.77, No.2, pp. 137-160, 2011, DOI:10.3970/cmes.2011.077.137

    Abstract In this paper, we present an efficient wavelet-based algorithm for solving a class of fractional vibration, diffusion and wave equations with strong nonlinearities. For this purpose, we first suggest a wavelet approximation for a function defined on a bounded interval, in which expansion coefficients are just the function samplings at each nodal point. As the fractional differential equations containing strong nonlinear terms and singular integral kernels, we then use Laplace transform to convert them into the second type Voltera integral equations with non-singular kernels. Certain property of the integral kernel and the ability of explicit 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 >

  • Open Access

    ARTICLE

    Simple "Residual-Norm" Based Algorithms, for the Solution of a Large System of Non-Linear Algebraic Equations, which Converge Faster than the Newton’s Method

    Chein-Shan Liu1, Satya N. Atluri2

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.3, pp. 279-304, 2011, DOI:10.3970/cmes.2011.071.279

    Abstract For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or Fi(xj) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂Fi/∂xj. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. More >

  • Open Access

    ARTICLE

    A New Insight into the Differential Quadrature Method in Solving 2-D Elliptic PDEs

    Ying-Hsiu Shen1, Chein-Shan Liu1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.2, pp. 157-178, 2011, DOI:10.3970/cmes.2011.071.157

    Abstract When the local differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by requiring to solve the inversions of ill-posed matrices. Previously, when one uses (n-1)th order polynomial test functions to determine the weighting coefficients with n grid points, the resultant n ×n Vandermonde matrix is highly ill-conditioned and its inversion is hard to solve. Now we use (m-1)th order polynomial test functions by n grid points that the size of Vandermonde matrix is m×n, of which m is much less than n. We More >

  • Open Access

    ARTICLE

    Numerical Simulations for Coupled Pair of Diffusion Equations by MLPG Method

    S. Abbasbandy1,2, V. Sladek3, A. Shirzadi1, J. Sladek3

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.1, pp. 15-38, 2011, DOI:10.3970/cmes.2011.071.015

    Abstract This paper deals with the development of a new method for solution of initial-boundary value problems governed by a couple of nonlinear diffusion equations occurring in the theory of self-organization in non-equilibrium systems. The time dependence is treated by linear interpolation using the finite difference method and the semi-discrete partial differential equations are considered in a weak sense by using the local integral equation method with approximating 2-d spatial variations of the field variables by the Moving Least Squares. The evaluation techniques are discussed and the applicability of the presented method is demonstrated on two More >

  • Open Access

    ABSTRACT

    General ray method for solution of the Dirichlet boundary value problems for elliptic partial differential equations in domains with complicated geometry

    A. Grebennikov1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.15, No.3, pp. 85-90, 2010, DOI:10.3970/icces.2010.015.085

    Abstract New General Ray (GR) method for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. GR-method consists in application of the Radon transform directly to the PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). The class of the PDE includes the Laplace, Poisson and Helmgoltz equations. GR-method presents the solution of the Dirichlet boundary value problem for this type of equations by explicit analytical formulas that use the direct and inverse Radon transform. Proposed version of GR-method justified theoretically, realized by fast algorithms and More >

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