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 >

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 >

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 >

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 >

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_{_α},γ⁻_α], [λ]^α = [λ_{_α},λ^¯_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 >

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 >

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 >

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 T_t(x, t) = T_xx(x, t)-α(x)T(x, t), and T_t(x, y, t) = T_xx(x, y, t) + T_yy(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 >

Zhou YH^1,2, Wang XM², Wang JZ^1,2 , 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 >

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)B^TF], where λ and α are scalars and B_ij = ∂F_i/∂x_j. 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 >