Browsing by Author "A. H., Bhrawy"

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A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order
(PERGAMON-ELSEVIER SCIENCE LTD, 2011) E. H., Doha; A. H., Bhrawy; S. S., Ezz-Eldien
We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs. (C) 2011 Elsevier Ltd. All rights reserved.
Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations
(ELSEVIER SCIENCE INC, 2011) E. H., Doha; A. H., Bhrawy; S. S., Ezz-Eldien
In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T(L,n)(x) with x is an element of (0,L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev-Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results. (C) 2011 Elsevier Inc. All rights reserved.
A new Jacobi operational matrix: An application for solving fractional differential equations
(ELSEVIER SCIENCE INC, 2012) E. H., Doha; A. H., Bhrawy; S. S., Ezz-Eldien
In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods. (C) 2011 Elsevier Inc. All rights reserved.