an:05989371
Zbl 1228.65126
Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S.
Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations
EN
Appl. Math. Modelling 35, No. 12, 5662-5672 (2011).
00285414
2011
j
65L60 34A08 26A33 45J05
multi-term fractional differential equations; nonlinear fractional differential equations; tau method; collocation method; shifted Chebyshev polynomials; Gauss quadrature
Summary: 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 \(x \in (0, L), L > 0\) 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.