an:06198514
Zbl 1278.65096
Bhrawy, A. H.; Tharwat, M. M.; Yildirim, A.
A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations
EN
Appl. Math. Modelling 37, No. 6, 4245-4252 (2013).
00315507
2013
j
65L05 65L03 34A08 34A30
tau method; shifted Chebyshev polynomials; Chebyshev-Gauss quadrature; fractional differential equation; initial value problem; Fourier expansion of solution; Riemann-Liouville fractional derivative; numerical examples
The authors construct an approximate method to solve the differential problem of the form
\[
D^{\nu}u(x) + \sum^{r-1}_{i=1}\gamma_{i}D^{\beta_{i}}u(x)+ \gamma_{r}u(x)=g(x) \qquad \text{in }(0,L),
\]
\[
u^{(i)}(x)=d_{i}, \quad i=0,1,\dots,m-1,\quad 0<\beta_{i}<\nu,\quad m-1<\nu\leq m,
\]
where the derivatives \( D^{\nu}\) and \( D^{\beta_{i}}\) denote the Riemann-Liouville fractional derivatives. The approximate method is constructed using the expansion of the solution by a system of Chebyshev orthogonal polynomials and the notion of integration of fractional order.
Reviewer's remark: The numerical examples given in this article are not sufficient to prove the convergence of the method.
Ivan Secrieru (Chi??in??u)