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A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations. (English) Zbl 1278.65096
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.

MSC:
65L05 Numerical methods for initial value problems
65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
34A30 Linear ordinary differential equations and systems, general
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