Department of Mathematics, Florida State University
"Hybrid functions in fractional calculus"
In this talk, a new numerical method for solving the fractional dynamical systems is presented. The advantage of using the hybrid functions as a trial function in spectral method rather than classic trial functions will be given. The Riemann-Liouville fractional integral operator for hybrid functions is introduced and the spectral accuracy of the present method for solving nonlinear fractional integro-differential equations and distributed order fractional differential equations is given. We then use the present method for solving fractional-order differential equations, fractional Bagley-Torvik equation, two-dimensional fractional partial differential equations, and fractional optimal control problems. In all cases the convergence of method is given. Illustrative examples are included to demonstrate the validity and applicability of the technique and we show that the accuracy of present method is more than existing numerical methods.