| Summary: | <p>In this dissertation, a new numerical method for solving the fractional dynamical systems,
is presented. We first introduce Riemann-Liouville fractional integral operator for
hybrid functions. Then we will show the spectral accuracy of the present method for solving
fractional-order differential equations, and we will extend the present method for solving
nonlinear fractional integro-differential equations, fractional Bagley-Torvik equation,
distributed order fractional differential equations, two-dimensional fractional partial differential
equations, and fractional optimal control problems. In all cases, we will show the rate
of convergence is more than some existing numerical methods which were used to solve
these kind of problems in the literature. Illustrative examples are included to demonstrate
the validity and applicability of the technique.</p>
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