In chapter one, a brief review of some work associated with the computational methods: direct and indirect are given for optimal control problem is discussed. Chapter two present and derives some new important properties and formulas concerning Laguerre and Hermite functions. Chapter three gives some modified algorithms are proposed depending on the pontryagin minimum principle. Chapter four describes a method for approximate the solution of OCP using state vector parameterizations. In chapter five the idea of spectral method is applied to propose four algorithms for OCP. In Chapter six Conclusions and further research directions are included.

In chapter one, some new formulas for Caputo fractional derivatives of some elementary functions is given. The system of M-linear Voltera integro-fractional differential equations is reduced into a system of Voltera integral equations and the global and semi-global fundamental existence and uniquenas theorems and presented in Chapter two. In chapter three, some analytic and approximate methods are applied to treat such a system. In chapter four, Runge-Kutta methods with different orders are given to treat such a system. The convergence and stability are also investigated. In chapter five, special Chebyshev method is considered. In chapter six, conclusions and recommendations with comparisons between the methods are included.