In this paper, the optimized decomposition method, which was developed
to solve integer-order differential equations, will be modified and
extended to handle nonlinear fractional differential equations.
Fractional derivatives will be considered in terms of Caputo sense. The
suggested modifications design new optimized decompositions for the
series solutions depending on linear approximations of the nonlinear
equations. Two optimized decomposition algorithms have been introduced
to obtain approximate solutions of broad classes of IVPs consisting of
nonlinear fractional ODEs and PDEs. A comparative study of the suggested
algorithms with the Adomian decomposition method was performed by means
of some test illustration problems. The executed numerical simulation
results demonstrated that the proposed algorithms give better accuracy
and convergence compared with Adomian's approach. This confirms the
belief that the optimized decomposition method will be effectively and
widely used in solving various functional equations.