Abstract
An optimization method standing on a new basis formed by the transcendental Bernstein series (TBS) is proposed. The TBS includes the unknown free coefficients and control parameters for solving nonlinear variable-order fractional functional boundary value problems (NV-FFBVP). The corresponding operational matrices for variable-order fractional derivatives are derived for expanding the solution by means of TBS. The TBS is a generalization of the Bernstein polynomials (BP) and represent a superset of BP. In the particular cases where all the control parameters are zero, the TBS method is equivalent to the BP method. The proposed technique reduces the NV-FFBVP to a system of algebraic equations and, subsequently, to the problem of finding the free coefficients and control parameters using an optimization technique. Several numerical results reveal the computational performance and reliability of the method.
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