Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations

A new and novel numerical method has been developed based on the fractional-order hybrid functions combining simulated annealing (SA) algorithm for the solutions of fractional pantograph delay differential equations. First of all, a fractional-order hybrid of block-pulse functions and Chebyshev polynomials (FOHBPCs) is defined. With the aid of regularized beta function, the exact formulas of FOHBPCs are derived under the definition of Riemann-Liouville fractional integral. And then, by the properties of FOHBPCs and the exact formulas together with the collocation method, the problem under consideration is simplified into algebraic equations, which are solved by Gaussian elimination method and Picard iteration method for linear and nonlinear cases, respectively. The error analysis of the proposed method is investigated. Due to the importance of the parameter alpha in the fractional-order hybrid functions method, SA algorithm is considered to find the optimal parameter alpha. Finally, the effectiveness and applicability of the suggested method are verified through some numerical examples.