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The method based on block pulse functions (BPFs) has been proposed to solve different kinds of fractional differential equations (FDEs). However, high accuracy requires considerable BPFs because they are piecewise constant and not so smooth. As a result, it increases the dimension of operational matrix and computational burden. To overcome this deficiency, a novel numerical method is developed to solve fractional differential equations. The method is based upon hybrid of BPFs and Bernstein polynomials (HBBPs), which are piecewise smooth. The HBBPs operational matrix of fractional‐order integral is derived to reduce the FDEs to a system of algebraic equations. Then the numerical solution of the FDEs is obtained through solving the system of algebraic equations. The convergence analysis is conducted for the suggested scheme, and the upper bound of error of the solution is given. Finally, illustrative examples are presented to demonstrate the validity, applicability, and efficiency of the proposed technique in contrast with other approaches.

In this paper, the estimation issue for fractional order system with unknown parameter and order is addressed. The Bayesian method is used for system identification. First, the framework of parameter identification algorithm is built and the conjugate prior is constructed. Then, an improved descent method is used to update the posterior distribution based on the idea of maximum likelihood function. The proposed method solves the problem of overall identification of order and parameters, and can also estimate the noise in the system. Finally, numerical experiments comparing with the optimization algorithm are completed. Experiments on prediction performance, convergence ability and time complexity verify the effectiveness and speed of the proposed algorithm.