Bernoulli wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications in solving fractional order differential equations

In this paper, a new numerical method based on Bernoulli wavelets operational matrix for solving fractional differential equations (FDEs) is presented. The fractional derivative is described in the Caputo sense. The Bernoulli wavelets are first presented, then an operational matrix of fractional order derivative is derived through a wavelet-polynomial transformation matrix and is utilized together with collocation methods to reduce the linear and non-linear FDEs to system of algebraic equations. Some numerical examples are illustrated to demonstrate the validity and applicability of the technique.