Fractional-order Bernoulli wavelets and their applications

In this paper, we define a new fractional function based on the Bernoulli wavelet to obtain a solution for systems of fractional differential equations (FDEs). The fractional derivative in these problems is in the Caputo sense. The method consists of expanding the required approximate solution as the elements of the fractional-order Bernoulli wavelets (FBWs). The fractional-order Bernoulli wavelets are constructed. The operational matrices of fractional order integration and derivative for FBWs are derived. The operational matrix of fractional integration and collocation method are utilized to reduce the initial value problems to system of algebraic equations. The error analysis shows that the method is convergent. Numerical examples are examined to highlight the significant features of the new technique.