Application of Shannon wavelet for solving boundary value problems of fractional differential equations

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique as a solution is regarded.This paper develops approximate solutions for boundary value problems ofdifferential equations with non integer order by using the Shannon waveletbases. Wavelet bases have different resolution capability for approximatingof different functions. Since for Shannon type wavelets, the scaling functionand the mother wavelet are not necessarily absolutely integrable, the partialsums of the wavelet series behave differently and a more stringent condition,such as bounded variation, is needed for convergence of Shannon waveletseries. With nominate Shannon wavelet operational matrices of integration,the solutions are approximated in the form of convergent series with easilycomputable terms. Also, by applying collocation points the exact solutionsof fractional differential equations can be achieved by well known series solutions. Illustrative exles are presented to demonstrate the applicabilityand validity of the wavelet base technique. To highlight the convergence,the numerical experiments are solved for different values of bounded series approximation.