An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems

Abstract

In this paper, we propose the numerical approximation of fractional initial and boundary value problems using Haar wavelets. In contrast to the Haar wavelet methods available in literature, where the fractional derivative of the function is approximated using the Haar basis, we approximate the function and its classical derivatives using Haar basis functions. Moreover, error bounds in the approximation of fractional integrals and the fractional derivatives are derived, which depend on the index J of the approximation space VJ and the fractional order α. A neural network problem modeled by a system of nonlinear fractional differential equations is also solved using the proposed method. The numerical results show that the proposed numerical approach is efficient.

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