Progress in Fractional Differentiation & Applications

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Fractional calculus has achieved a great interest in the last decades since many physical problems are modeled with fractional differential equations. The definition of fractional derivatives involves integral operators, some of them having singular kernel, and its calculation is not easy. For that reason, in addition to theoretical developments, it is important to look for accurate numerical approximations to these operators. In this work we propose a new and simple numerical scheme to approximate the solutions to initial value problems involving Caputo-Fabrizio fractional derivatives. Following some previous results, we choose a wavelet basis with special properties, apply the wavelet decomposition to the data, calculate the fractional derivatives of the wavelet basis and combine them by means of a Galerkin-type scheme to reconstruct the unknown from its wavelet coefficients. The properties of the chosen basis guarantee that the numerical scheme is simple, stable and its accuracy can be easily improved. It could be adapted to solve initial value problems combining other fractional and natural order derivatives and fractional partial differential equations.

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