A hybrid approach for piecewise fractional reaction–diffusion equations

Abstract

In this paper, the Caputo and Atangana–Baleanu fractional derivatives are handled to introduce a type of piecewise fractional derivative. More precisely, a linear combination of the Caputo and Atangana–Baleanu fractional derivatives are considered in each sub-interval to define this fractional derivative. It is employed to generate another form of nonlinear reaction–diffusion equations. The orthonormal Legendre polynomials together with the orthonormal piecewise Legendre functions are used to make a hybrid algorithm for this new problem. In this way, an explicit formula for computing the piecewise fractional differentiation of the stated piecewise basis functions is obtained and applied in generating the method. The applicability and validity of the adopted procedure are examined through three examples.