Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.