Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations

This work deals with the distributed-order time fractional nonlinear diffusion-wave equations. These equations are generated by replacing the first-and second-order time derivative terms with the distributed-order fractional derivative terms. The distributed-order fractional derivatives used in these problems are in the Caputo sense. The Chelyshkov polynomials as a well-known family of basis functions are used to develop a spectral collocation method for these problems. Through the way, some operational matrices regarding the classical and distribute-order fractional derivatives for these polynomials are extracted. In the proposed method, after approximating the solution of the problem in terms of the Chelyshkov polynomials and employing the expressed matrices, solving the primary equations transforms into solving algebraic systems which can be easily solved. Some numerical examples are studied to show the adequacy of the approach.