Orthonormal Bernstein polynomials for solving nonlinear variable-order time fractional fourth-order diffusion-wave equation with nonsingular fractional derivative

This article defines a novel version of nonlinear fourth-order diffusion-wave equation, which involves variable-fractional differentiations with nonsingular kernel. This newly defined equation entails a highly accurate method for its numerical solution. The proposed scheme is based upon the orthonormal Bernstein polynomials and their operational matrices of variable-order fractional differentiation together with the collocation method. In this paper, the formulation for obtaining these operational matrices are provided in details. The method converts the original problem to a system of algebraic equations that can be promptly solved using a software program. To clarify the validity and the high precision of the devised method, numerous illustrative examples with various boundary conditions are examined.