New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq–Burger’s equations

Abstract

This paper introduces the variable-order (VO) fractional version of the coupled Boussinesq–Burger’s equations by using the concept of VO fractional derivative in the Caputo form. A numerical algorithm based on the shifted orthonormal Bernoulli polynomials (OBPs) is developed for solving this system. To carry out this end, a new formula for generating these polynomials is introduced. Moreover, some matrix relations are extracted for these basis functions and used in the established method. The method transforms the introduced VO fractional system into a system of algebraic equations by expanding solution of the problem in terms of the shifted OBPs. Some test problems are studied to examine the accuracy of the algorithm.