Orthonormal Bernoulli polynomials for space–time fractal-fractional modified Benjamin–Bona–Mahony type equations

In this study, a new fractal-fractional (FF) derivative is defined by coupling the local conformable derivative and non-local Caputo fractional derivative. Using the defined derivative, a space–time FF version of the modified Benjamin–Bona–Mahony type equations is introduced. A collocation technique based on the orthonormal Bernoulli polynomials and their derivative matrices (including the ordinary and FF derivative matrices obtained in this study) is adopted for solving such equations. The presented method converts solving this equation to solve a simple system of algebraic equations. Some numerical problems are provided to show the accuracy of the expressed scheme.