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.

在这项研究中，通过耦合局部适形导数和非局部Caputo分数导数定义了一种新的分形-分数（FF）导数。使用定义的导数，引入了改进的本杰明-波纳-马洪尼型方程的时空FF版本。采用基于正交伯努利多项式及其导数矩阵（包括本研究中获得的普通和FF导数矩阵）的搭配技术来求解此类方程。提出的方法将求解此方程式转换为简单的代数方程式系统。提供了一些数值问题，以显示所表示方案的准确性。