A new operational matrix method to solve nonlinear fractional differential equations

Abstract

Abstract This study aims to propose novel Zernike wavelets and a new method based on the operational matrices for solving nonlinear fractional differential equations. First, non-orthogonal Zernike wavelets are introduced using the Zernike polynomials. Then, a new technique based on combining these wavelets with the block pulse functions is presented to derive the operational matrix of fractional integration and to solve nonlinear fractional differential equations. Moreover, an error analysis is conducted by providing required theorems. Besides, the proposed method is employed to solve a nonlinear fractional competition model of breast cancer. Finally, a parametric study is performed to consider the effect of fractional order on the population of healthy, cancer stem, tumour, and immune cells, as well as the excess estrogen.