A novel and efficient operational matrix method for solving multi-term variable-order fractional differential equations

The main aim of this research is to present a novel and efficient method based on the Müntz-Legendre polynomials for solving differential equations involving variable-order fractional Caputo derivatives. For the first time, based on the M\rm{\ddot{u}}ntz-Legendre polynomials, a formula for the operational matrix of the variable-order fractional Caputo differential operator is derived. By using this operational matrix via the collocation method, we convert the proposed problem into a system of equations. Then, we solve the obtained system by the Newton method to provide an approximate solution for the problem. Furthermore, we obtain an error bound for the approximation. Finally, we solve four test problems to confirm the reliability and effectiveness of the proposed method.

Keywords:

Multi-term variable-order fractional differential equations , variable-order fractional Caputo derivative , M\rm{\ddot{u}}ntz-Legendre polynomials , operational matrix , error bound