A matrix approach to multi-term fractional differential equations using two new diffusive representations for the Caputo fractional derivative

In the last decade, there has been a surge of interest in application of fractional calculus in various areas such as, mathematics, physics, engineering, mechanics and etc. So, numerical methods have rapidly been developed to handle problems containing fractional derivatives (or integrals). Due to the fact that all the operators which appear in fractional calculus are non-local, so, the classical linear multi-step methods have some difficulties from the (time/space) computational complexity point of view. Recently, two new non-classical methods or diffusive based methods have been proposed by the authors to approximate the Caputo fractional derivatives. Here, the main aim of this paper is to use these methods to solve linear multi-term fractional differential equations numerically. To reach our aim, an efficient matrix approach has been provided to solve some well-known multi-term fractional differential equations.