Numerical solution of the mixed Volterra-Fredholm integro-differential multi-term equations of fractional order

This paper aims to introduce two effective numerical approaches for solving the mixed Volterra-Fredholm integro-differential equation of fractional order and multi-order with initial values. Despite the fact that the second approach transfers the integro-differential equations into a system of algebraic equations through the usage of operational matrices, these problems can be transferred to a system of algebraic equations by expanding the solution's highest order derivative through the block pulse functions (BPFs). This is done by using the generalized operational matrices of BPFs for differentiation along with the fractional calculus properties in the first scheme, in which convergence of the solution obtained has been shown in the following. The accuracy and applicability of two proposed approaches will be compared by some relevant numerical examples, for which the exact solution is known. (C) 2020 Elsevier B.V. All rights reserved.