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.

本文旨在介绍两种有效的数值方法，用于求解带初始值的分数阶和多阶Volterra-Fredholm混合微分方程。尽管第二种方法通过使用运算矩阵将积分微分方程式转换为代数方程式系统，但可以通过使用块脉冲函数扩展解的最高阶导数，将这些问题转移到代数方程式系统中（ BPF）。这是通过使用BPF的广义运算矩阵进行微分以及第一种方案中的分数演算属性来完成的，其中获得的解的收敛性如下所示。

将通过一些相关的数值示例来比较两种提出的方​​法的准确性和适用性，对于这些示例，确切的解决方案是已知的。