In this paper, we use hybrid parabolic and block-pulse functions (2D-PBPFs) to provide an approximate solution of nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order. To reach this goal, we present the Volterra integral operational matrix, operational matrix of fractional integral and operational matrix of mixed Volterra–Fredholm integral by 2D-PBPFs. Using the proposed method, nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order become into a nonlinear system of algebraic equations. Moreover, we provide some theorems for convergence analysis and we demonstrate that the convergence order of the suggested approximate approach is \(O(h^{3})\). Finally, we solve two numerical examples to prove the accuracy of the proposed method.

中文翻译：

---

分数阶非线性部分混合积分-微分方程的混合方法

在本文中，我们使用混合抛物线函数和块脉冲函数（2D-PBPF）来提供分数阶Volterra-Fredholm非线性部分混合积分微分方程的近似解。为了达到这个目标，我们介绍了Volterra积分运算矩阵，分数积分运算矩阵和2D-PBPF混合Volterra-Fredholm积分的运算矩阵。使用提出的方法，分数阶非线性部分Volterra-Fredholm分数阶混合微分方程变成一个非线性代数方程组。此外，我们为收敛性分析提供了一些定理，并且证明了所建议的近似方法的收敛阶为\（O（h ^ {3}）\）。最后，我们通过两个数值例子证明了所提方法的准确性。