Integro-differential equations are developed as models in enormous fields of engineering and science such as biological models, population growth, aerospace systems and industrial mathematics. In this work, we consider a general class of nonlinear fractional integro-differential equations with variable order derivative. We use the operational matrices based on the Bernstein polynomials to obtain numerical solution of this type of equations. By utilizing the operational matrices along with the Newton–Cotes collocation points, the problem under study is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.

中文翻译：

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使用 Bernstein 多项式求解多变量阶分数阶积分微分方程

积分微分方程被开发为工程和科学领域的模型，例如生物模型、人口增长、航空航天系统和工业数学。在这项工作中，我们考虑一类具有变阶导数的非线性分数阶积分微分方程。我们使用基于伯恩斯坦多项式的运算矩阵来获得此类方程的数值解。通过利用运算矩阵和 Newton-Cotes 搭配点，研究中的问题被简化为非线性代数方程组。证明了数值解的误差估计。最后，通过一些例子来说明所提出方法的准确性和有效性。