In this work, we introduce an integration matrix operator that is fully consistent with the differentiation matrix operator defined by the differential quadrature method (DQM). Using these operators, a generic differential-integral quadrature method “DIQM” is proposed to directly discretize an integro-differential equation as a system of algebraic equations. To extend the applicability of the proposed DIQM to solve nonlinear and/or variable coefficients integro-differential equation, some matrix manipulations are introduced. A stability analysis for Volterra integro-partial-differential equations is presented and the exponential convergence of the proposed method is examined. Various types of integro-differential equations are solved including ordinary/partial, linear/nonlinear, Volterra parabolic/hyperbolic integro-differential equations with different boundary and initial conditions. Numerical results demonstrate the exponential convergence and the applicability of DIQM.

中文翻译：

---

求解非线性积分微分方程的一种新的微分积分求积法

在这项工作中，我们引入了一个积分矩阵算子，它与微分正交法（DQM）定义的微分矩阵算子完全一致。使用这些算子，提出了一种通用的微分积分正交方法“DIQM”，以将积分微分方程直接离散为代数方程组。为了扩展所提出的 DIQM 解决非线性和/或可变系数积分微分方程的适用性，引入了一些矩阵操作。提出了对 Volterra 积分偏微分方程的稳定性分析，并检验了所提出方法的指数收敛性。求解各种类型的积分微分方程，包括普通/偏分、线性/非线性、具有不同边界和初始条件的 Volterra 抛物线/双曲线积分微分方程。数值结果证明了 DIQM 的指数收敛性和适用性。