Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

工程系统通常由高阶微分方程系统控制，这些系统需要有效的数值方法来提供可靠的解决方案，并受到强加的约束。由于高阶数值微分对噪声的高度敏感性，直接逼近系统变量的传统方法可能会产生相当大的误差，因此需要改进技术，以更好地满足高阶系统求解中所需的数值精度要求。为此，提出了一种新颖的逆微分求积法（iDQM），用于工程系统的逼近。基于积分和 DQM 反演的 iDQM 的详细公式是单独开发的，用于从高阶导数逼近任意低阶函数。误差公式被进一步开发以评估所提出方法的性能，而通过收敛性、鲁棒性和数值稳定性的准确性是通过 iDQM 方案的两个独特概念（称为混合 iDQM 和完整 iDQM）的表达来呈现的。通过将从传热和力学问题中提取的线性和非线性系统的高阶微分方程的 iDQM 解与精确解和 DQM 解进行基准比较，证明了 iDQM 近似在提供准确解的同时不损失计算效率是稳健的，并提供卓越的数值稳定性超过 DQM 解决方案。称为混合 iDQM 和完整 iDQM。通过将从传热和力学问题中提取的线性和非线性系统的高阶微分方程的 iDQM 解与精确解和 DQM 解进行基准比较，证明了 iDQM 近似在提供准确解的同时不损失计算效率是稳健的，并提供卓越的数值稳定性超过 DQM 解决方案。称为混合 iDQM 和完整 iDQM。通过将从传热和力学问题中提取的线性和非线性系统的高阶微分方程的 iDQM 解与精确解和 DQM 解进行基准比较，证明了 iDQM 近似在提供准确解的同时不损失计算效率是稳健的，并提供卓越的数值稳定性超过 DQM 解决方案。