The transformed differential quadrature method for solving time-dependent partial differential equations: Framework and examples

This paper proposes a general framework of the transformed differential quadrature method (TDQM) for solving partial differential equations (PDEs). First, the basic mathematical concepts and key steps of the TDQM are presented. Then, several examples including homogeneous/inhomogeneous heat conduction equations and wave propagation equations are solved by the proposed method to show its detailed scheme. The influences of sampling point types, discrete point numbers and numerical Laplace transform inversion on the convergence and error of computational results are also discussed. Besides, we show the accuracy and efficiency of the proposed method by comparing the TDQM with the alternating direction implicit (ADI) scheme of the finite difference method. Compared with the traditional differential quadrature method (DQM), the TDQM introduces the integral transform theorem to decompose the solution process into a small-scale matrix inversion and a transform inversion. The number of algebraic equations is greatly reduced so that the difficulty of large-scale matrix inversion is avoided, and the computational efficiency is improved. Besides, the TDQM is no longer limited to finite entities, and reduces the limitation and influence of boundary conditions on computational results. This strategy is especially suitable for multi-dimensional problems with complex variables such as non-axisymmetric or three-dimensional problems.