## 1 Introduction
Later on, they employed [PERSON]'s expansion to form a unified set of single-step algorithms [9], discussed the accuracy and the stability, and applied the method of backward error analysis to obtain a few general rules to choose parameters. By separately approximating acceleration, velocity and displacement as linear, quadratic, and cubic Taylor series, [PERSON] and [PERSON] developed a single-step TFEM based on a weighted residual method, and applied it to linear [25] and nonlinear [22] structural dynamics. Numerical results showed that the method presents some advantages, especially in accuracy and overshooting behavior. Additionally, [PERSON] and [PERSON] presented three and four steps of TFEM using the least squares method as well as Lagrange interpolations for first-order initial-value problems [18]. The drawback of the weighted residual approaches based on TFEM is the determination of the optimum weighting functions. TFEMs were based on the assumption that state variables are continuous in the time domain. However, in the Time-Discontinuous Galerkin Method (TDGM) which was originated by [PERSON] and [PERSON] [26] and [PERSON] and [PERSON] [27] based on Galerkin weighted residual approach, and in the TDGM, the unknown variables are continuous within the time element but discontinuous at time nodes. The TDGM has been employed to solve periodic differential equations and Davey-Stewartson equations [28; 29]. It was first shown by [PERSON] and [PERSON] [27] and [PERSON] et al. [30] that TDGM leads to A-stable, higher-order accurate TFEMs. Subsequently, [PERSON] and [PERSON] first applied TDGM to elastodynamics and structural dynamics to develop space-time finite element methods [31; 32; 33; 34] and TFEMs [5; 35], respectively. According to their results, TDGM stability is easily proved, but the convergence had been proved only for linear elements, therefore the Time-Discontinuous Galerkin/Least Squares Method (TDGLSM) [36] was developed to solve this problem [31; 32; 35]. Compared with the continuous Galerkin FEM, the drawback of TDGMs is the usage of a higher number of degrees, which has been overcome by designing an improved predictor/multi-corrector algorithm [37] or a suitable adaptive strategy for iterative mesh refinement [38; 39]. According to the above review, the TFEMs in existence certainly have limitations. As is well known, researchers are always seeking new numerical methods with higher accuracy, efficiency, and flexibility to promote modeling capabilities. [PERSON] and [PERSON] proposed the Differential Quadrature Method (DQM) as a simply and highly efficient numerical technique for initial-value problems [40], which was regarded as a global and higher-order FD scheme [17; 41; 42; 43] and could yield accurate results using a considerably smaller number of sampling points. The general versatility of DQM has been established in a variety of physical problems. A recent survey refers to publications [44; 45], and an exhaustive list of the literature on DQM up to 1996 may be found in the survey paper [46]. Although DQM's numerical accuracy and computational efficiency has been well demonstrated, its limitations are also significant [46]. In 2009, based on the Differential Quadrature (DQ) rule, the Gauss-Lobatto quadrature rule, and the total potential energy variational principle, a DQ Finite Element Method (DQFEM) was proposed by the author and co-workers for free vibration analysis of thin plates [47], and the authors formulated explicit one- to three-dimensional (1-D to 3-D) element matrices [48]. The DQFEM uses Lagrange trial functions for C\({}^{0}\) and C\({}^{1}\) FEs and possesses high accuracy of DQM and the generality of standard FEM. In this context, similar to DQFEM for spatial domain [47; 48], [PERSON]'s variational principle together with the DQ rule and the Gauss-Lobatto quadrature rule are used to develop a TFEM in the present work, where the initial conditions can be readily imposed as in standard FEM. This novel method is called the DQ Time Finite Element Method (DQFEM).