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1

温, 燕静. "An Approach of High Order Numerical Flux." Advances in Applied Mathematics 08, no. 05 (2019): 990–97. http://dx.doi.org/10.12677/aam.2019.85113.

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2

Dyson, Rodger W., and John W. Goodrich. "Automated Approach to Very High-Order Aeroacoustic Computations." AIAA Journal 39, no. 3 (2001): 396–406. http://dx.doi.org/10.2514/2.1349.

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3

NAM, Taek-Kun, and Tsutomu MITA. "High Order Nonholonomic Systems Control by Backstepping Approach." Transactions of the Society of Instrument and Control Engineers 38, no. 6 (2002): 528–36. http://dx.doi.org/10.9746/sicetr1965.38.528.

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4

Wang, Jin, and Ruxun Liu. "A new approach to design high-order schemes." Journal of Computational and Applied Mathematics 134, no. 1-2 (2001): 59–67. http://dx.doi.org/10.1016/s0377-0427(00)00528-8.

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5

Navarrina, F., S. López-Fontán, I. Colominas, E. Bendito, and M. Casteleiro. "High order shape design sensitivity: a unified approach." Computer Methods in Applied Mechanics and Engineering 188, no. 4 (2000): 681–96. http://dx.doi.org/10.1016/s0045-7825(99)00355-2.

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6

AGNON, Y., P. A. MADSEN, and H. A. SCHÄFFER. "A new approach to high-order Boussinesq models." Journal of Fluid Mechanics 399 (November 25, 1999): 319–33. http://dx.doi.org/10.1017/s0022112099006394.

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An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.
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7

Kamiński, J. Z., and F. Ehlotzky. "Effective action approach to high-order harmonic generation." Physical Review A 54, no. 4 (1996): 3678–81. http://dx.doi.org/10.1103/physreva.54.3678.

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8

Dyson, Rodger W., and John W. Goodrich. "Automated approach to very high-order aeroacoustic computations." AIAA Journal 39 (January 2001): 396–406. http://dx.doi.org/10.2514/3.14744.

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9

Yang, Bo. "High-Order Consensus Seeking: A Frequency Domain Approach." Applied Mathematics & Information Sciences 8, no. 4 (2014): 1829–35. http://dx.doi.org/10.12785/amis/080440.

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10

Gao, Liang-hui, Xiao-feng Li, Dong-sheng Guo, and Pan-ming Fu. "Formal Scattering Approach to High-Order Harmonic Generation." Chinese Physics Letters 16, no. 7 (1999): 502–4. http://dx.doi.org/10.1088/0256-307x/16/7/012.

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11

Kasza, József, Péter Dombi, and Péter Földi. "Sturmian–Floquet approach to high-order harmonic generation." Journal of the Optical Society of America B 35, no. 5 (2018): A126. http://dx.doi.org/10.1364/josab.35.00a126.

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12

Levant, Arie. "Homogeneity approach to high-order sliding mode design." Automatica 41, no. 5 (2005): 823–30. http://dx.doi.org/10.1016/j.automatica.2004.11.029.

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13

Dhib, Rayan, Firas Ben Ameur, Vatsalya Sharma, Andrea Lani, and Stefaan Poedts. "Toward High-order Solar Corona Simulations: A High-order Hyperbolized Poisson Approach for Magnetic Field Initialization." Astrophysical Journal 980, no. 2 (2025): 163. https://doi.org/10.3847/1538-4357/adace5.

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Abstract Proper initialization of the solar corona magnetic field is important for easing the iterative process of realistic and efficient global magnetohydrodynamics (MHD) simulations. This study introduces a high-order flux reconstruction (FR) framework for solving the Poisson equation, a necessary step for computing a potential-field source-surface (PFSS) to initialize the magnetic field for global solar corona simulations with MHD. By hyperbolizing the elliptic Poisson equation into a set of hyperbolic equations, we develop an efficient and robust high-order PFSS solver. Our contributions include developing a Q2 (i.e., quadratic) geometrical representation using prismatic elements for the computational domain, which enables high-order mesh generation. Such a hyperbolized Poisson solver effectively relaxes magnetic fields extrapolated from solar magnetograms, producing scalar potential fields that align well with theoretical expectations. Extensive verification was conducted on the high-order FR solver for polynomial orders up to P3, achieving fourth-order spatial accuracy. The hyperbolized solver demonstrates comparable accuracy to reference solutions (both analytical and numerical) while offering efficient performance, particularly on coarser meshes, making it competitive with state-of-the-art low-order finite volume solvers, which are mostly used for solar MHD simulations. The described developments are a milestone for enabling high-order global solar corona simulations on 3D unstructured grids.
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14

BISWAS, Debasish, and Aya KITOH. "J055041 A High-Order LES Approach to Study Unsteady Flow Separation Control in a Low Pressure Turbine." Proceedings of Mechanical Engineering Congress, Japan 2013 (2013): _J055041–1—_J055041–5. http://dx.doi.org/10.1299/jsmemecj.2013._j055041-1.

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15

LU, Xiao-lei, Fu-rong WANG, and Ben-xiong HUANG. "Scoring matching approach: Learning high order Markov random fields." Journal of Computer Applications 28, no. 10 (2009): 2529–32. http://dx.doi.org/10.3724/sp.j.1087.2008.02529.

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16

Qian, J. "Closure Approach to High-Order Structure Functions of Turbulence." Physical Review Letters 84, no. 4 (2000): 646–49. http://dx.doi.org/10.1103/physrevlett.84.646.

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17

Cruz-Zavala, Emmanuel, and Jaime A. Moreno. "Homogeneous High Order Sliding Mode design: A Lyapunov approach." Automatica 80 (June 2017): 232–38. http://dx.doi.org/10.1016/j.automatica.2017.02.039.

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18

Lysenko, W. P. "The moment approach to high-order accelerator beam optics." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 363, no. 1-2 (1995): 90–99. http://dx.doi.org/10.1016/0168-9002(95)00359-2.

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19

Rongpeng, Zhang* Omer T. Karaguzel. "DEVELOPMENT AND CALIBRATION OF REDUCED-ORDER BUILDING ENERGY MODELS BY COUPLING WITH HIGH-ORDER SIMULATIONS." GLOBAL JOURNAL OF ADVANCED ENGINEERING TECHNOLOGIES AND SCIENCES 7, no. 2 (2020): 1–18. https://doi.org/10.5281/zenodo.3689397.

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Building energy modeling and simulation is an effective approach to evaluate building performance and energy system operations to achieve higher building energy efficiency. The high-order building models can offer exceptional simulation capacity and accuracy, however, its high level of complexity does not allow it to directly work with the optimization algorithms and methods that require a complete differential-algebraic-equations-based mathematical description of the physical model. In order to fill in the gap, the study presents a systematic approach to develop and calibrate the reduced-order building models. A notable feature of the approach is its coupling with high-order building simulations in order to pre-process the input information and support the calibration of the reduced model. A case study on a representative office building shows that the developed reduced-order model can present acceptable simulation accuracy compared with high-order simulations and significantly reduce the modeling complexity.     
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20

Famelis, I. Th, and Z. Jackiewicz. "A new approach to the construction of DIMSIMs of high order and stage order." Applied Numerical Mathematics 119 (September 2017): 79–93. http://dx.doi.org/10.1016/j.apnum.2017.03.015.

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21

Mazaheri, Alireza, and Hiroaki Nishikawa. "Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection–diffusion system approach." Journal of Computational Physics 321 (September 2016): 729–54. http://dx.doi.org/10.1016/j.jcp.2016.06.006.

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22

Nwaigwe, Chinedu, and Azubuike Weli. "Hermite Interpolation Approach to High-Order Approximation of Heat Equations." European Journal of Mathematics and Statistics 4, no. 1 (2023): 32–37. http://dx.doi.org/10.24018/ejmath.2023.4.1.208.

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 It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
 
 
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23

Mendoza, M., S. Succi, and H. J. Herrmann. "High-order kinetic relaxation schemes as high-accuracy Poisson solvers." International Journal of Modern Physics C 26, no. 05 (2015): 1550055. http://dx.doi.org/10.1142/s0129183115500552.

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We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher-order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knudsen numbers. In order to test our kinetic approach, we discretize the Boltzmann equation and solve the Poisson equation, spending up to six order of magnitude less computational time for a given precision than standard lattice Boltzmann methods (LBMs).
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24

Fuqing, Yuan, and Jinmei Lu. "A Slow-Motion Detecting Algorithm using High Order Statistic Approach." Journal of Physics: Conference Series 1827, no. 1 (2021): 012152. http://dx.doi.org/10.1088/1742-6596/1827/1/012152.

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25

K.I, Falade. "A Numerical Approach for Solving High-Order Boundary Value Problems." International Journal of Mathematical Sciences and Computing 5, no. 3 (2019): 1–16. http://dx.doi.org/10.5815/ijmsc.2019.03.01.

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26

Zhang, Jian, Jinjiao Hou, Jing Niu, Ruifeng Xie, and Xuefei Dai. "A high order approach for nonlinear Volterra-Hammerstein integral equations." AIMS Mathematics 7, no. 1 (2021): 1460–69. http://dx.doi.org/10.3934/math.2022086.

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<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>
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27

Golestan, Saeed, Francisco D. Freijedo, and Josep M. Guerrero. "A Systematic Approach to Design High-Order Phase-Locked Loops." IEEE Transactions on Power Electronics 30, no. 6 (2015): 2885–90. http://dx.doi.org/10.1109/tpel.2014.2351262.

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28

Boukabou, Abdelkrim, and Naim Mekircha. "Generalized chaos control and synchronization by nonlinear high-order approach." Mathematics and Computers in Simulation 82, no. 11 (2012): 2268–81. http://dx.doi.org/10.1016/j.matcom.2012.07.005.

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29

Ionita, A. D., I. Dumitrache, and T. Ionescu. "Noise Reduction in Biomedical Applications. A High Order Spectra Approach." IFAC Proceedings Volumes 29, no. 1 (1996): 944–49. http://dx.doi.org/10.1016/s1474-6670(17)57785-x.

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30

Tian, Ye, and Jr-Fong Dang. "MIMO Detection for High Order QAM by Canonical Dual Approach." Journal of Applied Mathematics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/201369.

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We develop a canonical dual approach for solving the MIMO problem. First, a special linear transformation is introduced to reformulate the original problem into a{−1,1}constrained quadratic programming problem. Then, we derive a canonical dual problem which is piecewise continuous problem with no duality gap. Under certain conditions, the canonical problem becomes a concave maximization dual problem over a convex feasible domain. By getting the stationary point of the canonical dual problem, we can find either an optimal or approximate solution of the original problem. A gradient decent algorithm is proposed to solve the MIMO problem and simulation results are provided to demonstrate the effectiveness of the method.
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31

Viaño, J. M., C. Ribeiro, J. Figueiredo, and Á. Rodríguez-Arós. "A high order model for piezoelectric rods: An asymptotic approach." International Journal of Solids and Structures 81 (March 2016): 294–310. http://dx.doi.org/10.1016/j.ijsolstr.2015.12.005.

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32

Nastase, Cristian R., and Dimitri J. Mavriplis. "High-order discontinuous Galerkin methods using an hp-multigrid approach." Journal of Computational Physics 213, no. 1 (2006): 330–57. http://dx.doi.org/10.1016/j.jcp.2005.08.022.

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33

Özkurt, S. "A cosmological approach to a high-order effective field theory." Classical and Quantum Gravity 13, no. 2 (1996): 265–75. http://dx.doi.org/10.1088/0264-9381/13/2/014.

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34

Barua, Alok, and Kedar A. Choudhary. "EXSHOF: An artificial intelligence approach to high-order filter synthesis." Engineering Applications of Artificial Intelligence 6, no. 6 (1993): 533–47. http://dx.doi.org/10.1016/0952-1976(93)90050-8.

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35

Moxey, D., M. D. Green, S. J. Sherwin, and J. Peiró. "An isoparametric approach to high-order curvilinear boundary-layer meshing." Computer Methods in Applied Mechanics and Engineering 283 (January 2015): 636–50. http://dx.doi.org/10.1016/j.cma.2014.09.019.

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36

Weinberg, Kerstin, and Christian Hesch. "A high-order finite deformation phase-field approach to fracture." Continuum Mechanics and Thermodynamics 29, no. 4 (2015): 935–45. http://dx.doi.org/10.1007/s00161-015-0440-7.

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37

Zhang, Zhenzhong, Chunxiang Guo, Wenjie Ruan, Wei Wang, and Peng Zhou. "An intelligent stochastic optimization approach for stochastic order allocation problems with high-dimensional order uncertainties." Computers & Industrial Engineering 167 (May 2022): 108008. http://dx.doi.org/10.1016/j.cie.2022.108008.

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38

Luo, Yunhua, Bingji Zhao, Xiaolei Han, Robert Wang, Hongjun Song, and Yunkai Deng. "A Novel High-Order Range Model and Imaging Approach for High-Resolution LEO SAR." IEEE Transactions on Geoscience and Remote Sensing 52, no. 6 (2014): 3473–85. http://dx.doi.org/10.1109/tgrs.2013.2273086.

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39

Flayyih, Wameedh Nazar, Ahlam Hanoon Al-sudani, Basheera M. Mahmmod, Sadiq H. Abdulhussain, and Muntadher Alsabah. "High-Performance Krawtchouk Polynomials of High Order Based on Multithreading." Computation 12, no. 6 (2024): 115. http://dx.doi.org/10.3390/computation12060115.

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Orthogonal polynomials and their moments serve as pivotal elements across various fields. Discrete Krawtchouk polynomials (DKraPs) are considered a versatile family of orthogonal polynomials and are widely used in different fields such as probability theory, signal processing, digital communications, and image processing. Various recurrence algorithms have been proposed so far to address the challenge of numerical instability for large values of orders and signal sizes. The computation of DKraP coefficients was typically computed using sequential algorithms, which are computationally extensive for large order values and polynomial sizes. To this end, this paper introduces a computationally efficient solution that utilizes the parallel processing capabilities of modern central processing units (CPUs), namely the availability of multiple cores and multithreading. The proposed multi-threaded implementations for computing DKraP coefficients divide the computations into multiple independent tasks, which are executed concurrently by different threads distributed among the independent cores. This multi-threaded approach has been evaluated across a range of DKraP sizes and various values of polynomial parameters. The results show that the proposed method achieves a significant reduction in computation time. In addition, the proposed method has the added benefit of applying to larger polynomial sizes and a wider range of Krawtchouk polynomial parameters. Furthermore, an accurate and appropriate selection scheme of the recurrence algorithm is introduced. The proposed approach introduced in this paper makes the DKraP coefficient computation an attractive solution for a variety of applications.
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40

Ketcheson, David I., and Abhijit Biswas. "Approximation of arbitrarily high-order PDEs by first-order hyperbolic relaxation." Nonlinearity 38, no. 5 (2025): 055002. https://doi.org/10.1088/1361-6544/adc6e8.

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Abstract We present a framework for constructing a first-order hyperbolic system whose solution approximates that of a desired higher-order evolution equation. Constructions of this kind have received increasing interest in recent years, and are potentially useful as analytical or computational tools for understanding the corresponding higher-order equation. We perform a systematic analysis of a family of linear model equations and show that for each member of this family there is a stable hyperbolic approximation whose solution converges to that of the model equation in a certain limit. We then show through several examples that this approach can be applied successfully to a very wide range of nonlinear partial differential equations of practical interest.
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41

ITKIN, ANDREY. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs." International Journal of Theoretical and Applied Finance 18, no. 05 (2015): 1550031. http://dx.doi.org/10.1142/s0219024915500314.

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This paper is dedicated to the construction of high order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreassen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility (LSV) model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.
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42

Wahlberg, Bo. "Model reductions of high-order estimated models: the asymptotic ML approach." International Journal of Control 49, no. 1 (1989): 169–92. http://dx.doi.org/10.1080/00207178908559628.

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43

Cavallo, A., and C. Natale. "Output feedback control based on a high-order sliding manifold approach." IEEE Transactions on Automatic Control 48, no. 3 (2003): 469–72. http://dx.doi.org/10.1109/tac.2003.809152.

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44

Albin, Nathan. "High-order numerical methods for nonlinear acoustics: A Fourier Continuation approach." Journal of the Acoustical Society of America 132, no. 3 (2012): 1919. http://dx.doi.org/10.1121/1.4755041.

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45

Martin, Pablo, and Malcolm Haines. "Gyroinvariant high‐order orbit theory for unidirectional magnetostatic fields: New approach." Physics of Fluids B: Plasma Physics 2, no. 1 (1990): 11–21. http://dx.doi.org/10.1063/1.859522.

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46

Zhang, Sheng Guo, Kai Wang, and Xiao Ping Dang. "The Approach for Analyzing Motion Mode of High-Order Control Systems." Applied Mechanics and Materials 596 (July 2014): 594–97. http://dx.doi.org/10.4028/www.scientific.net/amm.596.594.

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This paper aims at exploring the modal analysis approach of a motion control system. Based on the inverse Laplace transformation, the step response of a control system is derived. Then this response is associated with the modal analyses in state space theory. And then the motion mode of a control system is analyzed with the modal analysis method. Application example indicates that this approach can be used to analyze the high-order control system successfully. This facilitates the motion mode analyses of high-order control system very much.
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47

Li, Jian Jun, Xi Bing Li, Hong You, and Cheng Liu. "An Approach of Signal Analysis Based on High-Order Spline Interpolation." Advanced Materials Research 507 (April 2012): 226–30. http://dx.doi.org/10.4028/www.scientific.net/amr.507.226.

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Empirical mode decomposition (EMD) provides a powerful tool for the nonlinear and nonstationary signals. This paper presents an application of the signature analysis based on high-order spline interpolation. The time-frequency analysis method based on EMD is introduced. The series data are separated into intrinsic mode functions (IMFs) with different time scale using EMD. The spectrum of Hilbert transformation is obtained by applying Hilbert transformation to every IMF. Based on cubic spline interpolation, high-order spline interpolation is used to improve the precision of the algorithm. Furthermore some strategies for improving the computational efficiency are proposed. The simulation result shows that the precision of the time-frequency analysis can be improved effectively using the proposed new algorithm.
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48

Wernerheim, C. "Understanding regional high-order service growth in Canada: a cointegration approach." Service Industries Journal 24, no. 1 (2004): 131–54. http://dx.doi.org/10.1080/02642060412331301172.

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49

Ma, Zixiao, Zhaoyu Wang, Dongbo Zhao, and Bai Cui. "High-fidelity large-signal order reduction approach for composite load model." IET Generation, Transmission & Distribution 14, no. 21 (2020): 4888–97. http://dx.doi.org/10.1049/iet-gtd.2020.0972.

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50

Nemirovsky, Danil. "Tensor approach to mixed high-order moments of absorbing Markov chains." Linear Algebra and its Applications 438, no. 4 (2013): 1900–1922. http://dx.doi.org/10.1016/j.laa.2011.08.027.

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