{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:42Z","timestamp":1760240382088,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T00:00:00Z","timestamp":1558915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51507091"],"award-info":[{"award-number":["51507091"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Project of Science and Technology Research Plan of Education Department of Hubei","award":["D20181204"],"award-info":[{"award-number":["D20181204"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A Chebyshev pseudospectral\u2013two-step three-order boundary value coupled method is proposed and presented for handling the issue associated with complicated calculation, low precision, and poor stability in the process of transient response of transmission line. The first order differential equation in time domain is obtained via dispersing the telegraph equation in space domain by utilizing the pseudospectral method (PSM) based on Chebyshev polynomial. Then the two-step three-order boundary value method (BVM3) is presented and employed to resolve the obtained differential equation, so the numerical solution of the space discrete points can be obtained. Furthermore, the Chebyshev pseudospectral\u2013two-step three-order boundary value coupled method (PSM-BVM3) is presented and compared with the Chebyshev pseudospectral\u2013two-step two order boundary value coupled method (PSM-BVM2), the pseudospectral\u2013differential quadrature method (PSM-DQM), and the pseudospectral method\u2013trapezoid rule (PSM-TR) to validate the feasibility of the new proposed method. Theoretical analysis and numerical simulation reveal that the proposed Chebyshev PSM-BVM3 has a higher performance than the conventional method. For the proposed Chebyshev PSM-BVM3, the higher precision, efficiency, and numerical stability can be obtained and achieved only with fewer discrete points in the space domain, which is suitable for solving the transient response of transmission line. The proposed PSM-BVM3 can improve the drawback of numerical instability of the PSM and can also improve the disadvantage of the BVM as it is not easy to change the latter\u2019s timestep size.<\/jats:p>","DOI":"10.3390\/sym11050721","type":"journal-article","created":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T11:19:27Z","timestamp":1558955967000},"page":"721","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fast Calculation Method for Transient Response of Transmission Line Based on Chebyshev Pseudospectral\u2013Two-Step Three-Order Boundary Value Coupled Method"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5229-9038","authenticated-orcid":false,"given":"Zhenhua","family":"Li","sequence":"first","affiliation":[{"name":"China Three Gorges University College of Electrical Engineering &amp; New Energy, Yichang 443002, China"},{"name":"Hubei Provincial Collaborative Innovation Center for New Energy Microgrid, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2961-3236","authenticated-orcid":false,"given":"Tinghe","family":"Hu","sequence":"additional","affiliation":[{"name":"China Three Gorges University College of Electrical Engineering &amp; New Energy, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Tao","sequence":"additional","affiliation":[{"name":"China Three Gorges University College of Electrical Engineering &amp; New Energy, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Zhang","sequence":"additional","affiliation":[{"name":"China Three Gorges University College of Electrical Engineering &amp; New Energy, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhenxing","family":"Li","sequence":"additional","affiliation":[{"name":"China Three Gorges University College of Electrical Engineering &amp; New Energy, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1109\/TIM.2018.2854939","article-title":"A New Vibration Testing Platform for Electronic Current Transformers","volume":"68","author":"Li","year":"2019","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lee, S. (2018). Dynamics of Trapped Solitary Waves for the Forced KdV Equation. Symmetry, 10.","DOI":"10.3390\/sym10050129"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1109\/TMTT.2005.845197","article-title":"Periodic finite-difference time-domain analysis of loaded transmission-linenegative-refractive-index metamaterials","volume":"53","author":"Kokkinos","year":"2005","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1109\/TAP.2004.823889","article-title":"Finite-difference time-domain analysis of integrated ceramic ball grid array package antenna for highly integrated wireless transceivers","volume":"52","author":"Zhang","year":"2004","journal-title":"IEEE Trans. Antennas Propagat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1109\/LMWC.2007.899287","article-title":"On the numerical stability of the Precise Integration Time-Domain (PITD) method","volume":"17","author":"Jiang","year":"2007","journal-title":"IEEE Microw. Wirel. Compon. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1109\/TEMC.2007.913222","article-title":"A precise Time-Step Integration method for transient analysis of lossy nonuniform transmission lines","volume":"50","author":"Tang","year":"2008","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1109\/TEMC.2013.2272041","article-title":"Fast and accurate transient analysis of buried wires and its applications","volume":"56","author":"Feng","year":"2014","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/75.763242","article-title":"Modeling of transmission lines by the differential quadrature method","volume":"9","author":"Xu","year":"1999","journal-title":"IEEE Microw. Guided Wave Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1109\/TEMC.2009.2036602","article-title":"An FFT-accelerated time domain multiconductor transmission line simulator","volume":"52","author":"Bagci","year":"2010","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1109\/15.917924","article-title":"Fast transmission line coupling calculation using a convolution technique","volume":"43","author":"Thielen","year":"2001","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1448","DOI":"10.1016\/j.apnum.2007.08.007","article-title":"Composite generalized Laguerre\u2013Legendre pseudospectral method for Fokker-Planck equation in an infinite channel","volume":"58","author":"Wang","year":"2008","journal-title":"Appl. Numer. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ghoshi, S. (2017). Relative Effects of Asymmetry and Wall Slip on the Stability of Plane Channel Flow. Fluids, 2.","DOI":"10.3390\/fluids2040066"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1002\/mma.4656","article-title":"An extension of the Gegenbauer pseudospectral method for the time fractional Fokker-Planck equation","volume":"41","author":"Izadkhah","year":"2018","journal-title":"Math. Method. Appl. Sci."},{"key":"ref_14","first-page":"192","article-title":"Optimization of path-constrained systems using pseudospectral methods applied to aircraft trajectory planning","volume":"48","author":"Arribas","year":"2015","journal-title":"IFIC Pap."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., Shakhno, S., and Yarmola, H. (2019). Two-Step Solver for Nonlinear Equations. Symmetry, 11.","DOI":"10.3390\/sym11020128"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1002\/mma.786","article-title":"On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow","volume":"30","author":"Zhang","year":"2011","journal-title":"Math. Method Appl. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0021-9991(72)90094-0","article-title":"Finite difference boundary value method for solving one-dimensional eigenvalue equations","volume":"10","author":"Truhlar","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1109\/TEC.2018.2876303","article-title":"A dual-input high step-up DC\/DC converter with ZVT auxiliary circuit","volume":"34","author":"Zhu","year":"2019","journal-title":"IEEE Trans. Energy Convers."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2591","DOI":"10.1007\/s10910-018-0905-6","article-title":"New three\u2013stages symmetric two step method with improved properties for second order initial\/boundary value problems","volume":"56","author":"Chen","year":"2018","journal-title":"J. Math. Chem."},{"key":"ref_20","unstructured":"Jagannadha Rao, G.V.V., Padhan, S.K., and Postolache, M. (2019). Application of Fixed Point Results on Rational F-Contraction Mappings to Solve Boundary Value Problems. Symmetry, 11."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1750020","DOI":"10.1142\/S0219455417500201","article-title":"Chebyshev Collocation Solutions for Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions","volume":"17","author":"Wattanasakulpong","year":"2017","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1023\/A:1017972217727","article-title":"Finite-Difference Method for a System of Third-Order Boundary-Value Problems","volume":"112","author":"Noor","year":"2002","journal-title":"J. Optim. Theory Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.amc.2006.10.008","article-title":"Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration","volume":"188","author":"Babolian","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Qiu, L., Yu, Y., Xiong, Q., Deng, C., Cao, Q., Han, X., and Li, L. (2018). Analysis of electromagnetic force and deformation behavior in electromagnetic tube expansion with concave coil based on finite element method. IEEE Trans. Appl. Supercond., 28.","DOI":"10.1109\/TASC.2017.2789287"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.newast.2013.03.002","article-title":"Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane\u2013Emden type","volume":"24","author":"Doha","year":"2013","journal-title":"New Astron."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Xiong, Q., Tang, H., Wang, M., Huang, H., Qiu, L., Yu, K., and Chen, Q. (2019). Design and implementation of tube bulging by an attractive electromagnetic force. J. Mater. Process. Tech.","DOI":"10.1016\/j.jmatprotec.2019.05.021"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"561015","DOI":"10.1155\/2011\/561015","article-title":"An operator-difference method for telegraph equations arising in transmission lines","volume":"2011","author":"Koksa","year":"2011","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Yang, N., Huang, Y., Hou, D., Liu, S., Ye, D., Dong, B., and Fan, Y. (2019). Adaptive Nonparametric Kernel Density Estimation Approach for Joint Probability Density Function Modeling of Multiple Wind Farms. Energies, 12.","DOI":"10.3390\/en12071356"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.mechrescom.2011.09.004","article-title":"Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method","volume":"39","author":"Danesh","year":"2012","journal-title":"Mech. Res. Commun."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/721\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:29Z","timestamp":1760187329000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/721"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,27]]},"references-count":29,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050721"],"URL":"https:\/\/doi.org\/10.3390\/sym11050721","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,5,27]]}}}