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Volume 16, Issue 2
Implicit Runge-Kutta-Nyström Methods with Lagrange Interpolation for Nonlinear Second-Order IVPs with Time-Variable Delay

Chengjian Zhang, Siyi Wang & Changyang Tang

Adv. Appl. Math. Mech., 16 (2024), pp. 423-436.

Published online: 2024-01

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  • Abstract

This paper deals with nonlinear second-order initial value problems with time-variable delay. For solving this kind of problems, a class of implicit Runge-Kutta-Nyström (IRKN) methods with Lagrange interpolation are suggested. Under the suitable condition, it is proved that an IRKN method is globally stable and has the computational accuracy $\mathcal{O}(h^{min\{p,\mu+ν+1\}}),$ where $p$ is the consistency order of the method and $\mu, ν ≥0$ are the interpolation parameters. Combining a fourth-order compact difference scheme with IRKN methods, an initial-boundary value problem of nonlinear delay wave equations is solved. The presented experiments further confirm the computational effectiveness of the methods and the theoretical results derived in previous.

  • AMS Subject Headings

65L03, 65L04, 65L80

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-16-423, author = {Zhang , ChengjianWang , Siyi and Tang , Changyang}, title = {Implicit Runge-Kutta-Nyström Methods with Lagrange Interpolation for Nonlinear Second-Order IVPs with Time-Variable Delay}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {16}, number = {2}, pages = {423--436}, abstract = {

This paper deals with nonlinear second-order initial value problems with time-variable delay. For solving this kind of problems, a class of implicit Runge-Kutta-Nyström (IRKN) methods with Lagrange interpolation are suggested. Under the suitable condition, it is proved that an IRKN method is globally stable and has the computational accuracy $\mathcal{O}(h^{min\{p,\mu+ν+1\}}),$ where $p$ is the consistency order of the method and $\mu, ν ≥0$ are the interpolation parameters. Combining a fourth-order compact difference scheme with IRKN methods, an initial-boundary value problem of nonlinear delay wave equations is solved. The presented experiments further confirm the computational effectiveness of the methods and the theoretical results derived in previous.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0290}, url = {http://global-sci.org/intro/article_detail/aamm/22338.html} }
TY - JOUR T1 - Implicit Runge-Kutta-Nyström Methods with Lagrange Interpolation for Nonlinear Second-Order IVPs with Time-Variable Delay AU - Zhang , Chengjian AU - Wang , Siyi AU - Tang , Changyang JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 423 EP - 436 PY - 2024 DA - 2024/01 SN - 16 DO - http://doi.org/10.4208/aamm.OA-2022-0290 UR - https://global-sci.org/intro/article_detail/aamm/22338.html KW - Nonlinear second-order initial value problems, time-variable delay, Lagrange interpolation, implicit Runge-Kutta-Nyström methods, error analysis, global stability. AB -

This paper deals with nonlinear second-order initial value problems with time-variable delay. For solving this kind of problems, a class of implicit Runge-Kutta-Nyström (IRKN) methods with Lagrange interpolation are suggested. Under the suitable condition, it is proved that an IRKN method is globally stable and has the computational accuracy $\mathcal{O}(h^{min\{p,\mu+ν+1\}}),$ where $p$ is the consistency order of the method and $\mu, ν ≥0$ are the interpolation parameters. Combining a fourth-order compact difference scheme with IRKN methods, an initial-boundary value problem of nonlinear delay wave equations is solved. The presented experiments further confirm the computational effectiveness of the methods and the theoretical results derived in previous.

Chengjian Zhang, Siyi Wang & Changyang Tang. (2024). Implicit Runge-Kutta-Nyström Methods with Lagrange Interpolation for Nonlinear Second-Order IVPs with Time-Variable Delay. Advances in Applied Mathematics and Mechanics. 16 (2). 423-436. doi:10.4208/aamm.OA-2022-0290
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