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Volume 15, Issue 4
A Novel Wavelet-Homotopy Galerkin Method for Unsteady Nonlinear Wave Equations

Yue Zhou & Hang Xu

Adv. Appl. Math. Mech., 15 (2023), pp. 964-983.

Published online: 2023-04

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  • Abstract

The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time. The Korteweg-de Vries (KdV) equation, the Burgers equation and the Korteweg-de Vries-Burgers (KdVB) equation are examined as illustrative examples. Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm. Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies. Furthermore, it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter. It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems. It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.

  • AMS Subject Headings

76B15, 65Mxx, 76Mxx

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-15-964, author = {Zhou , Yue and Xu , Hang}, title = {A Novel Wavelet-Homotopy Galerkin Method for Unsteady Nonlinear Wave Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2023}, volume = {15}, number = {4}, pages = {964--983}, abstract = {

The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time. The Korteweg-de Vries (KdV) equation, the Burgers equation and the Korteweg-de Vries-Burgers (KdVB) equation are examined as illustrative examples. Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm. Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies. Furthermore, it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter. It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems. It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2022-0046}, url = {http://global-sci.org/intro/article_detail/aamm/21598.html} }
TY - JOUR T1 - A Novel Wavelet-Homotopy Galerkin Method for Unsteady Nonlinear Wave Equations AU - Zhou , Yue AU - Xu , Hang JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 964 EP - 983 PY - 2023 DA - 2023/04 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2022-0046 UR - https://global-sci.org/intro/article_detail/aamm/21598.html KW - Coiflet wavelet, homotopy analysis method, wavelet-homotopy method, wave equations, unsteady. AB -

The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time. The Korteweg-de Vries (KdV) equation, the Burgers equation and the Korteweg-de Vries-Burgers (KdVB) equation are examined as illustrative examples. Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm. Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies. Furthermore, it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter. It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems. It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.

Yue Zhou & Hang Xu. (2023). A Novel Wavelet-Homotopy Galerkin Method for Unsteady Nonlinear Wave Equations. Advances in Applied Mathematics and Mechanics. 15 (4). 964-983. doi:10.4208/aamm.OA-2022-0046
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