Numerical solution of a highly nonlinear and non-integrable equation using integrated radial basis function network method

Article


Bhanot, Rajeev P., Strunin, Dmitry V. and Ngo-Cong, Duc. 2020. "Numerical solution of a highly nonlinear and non-integrable equation using integrated radial basis function network method." Chaos: an interdisciplinary journal of nonlinear science. 30 (8). https://doi.org/10.1063/5.0009215
Article Title

Numerical solution of a highly nonlinear and non-integrable equation using integrated radial basis function network method

ERA Journal ID87
Article CategoryArticle
AuthorsBhanot, Rajeev P. (Author), Strunin, Dmitry V. (Author) and Ngo-Cong, Duc (Author)
Journal TitleChaos: an interdisciplinary journal of nonlinear science
Journal Citation30 (8)
Article Number083119
Number of Pages14
Year2020
PublisherAIP Publishing
Place of PublicationUnited States
ISSN1054-1500
1089-7682
Digital Object Identifier (DOI)https://doi.org/10.1063/5.0009215
Web Address (URL)https://pubs.aip.org/aip/cha/article-abstract/30/8/083119/341980/Numerical-solution-of-a-highly-nonlinear-and-non?redirectedFrom=fulltext
Abstract

In this paper, we investigate a wide range of dynamical regimes produced by the nonlinearly excited phase (NEP) equation (a single sixth-order nonlinear partial differential equation) using a more advanced numerical method, namely, the integrated radial basis function network method. Previously, we obtained single-step spinning solutions of the equation using the Galerkin method. First, we verify the numerical solver through an exact solution of a forced version of the equation. Doing so, we compare the numerical results obtained for different space and time steps with the exact solution. Then, we apply the method to solve the NEP equation and reproduce the previously obtained spinning regimes. In the new series of numerical experiments, we find regimes in the form of spinning trains of steps/kinks comprising one, two, or three kinks. The evolution of the distance between the kinks is analyzed. Two different kinds of boundary conditions are considered: homogeneous and periodic. The dependence of the dynamics on the size of the domain is explored showing how larger domains accommodate multiple spinning fronts. We determine the critical domain size (bifurcation size) above which non-trivial settled regimes become possible. The initial condition determines the direction of motion of the kinks but not their sizes and velocities.

Keywordsnon-linear partial differential equation, active, dissipative, spinning fronts
ANZSRC Field of Research 2020490303. Numerical solution of differential and integral equations
490105. Dynamical systems in applications
Public Notes

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 083119 (2020) and may be found at https://doi.org/10.1063/5.0009215

Byline AffiliationsSchool of Sciences
Institute for Advanced Engineering and Space Sciences
Institution of OriginUniversity of Southern Queensland
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Strunin, D. V.. 2007. "Universal regimes of a free turbulent jet." Jacobs, Peter A., McIntyre, Tim, Cleary, Matthew J., Buttsworth, David R., Mee, David, Clements, Rose, Morgan, Richard and Lemckert, Charles (ed.) 16th Australasian Fluid Mechanics Conference (AFMC 2007). Gold Coast, Australia 03 - 07 Dec 2007 Brisbane, Australia.
Attractors in confined source problems for coupled nonlinear diffusion
Strunin, Dmitry V.. 2007. "Attractors in confined source problems for coupled nonlinear diffusion." SIAM Journal on Applied Mathematics. 67 (6), pp. 1654-1674. https://doi.org/10.1137/060657923
A numerical model for the confinement of oil spill with floating booms
Zhu, SongPing and Strunin, Dmitry V.. 2002. "A numerical model for the confinement of oil spill with floating booms." Spill Science and Technology Bulletin. 7 (5-6), pp. 249-255. https://doi.org/10.1016/S1353-2561(02)00042-7
Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model
Georgiev, Dian J., Roberts, A. J. and Strunin, Dmitry V.. 2007. "Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model." Discrete and Continuous Dynamical Systems Series A.
The dynamics of the vertical structure of turbulence in flood flows
Georgiev, D. J., Roberts, A. J. and Strunin, D. V.. 2007. "The dynamics of the vertical structure of turbulence in flood flows." Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal. 48, pp. C573-C590. https://doi.org/10.0000/anziamj.v48i0.124
Models encompassing hydraulic jumps in radial flows over a horizontal plate
Strunin, Dmitry and Roberts, Tony. 2001. "Models encompassing hydraulic jumps in radial flows over a horizontal plate." Kluev, Vitaly and Mastorakis, Nikos (ed.) 2nd WSEAS Multiconference on Applied and Theoretical Mathematics (WSEAS 2001). Cairns, Australia 17 - 23 Dec 2001 Greece.
Coupled thermomechanical waves in hyperbolic thermoelasticity
Strunin, D. V., Melnik, R. V. N. and Roberts, A. J.. 2001. "Coupled thermomechanical waves in hyperbolic thermoelasticity." Journal of Thermal Stresses. 24 (2), pp. 121-140. https://doi.org/10.1080/01495730150500433
Nonlinear instability in generalized nonlinear phase diffusion equation
Strunin, Dmitry V.. 2003. "Nonlinear instability in generalized nonlinear phase diffusion equation." Progress of Theoretical Physics Supplement.
Attractors and centre manifolds in nonlinear diffusion
Strunin, Dmitry V.. 2005. "Attractors and centre manifolds in nonlinear diffusion." 2nd International Conference on Scientific Computing and Partial Differential Equations & The First East Asian SIAM Symposium. Hong Kong, China 12 - 16 Dec 2005 Hong Kong.
Nonlinear analysis of rubber-based polymeric materials with thermal relaxation models
Melnik, R. V. N., Strunin, D. V. and Roberts, A. J.. 2005. "Nonlinear analysis of rubber-based polymeric materials with thermal relaxation models." Numerical Heat Transfer Part A: Applications. 47 (6), pp. 549-569. https://doi.org/10.1080/10407780590891236