New travelling wave solutions of the Porous–Fisher model with a moving boundary

Fadai, Nabil T.; Simpson, Matthew J.

Abstract

We examine travelling wave solutions of the Porous-Fisher model, ϑtu(x,t) = u(x,t)[1 u(x,t)] + ϑx [u(x,t)ϑxu(x,t)], with a Stefan-like condition at the moving front, x = L(t). Travelling wave solutions of this model have several novel characteristics. These travelling wave solutions: (i) move with a speed that is slower than the more standard Porous-Fisher model, c < 1/√2; (ii) never lead to population extinction; (iii) have compact support and a well-defined moving front, and (iv) the travelling wave profiles have an infinite slope at the moving front. Using asymptotic analysis in two distinct parameter regimes, c → 0+ and c → 1/√2-, we obtain closed-form mathematical expressions for the travelling wave shape and speed. These approximations compare well with numerical solutions of the full problem.

Citation

Fadai, N. T., & Simpson, M. J. (2020). New travelling wave solutions of the Porous–Fisher model with a moving boundary. Journal of Physics A: Mathematical and Theoretical, 53(9), Article 095601. https://doi.org/10.1088/1751-8121/ab6d3c

• Journal Article Type: Article
• Acceptance Date: Jan 17, 2020
• Online Publication Date: Feb 7, 2020
• Publication Date: Mar 6, 2020
• Deposit Date: Feb 22, 2021
• Publicly Available Date: Mar 1, 2021
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 9
• Article Number: 095601
• DOI: https://doi.org/10.1088/1751-8121/ab6d3c
• Keywords: Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/4838938
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/ab6d3c
• Additional Information: This is the Accepted Manuscript version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1751-8121/ab6d3c