The Stability of a Modified Form of Reaction Diffusion Equation in Phase Plane

Yıl 2021, Sayı: 25, 702 - 706, 31.08.2021

Esen Hanaç

https://doi.org/10.31590/ejosat.920615

Cited By: 1

Öz

We examine the dynamics of nonlinear system related in the following equation namely,
𝑢𝑡 + 𝑢𝑢𝑥 = 𝑢𝑥𝑥 − (1 − 𝑢2),
where 𝑥 ≠ 0, represents distance, 𝑡 represents time. As a beginning we start to get ordinary differential equation form of above equation after substituting of a new transformation into it. Then dynamical system of ordinary differential equation form is indicated depend on selected variables. According to the critical points of the dynamical system of ordinary differential equation form, the structures of the eigenvalues of them are identified. We attempt to find a heteroclinic connection from unstable node to stable node in parallel with travelling wave solutions for the minimum wave speed and the structure of the other travelling wave solutions to be identified. Furthermore, by applying a matlab implementation of ode45 package the ordinary differential equation form is numerically solved in phase plane and applying parabolic method to compare analytic and numric results.

Anahtar Kelimeler

Phase plane analysis, stable node, unstable node

Modife Formdaki Reaksiyon Difüzyon Denkleminin Faz Diyagramındaki Kararlılığı

Öz

Aşağıdaki denklem
𝑢𝑡 + 𝑢𝑢𝑥 = 𝑢𝑥𝑥 − (1 − 𝑢2),
𝑥 ≠ 0, uzaklığı ve 𝑡 zamanı niteleyen, nonlineer dinamik sistemi içerisinde incelenmiştir. Başlangıç olarak yukarıdaki denleme yeni dönüşüm uygulanarak kısmi differansiyel formu elde edildi. Sonra oluşturulan denklemin seçilmiş değerlerine bağlı kalınarak kısmi diferansiyel denklemin dinamik sistemi tanımlandı. Oluşturan kısmi diferansiyel formdaki denklemin dinamik sisteminin kritik noktalarına bağlı kalınarak, sistemin özdğerlerinin yapısı tanımlandı. Amacımız unstable node dan stale node a dğoru bir heteroclinic yapı tanımlamak ve buna bağlı olarak dalgalanma hareketleri için gereken en küçük dalga hızını tanımlayıp başka dalgalanma hareketleri oluşumu varsa yapılarını belirlemek. Son olarak yapılan uygulamalara ek olarak matlab ode45 paketi kısmi diferansiyel formdaki denkleme uygulanarak faz diyagramında numerik çözümü elde edilmiştir ve parabolic method uygulanarak elde edilen numerik çözümlerle analitik çözüm karşılaştırılmıştır.

Anahtar Kelimeler

Faz diyagram analizi, stable node, unstable node

Kaynakça

• Behzadi, S.S and Araghi, M.A.F., (2011). Numerical Solution for Solving Burgers-Fisher Eguation by Using Iterative Methods.Mathematical and Computational Applications16, 443-455. https://doi.org/10.3390/mca16020443
• Bramson, M.D., (1983). Convergence of solutions of the Kolmogorov equation to travellingwaves. Mem. Amer. Math. Soc.44
• Burgers, J.M., (1939). Mathematical examples illustrating relations occurring in the theoryof turbulent fluid motion.Verh. Nederl. Akad. Wetensch. Afd. Natuurk.17, 1-53.
• Burgers, J.M., (1940). Application of a model system to illustrate some points of the statistical theory of free turbulence. Proc. Kon. Nederl. Akad. Wetensch.43, 2-12.
• Burgers, J.M., (1975). The Nonlinear Diffusion Equation. D. Reidel Publishing Company, Dordrecht, Holland.
• Canosa, J., (1973). On a nonlinear diffusion equation describing population growth. IBM Journal of Research and Development. 17 307-313. https://doi.org/10.1147/rd.174.0307
• Cross, M. C. and Hohenberg, P. C., (1993). Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 3, 851. https://doi.org/10.1103/RevModPhys.65.851
• Dee, G. and Langer, J. S., (1983). Propagation Pattern Selection, Rev. Lett. 50, 6, 383. https://doi.org/10.1103/PhysRevLett.50.383
• Edelstein-Keshet, L., (2005). Mathematical Models in Biology. SIAM, Philadelphia. https://doi.org/10.1137/1.9780898719147 Fisher, RA., (1937). The wave of advance of advantageous genes. Annals of Eugenics. 7 355-369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
• Griffiths, G., Schiesser, W. E., (2009). A Compendium of Partial Differential EquationModels. Cambridge University Press doi:10.1017/CBO9780511576270
• Griffiths, G., Schiesser, W. E., (2010). Travelling Wave Analysis of Partial DifferentialEquations. Academic Press. ISBN: 978-0-12-384652-5.
• Kolmogorov, AN, Petrovskii, PG, Piskunov, NS. (1937). A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Moscow University Mathematics Bulletin. 1 1-26.
• Kot, M., (2003). Elements of Mathematical Ecology. Camridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511608520. https://doi.org/10.1017/CBO9780511608520
• Landejuela, M., (2011). Burgers Equation. BCAM Internship report: Basque Center forApplied Mathematics.
• McKean, H.P., (1975) . Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math.28, 323-331. https://doi.org/10.1002/cpa.3160280302
• Murray, JD. (2002). Mathematical Biology I: An Introduction. Third edition, Springer, New York.ISBN 978-0-387-22437-4
• Van Saarloos, W., (2003). Front propagation into unstable states. Phys. Rep.386, 29-222.