Analysing of Nonlinear Advanced Equation in Dynamic System

Öz

We mainly examine the type of the structure of the solutions of the following equation namely, u_t+kuu_x=u_xx+u^2 (1-u),-∞0 where k≠0 is a parameter occurrence in the long term by using dynamical system theory and exhibiting a phase-space analysis of its stable points. The critical points are identified depend on the solution of above equation in dynamic system. Then in parallel with the ciritical points eigenvalues and eigenvectors are determined and thus general solutions are written by depending on those found eigenvalues and eigenvectors. Thus, the structure of the critical points can be named in the phase -space. After some minor calculations are done, from one equilibrium point that enhancing from 0 to decreasing to 1 into the other and thus heteroclinic trajectory is demonstrated that supports the travelling wave solution to the equation. Then all points are indicated depending on properties of the structure of eigenvalues of the critical points in phase-space by using a generated matlab implementation. The result of the our work illustrates that the equation can confirm shock-wave solutions

Anahtar Kelimeler

• Phase-space analysis
• Nonlinear advanced equation
• Shock-wave solutions.

Dinamik Sistemde İleri Nonlineer Denklemin Analizi

Öz

Aşağıdaki denklemin u_t+kuu_x=u_xx+u^2 (1-u),-∞0 k≠0 bir parametre, sonuçlar yapısını dinamik sistemi ve sabit noktaların faz-uzay analizlerini kullanarak, uzun zamandaki oluşumunu inceliyoruz. Dinamik sistemde yukarıdaki denklemin çözümüne bağlı olarak kritik noktalar belirlenir. Daha sonra kritik noktalara paralel olarak özdeğerler ve özvektörler belirlenir ve böylece bulunan özdeğerler ve özvektörlere bağlı olarak genel çözümler yazılır. Böylece kritik noktaların yapısı faz uzayında isimlendirilebilir. Bazı küçük hesaplamalar yapıldıktan sonra, 0'dan 1'e artan bir denge noktasından diğerine doğru artan ve böylece heteroclinic yörüngenin denkleme giden dalga çözümünü desteklediği gösterilmiştir. Daha sonra üretilen bir matlab uygulaması kullanılarak faz uzayındaki kritik noktaların özdeğer yapısının özelliklerine bağlı olarak tüm noktalar belirtilir. Çalışmamızın sonucu, denklemin şok dalgası çözümlerini doğrulayabildiğini göstermektedir.

Anahtar Kelimeler

• Faz-uzay analizi
• İleri nonlineer denklem
• Dalga çözümleri.

Kaynakça

1. 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
2. Bramson, M.D., (1983). Convergence of solutions of the Kolmogorov equation to travellingwaves. Mem. Amer. Math. Soc.44. ISBN: 978-1-4704-0695-0
3. Burgers, J.M., (1939). Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion.Verh. Nederl. Akad. Wetensch. Afd. Natuurk.17, 1-53. https://www.dwc.knaw.nl/DL/publications/PU00011461.pdf
4. 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.
5. Burgers, J.M., (1975). The Nonlinear Diffusion Equation. D. Reidel Publishing Company, Dordrecht, Holland.
6. 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
7. Griffiths, G., Schiesser, W. E., (2009). A Compendium of Partial Differential EquationModels. Cambridge University Press doi:10.1017/CBO9780511576270
8. Griffiths, G., Schiesser, W. E., (2010). Travelling Wave Analysis of Partial DifferentialEquations. Academic Press. ISBN: 978-0-12-384652-5

9. Hanaç, E. (2021). The Stability of A modified form of Reaction Diffusion Equation in Phase Plane. https://doi.org/10.31590/ejosat.920615
10. 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.
11. Kot, M., (2003). Elements of Mathematical Ecology. Camridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511608520
12. Landejuela, M., (2011). Burgers Equation. BCAM Internship report: Basque Center forApplied Mathematics.
13. 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
14. Murray, JD. (2002). Mathematical Biology I: An Introduction. Third edition, Springer, New York. ISBN 978-0-387-22437-4
15. Van Saarloos, W., (2003). Front propagation into unstable states. Phys. Rep.386, 29-222.