Modife Formdaki Reaksiyon Difüzyon Denkleminin Faz Diyagramındaki Kararlılığı

Aşağıdaki denklem $?_?$ + $??_?$ = $?_??$ − (1 − $?^2$ ), ? ≠ 0, uzaklığı ve ? zamanı niteleyen, nonlineer dinamik sistemi içerisinde incelenmiştir. Başlangıç olarak yukarıdaki denleme yenidönüşüm uygulanarak kısmi differansiyel formu elde edildi. Sonra oluşturulan denklemin seçilmiş değerlerine bağlı kalınarak kısmidiferansiyel denklemin dinamik sistemi tanımlandı. Oluşturan kısmi diferansiyel formdaki denklemin dinamik sisteminin kritiknoktalarına bağlı kalınarak, sistemin özdğerlerinin yapısı tanımlandı. Amacımız unstable node dan stale node a dğoru bir heteroclinicyapı tanımlamak ve buna bağlı olarak dalgalanma hareketleri için gereken en küçük dalga hızını tanımlayıp başka dalgalanmahareketleri oluşumu varsa yapılarını belirlemek. Son olarak yapılan uygulamalara ek olarak matlab ode45 paketi kısmi diferansiyelformdaki denkleme uygulanarak faz diyagramında numerik çözümü elde edilmiştir ve parabolic method uygulanarak elde edilennumerik çözümlerle analitik çözüm karşılaştırılmıştır.

The Stability of a Modified Form of Reaction Diffusion Equation inPhase Plane

We examine the dynamics of nonlinear system related in the following equation namely, $?_?$ + $??_?$ = $?_??$ − (1 − $?^2$ ), where ? ≠ 0, represents distance, ? represents time. As a beginning we start to get ordinary differential equation form of aboveequation after substituting of a new transformation into it. Then dynamical system of ordinary differential equation form is indicateddepend on selected variables. According to the critical points of the dynamical system of ordinary differential equation form, thestructures of the eigenvalues of them are identified. We attempt to find a heteroclinic connection from unstable node to stable nodein parallel with travelling wave solutions for the minimum wave speed and the structure of the other travelling wave solutions to beidentified. Furthermore, by applying a matlab implementation of ode45 package the ordinary differential equation form isnumerically solved in phase plane and applying parabolic method to compare analytic and numric results.

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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
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
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.
Murray, JD. (2002). Mathematical Biology I: An Introduction. Third edition, Springer, New York.
Van Saarloos, W., (2003). Front propagation into unstable states.Phys. Rep.386, 29-222.