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

Abstract

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.

Keywords

• 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

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