Analysing of Nonlinear Advanced Equation in Dynamic System

Year 2021, Issue: 31, 141 - 145, 31.12.2021

Esen Hanaç

https://doi.org/10.31590/ejosat.1005848

Abstract

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

Keywords

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

Dinamik Sistemde İleri Nonlineer Denklemin Analizi

Abstract

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.

Keywords

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

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