Akademik Veri Yönetim Sistemi
YILDIRIM Ö. (Yürütücü), Yıldız Ç.
Yükseköğretim Kurumları Destekli Proje, BAP Y.Lisans, 2024 - 2025
In this thesis, an unconditionally stable fourth-order accuracy difference scheme corresponding to a nonlinear system of sine-Gordon equations is studied. Many problems from real-world phenomena, such as the coupled sine-Gordon equations, are receiving increasing attention due to their weak-sense solvability resulting from the low regularity of the coefficients and source functions. A weak solution is a type of solution that does not satisfy the equations in the classical sense but instead satisfies a weaker set of conditions. The energy method is a mathematical technique that can be used to prove the existence and uniqueness of weak solutions of PDEs. The variational method, which is also called the energy-based approach, is a powerful tool in the theory of partial differential equations, allowing these equations to be analyzed and solved in Sobolev spaces in the context of distributions. In this thesis, in Chapter 1, the properties and application areas of the sine-Gordon equation are mentioned, then the relationship between Soliton and the sine-Gordon equation is mentioned. The chapter is completed by giving the relationship between DNA and the sine-Gordon equation. In Chapter 2, the basic definitions and theories of nonlinear PDEs, which play a major role in the formation of Chapter 3, are given. In Chapter 3, firstly using variational calculus techniques, the fourth order unconditionally stable difference scheme of the coupled sine-Gordon equation in Sobolev spaces is given, the existence and uniqueness of the weak solutions of the scheme are proven and it is shown that they are stable. The sine-Gordon equations are widely used in various fields of physics and mathematics to model wave phenomena and solitons. The findings of this study have implications for various scientific and technological disciplines such as physics, engineering and applied mathematics. The thesis argues that weak solutions and the energy method are powerful tools for solving PDEs.