Numerical Approximation Methods using Multiplicative Calculus - Aktöre, Hatice (Eastern Mediterranean University EMU, 2015)
In this thesis, the 2nd, 3rd and 4th order multiplicative Runge-Kutta Methods are developed in analogy to the classical Runge-Kutta Method. The error analysis is only carried out for the 4th order multiplicative Runge-Kutta method based on the convergence and stability analysis. The convergence behaviour of the developed multiplicative Runge-Kutta method is analysed by examining examples of initial value problems with closed form solutions, as well as problems without closed form solutions. The obtained results are also compared to the results obtained from the solutions of the classical Runge-Kutta method for the same examples. The error analysis shows that the solutions of the multiplicative Runge-Kutta methods give better results especially when the solution has an exponential nature. The modified quadratic Lorenz attractor is developed to examine the applicability of the proposed multiplicative Runge-Kutta method on the chaotic systems. The chaotic system is analysed numerically for its chaotic behaviour. Finally, the chaotic system is transformed into the corresponding system in terms of multiplicative calculus and the analysis are also done based on the rules of the multiplicative calculus. The results of the analysis show that the multiplicative Runge-Kutta method is also applicable to multiplicative chaotic systems. Keywords: Multiplicative calculus, complex multiplicative calculus, Runge-Kutta, differential equations, numerical approximation, dynamical systems.

ÖZ: Bu tezde, 2. ,3. ve 4. derece Runge-Kutta metodları temelinde çarpımsal analiz kuralları kullanılarak 2. ,3. ve 4. dereceden çarpımsal Runge-Kutta yöntemleri bulunmu¸s ve incelenmi¸stir. Bulunan yöntemlerin hata analizleri, yakınsaklık ve istikrarlılık analizleri temel alınarak yapılmı¸stır. Bulunan metodların yakınsaklık özellikleri, çözümleri bilinen ve bilinmeyen diferansiyel denklemler çözülerek gösterilmi¸stir. Çözümleri bilinen adi diferansiyel denklemler, çarpımsal Runge-Kutta ve Runge-Kutta yöntemleri kullanılarak çözülmü¸s ve hata analizleri yapılmı¸stır. Bu sonuçlara göre, özellikle çözümü eksponensiyel olan denklemlerde, çarpımsal Runge-Kutta metodunun bilinen Runge-Kutta metoduna göre daha iyi sonuçlar verdi˘gi görülmü¸stür. Son olarak da çarpımsal Runge-Kutta metodlarının karma¸sık sistemler üzerinde uygulanabildi˘gini göstermek için karma¸sık bir sistem bulunmu¸s ve numerik olarak incelenmi¸stir. Daha sonra bulunan sistem çarpımsal analiz kurallarına göre düzenlenmi¸s ve çarpımsal Runge- Kutta yöntemleri kullanılarak çözülmü¸stür. Elde edilen sonuçlar bulunan yöntemlerin karma¸sık sistemler üzerinde de kullanılabilece˘gini göstermi¸stir. AnahtarKelimeler: Çarpımsal analiz, kompleks çarpımsal analiz, Runge-Kutta, diferansiyel denklemler, numerik yakınsama, dinamik sistemler.
Doctor of Philosophy in Applied Mathematics and Computer Science. Thesis (Ph.D.)--Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2015. Supervisor: Assist. Prof. Dr. Mustafa Rıza.
Keywords: Mathematics; Applied Mathematics and Computer Science; Multiplicative calculus; complex multiplicative calculus; Runge-Kutta; differential equations; numerical approximation; dynamical systems