Numerical Solutions of Time fractional Klein Gordon Equation using Crank-Nicolson Finite Difference Method

Yıl 2024, , 1717 - 1730, 01.12.2024

Berat Karaağaç , Alaattin Esen , Muhammed Huzeyfe Uzunyol

https://doi.org/10.21597/jist.1496717

Öz

Finite difference methods are widely used numerical techniques used to solve partial differential equations observed in many fields, such as science and engineering. This research presents a study on the numerical solutions of the Klein-Gordon equation, which describes anomalous diffusion and wave propagation in quantum fields and possesses a fractional derivative in the Caputo sense. The content of the paper begins by discretizing the region of the problem while taking into account the fundamental characteristics of finite difference methods. Subsequently, the time derivative algorithm, and the other terms, are discretized using the Crank-Nicolson finite difference approach, resulting in a system of algebraic equations. Solving this algebraic equation system yields numerical solutions. The numerical results are calculated for various values of the parameters associated with the equation and fractional order derivatives , leading to the computation of error norms. Graphical findings illustrate the physical behavior of approximation solutions for a variety of fraction order values. Additionally, the stability analysis of the numerical scheme is investigated using von-Neumann stability analysis. The results of this paper will help other researchers studying in the field to apply the presented method to other problems modelling the natural phenomena.

Anahtar Kelimeler

Fractional Klein Gordon equation, Finite difference technique, Stability analysis

Zaman Kesirli Klein Gordon Denkleminin Crank-Nicolson Sonlu Farklar Yöntemi ile Sayısal Çözümleri

Öz

Sonlu fark yöntemleri fen ve mühendislik gibi birçok alanda gözlemlenen kısmi diferansiyel denklemlerin çözümünde yaygın olarak kullanılan sayısal bir yöntemdir. Bu araştırma, kuantum alanlarındaki anormal difüzyonu ve dalga yayılımını tanımlayan ve Caputo anlamında zamana göre kesirli türeve sahip Klein Gordon denkleminin nümerik çözümleri hakkında bir inceleme sunmaktadır. Araştırmanın içeriğinde sonlu fark yöntemlerinin temel karakteristiklerini göz önüne alınarak ilk olarak problemin çalışıldığı bölge ayrıklaştırılır. Daha sonra, zamana göre türev algoritması ve diğer terimler ise Crank-Nicolson sonlu fark yaklaşımı yardımıyla ayrıklaştırılarak bir cebirsel denklem sistemi elde edilir. Elde edilen Cebirsel denklem sisteminin çözülmesi ise nümerik çözümlerin üretilmesi ile sonuçlanır. Nümerik sonuçlar, denkleme ait parametrelerin ve kesirli mertebeden türevin çeşitli değerleri için hesaplanarak hata normları hesaplanır. Grafiksel bulgular ise kesirli mertebenin çeşitli değerleri için yaklaşık çözümlerin fiziksel davranışını sergilemektedir. Ayrıca, nümerik şemanın kararlılık analizi von- Neumann kararlılık analizi ile araştırılır. Bu çalışmanın sonuçları bu çalışmada sunulan yöntemi bu alanda çalışan diğer araştırmacıların doğadaki olayları modelleyen diğer problemlere uygulamalarına yardım edecektir.

Anahtar Kelimeler

Kesirli Klein Gordon denklemi, Sonlu Fark Yöntemleri, Kararlılık Analizi

Kaynakça

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