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

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

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

Kaynakça

1. Akram, T., Abbas, M., Riaz, M. B., Ismail, A. I. and Ali, N. M. (2020). Development and analysis of new approximation of extended cubic b-spline to the nonlinear time fractional klein–gordon equation. Fractals, 28(08) , 2040039.
2. Amin, M., Abbas, M., Iqbal, M. K., Baleanu, D.(2020). Numerical treatment of time-fractional Klein–Gordon equation using redefined extended cubic B-spline functions. Frontiers in Physic, 8, 288.
3. Bansu, H., Kumar, S.(2021). Numerical solution of space-time fractional Klein-Gordon equation by radial basis functions and Chebyshev polynomials. International Journal of Applied and Computational Mathematics, 7, 1-19.
4. Biswas B, (2024). Analytical Solutions of the D-dimensional Klein-Gordon equation with q-deformed modified Pöschl-Teller Potential. Electronic Journal of Applied Mathematics, 2,1,14-21.
5. Dehghan, M., Abbaszadeh, M. and Mohebbi, A. (2015). An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Engineering Analysis with Boundary Elements, 50, 412-434.
6. Ganji, R. M., Jafari, H., Kgarose, M. and Mohammadi, A. (2021). Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials. Alexandria Engineering Journal , 60.5, 4563-4571.
7. Habjia, A., Hajaji, A. E., Ghordaf, J. E., Hilal, K., & Charhabil, A. (2024). High-Precision Method for Space-Time-Fractional Klein-Gordon Equation. In Applied Mathematics and Modelling in Finance, Marketing and Economics, 1-14. Cham: Springer Nature Switzerland
8. Hashemizadeh, E., Ebrahimzadeh, A. (2018). An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics. Physica A: Statistical Mechanics and its Applications, 503, 1189-1203.

9. Korichi, Z., Souigat, A., Bekhouche, R., and Meftah, M. T. (2024). Solution of the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives in statistical mechanics. Theoretical and Mathematical Physics, 218 (2), 336-345.
10. Mirzaei, S., & Shokri, A. (2024). Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral Method. Computational Methods for Differential Equations.
11. Mulimani, M., Kumbinarasaiah, S.(2024). A numerical study on the nonlinear fractional Klein–Gordon equation. Journal of Umm Al-Qura University for Applied Sciences , 10.1, 178-199.
12. Mohebbi, A., Abbaszadeh, M., Dehghan, M. (2014). High-order difference scheme for the solution of linear time fractional Klein–Gordon equations. Numerical Methods for Partial Differential Equations, 30.4,1234-1253.
13. Nagy, A. M. (2017). Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method, Applied Mathematics and Computation. 310 ,139-148.
14. Odibat, Z., Momani, S.(2009). The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl., 58,2199–208.
15. Odibat, Z. (2024). Numerical simulation for an initial-boundary value problem of time-fractional Klein-Gordon equations. Applied Numerical Mathematics
16. Oldham, K., and Spanier, J. (1974). The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
17. Paredes, G. E. (2020). Fractional-order models for nuclear reactor analysis. Woodhead Publishing.
18. Rubin, S. G., Graves, R. A.(1975). A cubic spline approximation for problems in fluid mechanics, National aeronauticsand space administration. Technical Report, Washington, 1975.
19. Sahu, I., & Jena, S. R. (2024). An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics. Partial Differential Equations in Applied Mathematics, 100744.
20. Vivas-Cortez, M., Huntul, M. J., Khalid, M., Shafiq, M., Abbas, M., & Iqbal, M. K. (2024). Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics. Computation, 12(4), 80.
21. Yaseen M, Abbas M, Ahmad B(2021), Numerical simulation of the nonlinear generalized time-fractional Klein–Gordon equation using cubic trigonometric B-spline functions, Mathematical Methods in the Applied Sciences 44.1: 901-916.