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The B-spline Collocation Approach for Coupled Klein-Gordon Equation

Yıl 2019, , 295 - 300, 15.04.2019
https://doi.org/10.17714/gumusfenbil.427097

Öz

 This
research presents a new approach for obtaining numerical solutions of Coupled
Klein Gordon equation using the collocation method which based on cubic
B-spline base functions and finite element approximation.
The main advantage of the collocation
method is that the structure of the method is simple and the computational cost
is low. It also provides an easy and simpler procedure for solving various
problems involving differential equations that model real-world phenomena
. In the current research, the temporal
and spatial partial derivatives are discretized with using approximate solution
which is formed linear combination of B-spline basis and time dependent
parameters. With the help of the idea that approximate solution satisfy the PDE
at collocation points, a new numerical scheme is constructed. The newly
obtained numerical scheme tested on a model problem. Numerical results are
compared with exact solution with the aid of the error norms  Land Ls presented via tables .
Additionally, graphical simulations of numerical solutions are presented. 

Kaynakça

  • Alagesan, T., Chung Y. and Nakkeeran K., 2004. Soliton solutions of coupled nonlinear Klein–Gordon equations. Chaos, Solitons & Fractals, 21(4), 879-882.
  • Biswas, A., Kara, A. H., Moraru, L., Bokhari, A. H., and Zaman, F. D. 2014. Conservation laws of coupled Klein-Gordon equations with cubic and power law nonlinearities. Proceedings of the Romanian academy, Series A, 15(2), 123-129.
  • Dağ, I., Irk, D., and Saka, B. 2005. A numerical solution of the Burgers' equation using cubic B-splines. Applied Mathematics and Computation, 163(1), 199-211.
  • Doha, E. H., Bhrawy, A. H., Baleanu, D., and Abdelkawy, M. A. 2014. Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations. Rom. J. Phys, 59(3-4), 247-264.
  • Esen, A., Tasbozan, O., Ucar Y. and Yagmurlu, N. M. 2015. A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations. Tbilisi Mathematical Journal, 8.2, 181-193.
  • Khusnutdinova, K. R., and Pelinovsky, D. E. (2003). On the exchange of energy in coupled Klein–Gordon equations. Wave Motion, 38(1), 1-10.
  • Kutluay, S., Ucar, Y., and Yagmurlu, N. M. 2016. Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method. Bulletin of the Malaysian Mathematical Sciences Society , 39.4, 1603-1614.
  • Liu, S., Fu, Z., Liu, S., and Wang, Z. (2004). The periodic solutions for a class of coupled nonlinear Klein–Gordon equations. Physics Letters A, 323(5-6), 415-420.
  • Malomed, B. A., Mihalache, D., Wise, F., and Torner, L. 2005. Spatiotemporal optical solitons. Journal of Optics B: Quantum and Semiclassical Optics, 7(5), R53.
  • Mihalache, D. 2012. Linear and nonlinear light bullets: recent theoretical and experimental studies. Rom. J. Phys, 57(1-2), 352-371.
  • Porsezian, K., and Alagesan, T. 1995. Painlevé analysis and complete integrability of coupled Klein-Gordon equations. Physics Letters A, 198(5-6), 378-382.
  • Prenter, P. M. 2008. Splines and variational methods. Courier Corporation.

İkili Klein-Gordon Denklemi İçin B-spline Kollokasyon Yaklaşımı

Yıl 2019, , 295 - 300, 15.04.2019
https://doi.org/10.17714/gumusfenbil.427097

Öz

Bu çalışma, kübik B-spline baz fonksiyonları ve
sonlu eleman yaklaşımına temellenen kollokasyon yöntemi kullanılarak ikili
Klein-Gordon denkleminin nümerik çözümlerini elde etmek için yeni bir yaklaşım
sunmaktadır. Kollokasyon yönteminin başlıca avantajı, yöntemin yapısının basit
ve hesaplama maliyetinin düşük olmasıdır. Ayrıca, gerçek dünya olgularını
modelleyen diferansiyel denklemleri içeren çeşitli problemlerin çözümünde kolay
ve daha basit bir prosedür elde edilmesini sağlar.  Mevcut çalışmada, zamansal ve konumsal kısmi
türevler, B-spline bazların ve zamana bağlı parametrelerin doğrusal
birleşiminden oluşan yaklaşık çözüm kullanılarak ayrıştırılır. Yaklaşık çözümün
kısmi diferansiyel denklemi kollokasyon noktalarında sağlaması fikrinin yardımı
ile yeni bir sayısal şema oluşturulur. Yeni elde edilen şema bir model problem
üzerinde test edilir. Sayısal sonuçlar L2 
ve Ls hata normları yardımı ile tam çözümlerle
karşılaştırılır ve tablolar aracılığı ile sunulur.  Ayrıca sayısal çözümlerin grafik benzetimleri
sunulur.

Kaynakça

  • Alagesan, T., Chung Y. and Nakkeeran K., 2004. Soliton solutions of coupled nonlinear Klein–Gordon equations. Chaos, Solitons & Fractals, 21(4), 879-882.
  • Biswas, A., Kara, A. H., Moraru, L., Bokhari, A. H., and Zaman, F. D. 2014. Conservation laws of coupled Klein-Gordon equations with cubic and power law nonlinearities. Proceedings of the Romanian academy, Series A, 15(2), 123-129.
  • Dağ, I., Irk, D., and Saka, B. 2005. A numerical solution of the Burgers' equation using cubic B-splines. Applied Mathematics and Computation, 163(1), 199-211.
  • Doha, E. H., Bhrawy, A. H., Baleanu, D., and Abdelkawy, M. A. 2014. Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations. Rom. J. Phys, 59(3-4), 247-264.
  • Esen, A., Tasbozan, O., Ucar Y. and Yagmurlu, N. M. 2015. A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations. Tbilisi Mathematical Journal, 8.2, 181-193.
  • Khusnutdinova, K. R., and Pelinovsky, D. E. (2003). On the exchange of energy in coupled Klein–Gordon equations. Wave Motion, 38(1), 1-10.
  • Kutluay, S., Ucar, Y., and Yagmurlu, N. M. 2016. Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method. Bulletin of the Malaysian Mathematical Sciences Society , 39.4, 1603-1614.
  • Liu, S., Fu, Z., Liu, S., and Wang, Z. (2004). The periodic solutions for a class of coupled nonlinear Klein–Gordon equations. Physics Letters A, 323(5-6), 415-420.
  • Malomed, B. A., Mihalache, D., Wise, F., and Torner, L. 2005. Spatiotemporal optical solitons. Journal of Optics B: Quantum and Semiclassical Optics, 7(5), R53.
  • Mihalache, D. 2012. Linear and nonlinear light bullets: recent theoretical and experimental studies. Rom. J. Phys, 57(1-2), 352-371.
  • Porsezian, K., and Alagesan, T. 1995. Painlevé analysis and complete integrability of coupled Klein-Gordon equations. Physics Letters A, 198(5-6), 378-382.
  • Prenter, P. M. 2008. Splines and variational methods. Courier Corporation.
Toplam 12 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Berat Karaagac 0000-0002-6020-3243

Yayımlanma Tarihi 15 Nisan 2019
Gönderilme Tarihi 25 Mayıs 2018
Kabul Tarihi 1 Ekim 2018
Yayımlandığı Sayı Yıl 2019

Kaynak Göster

APA Karaagac, B. (2019). İkili Klein-Gordon Denklemi İçin B-spline Kollokasyon Yaklaşımı. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 9(2), 295-300. https://doi.org/10.17714/gumusfenbil.427097