Araştırma Makalesi
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Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü

Yıl 2018, Cilt: 20 Sayı: 1, 440 - 449, 25.04.2018
https://doi.org/10.25092/baunfbed.418501

Öz

Bu çalışmada, özellikle mühendislik, finans, fizik ve sismoloji gibi pek çok bilim dalında uygulamalara haiz başlangıç değer koşullarına sahip kesirli telegraf kısmi diferansiyel denklemi ele alındı.  Caputo kesirli kısmi türevli denklemin tanımı vasıtasıyla ele alınan kesirli telegraf kısmi diferansiyel denkleminin sonlu farklardaki ifadesi oluşturuldu. Aynı şekilde, ele alınan denklemin abstract formu ifade edildi.  Abstract formda verilen bu denklem için sonlu fark şemaları oluşturuldu.  Hilbert uzayı üzerinde tanımlanan norma göre denklemin oluşturulan bu sonlu fark şemalar için kararlılık kestirimleri gösterildi.  Kararlılık kestirimini ifade eden Teorem ispatıyla birlikte ifade edildi.  Sonlu fark şeması metodu kullanılarak α=0.1,0.5,0.9 un farklı değerleri için Caputo kesirli türevi vasıtası ile tanımlanan kesirli telegraf kısmi diferansiyel denkleminin nümerik çözümü elde edildi. Burada, kullanılan örnek problemlerin nümerik çözümleri Matlab programı kullanılarak oluşturuldu.  Laplace metodu veya geleneksel metotlar yardımıyla elde edilen tam çözüm ile yaklaşık çözümler mukayese edilerek hata analizi yapıldı.  Hata analizi tablosundan elde edilen çıkarsamaya göre önerilen metodun ne kadar etkili ve tutarlı olduğu gözlemlendi.

Kaynakça

  • Celik, C. ve Duman M., Crank-Nicholson method for the fractional equation with the Riezs fractional derivative, Journal of computational physics, 231, 1743-1750, (2012).
  • Gorial, I. I., Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative, Engineering. and Techology Journal, 29, 709-715, (2011).
  • Jafari, H. Ve Gejii, V. D., Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition, Applied Mathematics and Computation, 180,488-497, (2006).
  • Karatay, I., Bayramoglu, S. R. ve Sahin, A., Implicit difference approximation for the time fractional heat equation with the nonlocal condition, Applied Numerical Mathematics, 61, 1281-1288, (2011).
  • Su, L., Wang, W. ve Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Physics Letters A., 373, 4405-4408, (2009).
  • Tadjeran, C., Meerschaert, M. M. ve Scheffler, H. P., A Second-order Accurate Numerical Approximation for the Fractional Diffusion Equation, Journal of Computational Physics, 213, 205-213, (2006).
  • Karatay, I., Kale, N. ve Bayramoglu Erguner, S. R., Stability and Convergence of a Finite Partial Diferential Equations by Matrix Method, International Mathematical Forum, 9, 1757-1765, (2014).
  • Aslefallah, M., Rostamy, D. ve Hosseinkhani, K., Solving time-fractional differential diffusion equation by theta method, International Journal of Applied Mathematics and Mechanics, 2, 1-8, (2014).
  • Srivastava, V. K., Awasthi, M. K. ve Tamsir, M., RDTM solution of Caputo time fractional-order hyperbolic telegraph equation, AIP advances, 3, 032142, 1-11, (2013).
  • Ashyralyev, A. ve Dal, F., Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition, Discrete Dynamics in Nature and Society, 2012, 1-15, ( 2012).
  • Akgul, A., Inc, M. ve Baleanu, D., On solutions of variable-order fractional differential equations, An International Journal of Optimization and Control: Theories & Applications, 7(1), 112-116, (2017).
  • Chen, H., Xu, D., Cao, J. ve Zhou, J.,A backward Euler alternating direction implicit difference scheme for the three-dimensional fractional evolution equation, Numerical Methods for Partial Differential Equation, (2017).
  • Szekeres, B. ve Izsák, F., A finite difference method for fractional diffusion equations with Neumann boundary conditions, Open Mathematics, 13(1), pp. -. Retrieved 21 Feb. (2018), from doi:10.1515/math-2015-0056.
  • Changpin, L. ve Fanhai, Z., The Finite Difference Methods for Fractional Ordinary Differential Equations, Numerical Functional Analysis and Optimization, 34, 2, 149-179, (2013).
  • Modanli, M. ve Akgul, A., On solutions to the second-order partial differential equations by two accurate methods, Numerical Methods for Partial Differential Equations, 1-15, (2017).
  • Ashyralyev, A. ve Modanli, M., An operator method for telegraph partial differential and difference equations, Boundary Value Problems, 2015(1), 1-17, 2015).
  • Karatay, I., Bayramoglu, S. R. ve Sahin, A., A new difference scheme for time fractional heat equation based on the Crank-Nicholson method, Fractional Calculus and Applied Analysis, 16, 892-910, (2013).
  • Ashyralyev, A. ve Sobolevskii, P.E., New Difference Schemes for Partial Differential Equations, Birkhauser, Verlag, Basel, Boston, Berlin, (2004).
  • Kumar, S., A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling, 38, 3154-3163, (2014).

Numerical solution of fractional telegraph partial differential equations by difference scheme method

Yıl 2018, Cilt: 20 Sayı: 1, 440 - 449, 25.04.2018
https://doi.org/10.25092/baunfbed.418501

Öz

In this study, fractional telegraph partial differential equation with initial value condition having applications in physics, engineering, finance, seismology and other disciplines is discussed. By applying definition of Caputo fractional, difference scheme for fractional telegraph partial differential equation is obtained.  The abstract form of the considered equation is also stated.  The finite difference schemes of the abstract form for fractional telegraph partial differential equation are constructed.  The stability estimates of this finite difference scheme is proved with respect to the norm defined on the Hilbert space.  The proof of our main theorem determining the stability estimates is given in detail.  By using difference scheme method defined by Caputo fractional derivative, numerical solution of fractional telegraph partial differential equation is obtained  for different values of α=0.1,0.5,0.9.  Numerical solutions of our example is tested by using Matlab programming.  Error analysis was performed by comparing approximate solutions with exact solution obtained by Laplace or other traditional methods.  It is obvious that the proposed method is effective and consistent according to error analysis. 

Kaynakça

  • Celik, C. ve Duman M., Crank-Nicholson method for the fractional equation with the Riezs fractional derivative, Journal of computational physics, 231, 1743-1750, (2012).
  • Gorial, I. I., Numerical methods for fractional reaction-dispersion equation with Riesz space fractional derivative, Engineering. and Techology Journal, 29, 709-715, (2011).
  • Jafari, H. Ve Gejii, V. D., Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition, Applied Mathematics and Computation, 180,488-497, (2006).
  • Karatay, I., Bayramoglu, S. R. ve Sahin, A., Implicit difference approximation for the time fractional heat equation with the nonlocal condition, Applied Numerical Mathematics, 61, 1281-1288, (2011).
  • Su, L., Wang, W. ve Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Physics Letters A., 373, 4405-4408, (2009).
  • Tadjeran, C., Meerschaert, M. M. ve Scheffler, H. P., A Second-order Accurate Numerical Approximation for the Fractional Diffusion Equation, Journal of Computational Physics, 213, 205-213, (2006).
  • Karatay, I., Kale, N. ve Bayramoglu Erguner, S. R., Stability and Convergence of a Finite Partial Diferential Equations by Matrix Method, International Mathematical Forum, 9, 1757-1765, (2014).
  • Aslefallah, M., Rostamy, D. ve Hosseinkhani, K., Solving time-fractional differential diffusion equation by theta method, International Journal of Applied Mathematics and Mechanics, 2, 1-8, (2014).
  • Srivastava, V. K., Awasthi, M. K. ve Tamsir, M., RDTM solution of Caputo time fractional-order hyperbolic telegraph equation, AIP advances, 3, 032142, 1-11, (2013).
  • Ashyralyev, A. ve Dal, F., Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition, Discrete Dynamics in Nature and Society, 2012, 1-15, ( 2012).
  • Akgul, A., Inc, M. ve Baleanu, D., On solutions of variable-order fractional differential equations, An International Journal of Optimization and Control: Theories & Applications, 7(1), 112-116, (2017).
  • Chen, H., Xu, D., Cao, J. ve Zhou, J.,A backward Euler alternating direction implicit difference scheme for the three-dimensional fractional evolution equation, Numerical Methods for Partial Differential Equation, (2017).
  • Szekeres, B. ve Izsák, F., A finite difference method for fractional diffusion equations with Neumann boundary conditions, Open Mathematics, 13(1), pp. -. Retrieved 21 Feb. (2018), from doi:10.1515/math-2015-0056.
  • Changpin, L. ve Fanhai, Z., The Finite Difference Methods for Fractional Ordinary Differential Equations, Numerical Functional Analysis and Optimization, 34, 2, 149-179, (2013).
  • Modanli, M. ve Akgul, A., On solutions to the second-order partial differential equations by two accurate methods, Numerical Methods for Partial Differential Equations, 1-15, (2017).
  • Ashyralyev, A. ve Modanli, M., An operator method for telegraph partial differential and difference equations, Boundary Value Problems, 2015(1), 1-17, 2015).
  • Karatay, I., Bayramoglu, S. R. ve Sahin, A., A new difference scheme for time fractional heat equation based on the Crank-Nicholson method, Fractional Calculus and Applied Analysis, 16, 892-910, (2013).
  • Ashyralyev, A. ve Sobolevskii, P.E., New Difference Schemes for Partial Differential Equations, Birkhauser, Verlag, Basel, Boston, Berlin, (2004).
  • Kumar, S., A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling, 38, 3154-3163, (2014).
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Araştırma Makalesi
Yazarlar

Mahmut Modanlı

Yayımlanma Tarihi 25 Nisan 2018
Gönderilme Tarihi 1 Aralık 2017
Yayımlandığı Sayı Yıl 2018 Cilt: 20 Sayı: 1

Kaynak Göster

APA Modanlı, M. (2018). Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(1), 440-449. https://doi.org/10.25092/baunfbed.418501
AMA Modanlı M. Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. BAUN Fen. Bil. Enst. Dergisi. Temmuz 2018;20(1):440-449. doi:10.25092/baunfbed.418501
Chicago Modanlı, Mahmut. “Kesirli Telegraf kısmi Diferansiyel Denklemlerin Fark şeması Metodu Ile nümerik çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20, sy. 1 (Temmuz 2018): 440-49. https://doi.org/10.25092/baunfbed.418501.
EndNote Modanlı M (01 Temmuz 2018) Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20 1 440–449.
IEEE M. Modanlı, “Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü”, BAUN Fen. Bil. Enst. Dergisi, c. 20, sy. 1, ss. 440–449, 2018, doi: 10.25092/baunfbed.418501.
ISNAD Modanlı, Mahmut. “Kesirli Telegraf kısmi Diferansiyel Denklemlerin Fark şeması Metodu Ile nümerik çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 20/1 (Temmuz 2018), 440-449. https://doi.org/10.25092/baunfbed.418501.
JAMA Modanlı M. Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. BAUN Fen. Bil. Enst. Dergisi. 2018;20:440–449.
MLA Modanlı, Mahmut. “Kesirli Telegraf kısmi Diferansiyel Denklemlerin Fark şeması Metodu Ile nümerik çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 20, sy. 1, 2018, ss. 440-9, doi:10.25092/baunfbed.418501.
Vancouver Modanlı M. Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. BAUN Fen. Bil. Enst. Dergisi. 2018;20(1):440-9.