Araştırma Makalesi
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Lineer olmayan dinamik teorisi için (1/G') -açılım metodunu kullanarak Kuramoto-Sivashinsky denkleminin karmaşık hiperbolik yürüyen dalga çözümleri

Yıl 2019, Cilt: 21 Sayı: 2, 590 - 599, 28.06.2019
https://doi.org/10.25092/baunfbed.631193

Öz

Bu makalede lineer olmayan Kuramoto–Sivashinsky denkleminin yeni karmaşık hiperbolik yürüyen dalga çözümlerini elde etmek için  (1/G') metodunu kullanılmıştır. Elde edilen çözümlerdeki parametrelere özel değerler verilmiş ve grafikler çizilmiştir.  Bu grafikler özel paket programı kullanılarak sunulmuştur.  Bu yöntem, bu çalışma için belirlenen hedeflere ulaşmak için kullanılmıştır.

Kaynakça

  • Zhang, S., and Xia, T., A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Physics Letters A, 363 (5-6), 356-360, (2007).
  • Wang, M., Li, X., and Zhang, J., The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (4), 417-423, (2008).
  • Liao, S., On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147 (2), 499-513, (2004).
  • Jiong, S., Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309 (5-6), 387-396, (2003).
  • Raslan, K. R., The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynamics, 53 (4), 281-286, (2008).
  • Gurefe, Y., Misirli, E., Sonmezoglu, A., and Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations, Applied Mathematics and Computation, 219 (10), 5253-5260, (2013).
  • Liu, S., Fu, Z., Liu, S., and Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289 (1-2), 69-74, (2001).
  • Liu, W., and Chen, K., The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana, 81 (3), 377-384, (2013).
  • Yokuş, A., Comparison of caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32 (29), 1850365, (2018).
  • Yokus, A., and Kaya, D., Numerical and exact solutions for time fractional Burgers’ equation, Journal of Nonlinear Sciences and Applications, 3419-3428, 10 (2017).
  • Yokuş, A., An expansion method for finding traveling wave solutions to nonlinear pdes, İstanbul Ticaret Üniversitesi, (2015).
  • Yavuz, M., and Ozdemir, N., Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22 (1), 185-194, (2018).
  • Yavuz, M., Ozdemir, N., and Baskonus, H. M., Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133 (6), 215 (2018).
  • Kaya, D., An explicit solution of coupled viscous Burger’ equation by the decomposition method, International Journal of Mathematics and Mathematical Sciences, 27 (11), 675-680, (2001).
  • Aziz, I., and Ahmad, M., Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions, Computers Mathematics with Applications, 69 (3), 180-205, (2015).
  • Esen, A., Sulaiman, T. A., Bulut, H., and Baskonus, H. M., Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation, Optik, 150-156, 167 (2018).
  • Evirgen, F., and Özdemir, N. A fractional order dynamical trajectory approach for optimization problem with HPM. In Fractional Dynamics and Control (pp. 145-155). Springer, New York, NY., (2012).
  • Evirgen, F., and Özdemir, N. Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system. Journal of Computational and Nonlinear Dynamics, 6(2), 021003, (2011).
  • Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147 (5-6), 287-291, (1990).
  • Hyman, J. M., and Nicolaenko, B., The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems, Physica D: Nonlinear Phenomena, 18 (1-3), 113-126, (1986).
  • Jolly, M. S., Kevrekidis, I. G., and Titi, E. S., Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D: Nonlinear Phenomena, 44 (1-2), 38-60, (1990).
  • Rademacher, J. D., and Wittenberg, R. W., Viscous shocks in the destabilized Kuramoto-Sivashinsky equation, Journal of computational and nonlinear Dynamics, 1 (4), 336-347, (2006).
  • Conte, R., and Musette, M., Painleve analysis and Backlund transformation in the Kuramoto-Sivashinsky equation, Journal of Physics A: Mathematical and General, 22 (2), 169, (1989).
  • Zgliczynski, P., and Mischaikow, K., Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (3), 255-288, (2001).
  • Chen, H., and Zhang, H., New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation, Chaos, Solitons & Fractals, 19 (1), 71-76, (2004).
  • Chang, H. C., Traveling waves on fluid interfaces: normal form analysis of the Kuramoto–Sivashinsky equation, The Physics of fluids, 29 (10), 3142-3147, (1986).
  • Sneppen, K., Krug, J., Jensen, M. H., Jayaprakash, C., and Bohr, T., Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation. Physical Review A, 46 (12), R7351, (1992).
  • Abbasbandy, S., Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method, Nonlinear Dynamics, 52 (1-2), 35-40, (2008).
  • Xu, Y., and Shu, C. W., Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations, Computer methods in applied mechanics and engineering, 195 (25-28), 3430-3447, (2006).
  • Rademacher, J. D., & Wittenberg, R. W. (2006). Viscous shocks in the destabilized Kuramoto-Sivashinsky equation. Journal of computational and nonlinear dynamics, 1(4), 336-347.

Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G') expansion method for nonlinear dynamic theory

Yıl 2019, Cilt: 21 Sayı: 2, 590 - 599, 28.06.2019
https://doi.org/10.25092/baunfbed.631193

Öz

In this paper, it is (1/G') expansion method which are used to obtain new complex hyperbolic traveling wave solutions of the non-linear Kuramoto-Sivashinsky equation.   Special values are given to the parameters in the solutions obtained and graphs are drawn.  These graphs are presented using special package program.  This method is employed to achieve the goals set for this study.

Kaynakça

  • Zhang, S., and Xia, T., A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Physics Letters A, 363 (5-6), 356-360, (2007).
  • Wang, M., Li, X., and Zhang, J., The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (4), 417-423, (2008).
  • Liao, S., On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147 (2), 499-513, (2004).
  • Jiong, S., Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309 (5-6), 387-396, (2003).
  • Raslan, K. R., The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynamics, 53 (4), 281-286, (2008).
  • Gurefe, Y., Misirli, E., Sonmezoglu, A., and Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations, Applied Mathematics and Computation, 219 (10), 5253-5260, (2013).
  • Liu, S., Fu, Z., Liu, S., and Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289 (1-2), 69-74, (2001).
  • Liu, W., and Chen, K., The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana, 81 (3), 377-384, (2013).
  • Yokuş, A., Comparison of caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32 (29), 1850365, (2018).
  • Yokus, A., and Kaya, D., Numerical and exact solutions for time fractional Burgers’ equation, Journal of Nonlinear Sciences and Applications, 3419-3428, 10 (2017).
  • Yokuş, A., An expansion method for finding traveling wave solutions to nonlinear pdes, İstanbul Ticaret Üniversitesi, (2015).
  • Yavuz, M., and Ozdemir, N., Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22 (1), 185-194, (2018).
  • Yavuz, M., Ozdemir, N., and Baskonus, H. M., Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133 (6), 215 (2018).
  • Kaya, D., An explicit solution of coupled viscous Burger’ equation by the decomposition method, International Journal of Mathematics and Mathematical Sciences, 27 (11), 675-680, (2001).
  • Aziz, I., and Ahmad, M., Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions, Computers Mathematics with Applications, 69 (3), 180-205, (2015).
  • Esen, A., Sulaiman, T. A., Bulut, H., and Baskonus, H. M., Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation, Optik, 150-156, 167 (2018).
  • Evirgen, F., and Özdemir, N. A fractional order dynamical trajectory approach for optimization problem with HPM. In Fractional Dynamics and Control (pp. 145-155). Springer, New York, NY., (2012).
  • Evirgen, F., and Özdemir, N. Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system. Journal of Computational and Nonlinear Dynamics, 6(2), 021003, (2011).
  • Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Physics Letters A, 147 (5-6), 287-291, (1990).
  • Hyman, J. M., and Nicolaenko, B., The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems, Physica D: Nonlinear Phenomena, 18 (1-3), 113-126, (1986).
  • Jolly, M. S., Kevrekidis, I. G., and Titi, E. S., Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D: Nonlinear Phenomena, 44 (1-2), 38-60, (1990).
  • Rademacher, J. D., and Wittenberg, R. W., Viscous shocks in the destabilized Kuramoto-Sivashinsky equation, Journal of computational and nonlinear Dynamics, 1 (4), 336-347, (2006).
  • Conte, R., and Musette, M., Painleve analysis and Backlund transformation in the Kuramoto-Sivashinsky equation, Journal of Physics A: Mathematical and General, 22 (2), 169, (1989).
  • Zgliczynski, P., and Mischaikow, K., Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (3), 255-288, (2001).
  • Chen, H., and Zhang, H., New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation, Chaos, Solitons & Fractals, 19 (1), 71-76, (2004).
  • Chang, H. C., Traveling waves on fluid interfaces: normal form analysis of the Kuramoto–Sivashinsky equation, The Physics of fluids, 29 (10), 3142-3147, (1986).
  • Sneppen, K., Krug, J., Jensen, M. H., Jayaprakash, C., and Bohr, T., Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation. Physical Review A, 46 (12), R7351, (1992).
  • Abbasbandy, S., Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method, Nonlinear Dynamics, 52 (1-2), 35-40, (2008).
  • Xu, Y., and Shu, C. W., Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations, Computer methods in applied mechanics and engineering, 195 (25-28), 3430-3447, (2006).
  • Rademacher, J. D., & Wittenberg, R. W. (2006). Viscous shocks in the destabilized Kuramoto-Sivashinsky equation. Journal of computational and nonlinear dynamics, 1(4), 336-347.
Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Asıf Yokuş 0000-0002-1460-8573

Hülya Durur 0000-0002-9297-6873

Yayımlanma Tarihi 28 Haziran 2019
Gönderilme Tarihi 7 Nisan 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 21 Sayı: 2

Kaynak Göster

APA Yokuş, A., & Durur, H. (2019). Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21(2), 590-599. https://doi.org/10.25092/baunfbed.631193
AMA Yokuş A, Durur H. Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory. BAUN Fen. Bil. Enst. Dergisi. Haziran 2019;21(2):590-599. doi:10.25092/baunfbed.631193
Chicago Yokuş, Asıf, ve Hülya Durur. “Complex Hyperbolic Traveling Wave Solutions of Kuramoto-Sivashinsky Equation Using (1/G’) Expansion Method for Nonlinear Dynamic Theory”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21, sy. 2 (Haziran 2019): 590-99. https://doi.org/10.25092/baunfbed.631193.
EndNote Yokuş A, Durur H (01 Haziran 2019) Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21 2 590–599.
IEEE A. Yokuş ve H. Durur, “Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory”, BAUN Fen. Bil. Enst. Dergisi, c. 21, sy. 2, ss. 590–599, 2019, doi: 10.25092/baunfbed.631193.
ISNAD Yokuş, Asıf - Durur, Hülya. “Complex Hyperbolic Traveling Wave Solutions of Kuramoto-Sivashinsky Equation Using (1/G’) Expansion Method for Nonlinear Dynamic Theory”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21/2 (Haziran 2019), 590-599. https://doi.org/10.25092/baunfbed.631193.
JAMA Yokuş A, Durur H. Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory. BAUN Fen. Bil. Enst. Dergisi. 2019;21:590–599.
MLA Yokuş, Asıf ve Hülya Durur. “Complex Hyperbolic Traveling Wave Solutions of Kuramoto-Sivashinsky Equation Using (1/G’) Expansion Method for Nonlinear Dynamic Theory”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 21, sy. 2, 2019, ss. 590-9, doi:10.25092/baunfbed.631193.
Vancouver Yokuş A, Durur H. Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G’) expansion method for nonlinear dynamic theory. BAUN Fen. Bil. Enst. Dergisi. 2019;21(2):590-9.

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