Some Exact Solutions of Caudrey-Dodd-Gibbon (CDG) Equation and Dodd-Bullough-Mikhailov Equation

Yıl 2019, Cilt: 8 Sayı: 1, 60 - 65, 12.03.2019

Ünal İç

https://doi.org/10.17798/bitlisfen.480896

Öz

Nonlinear partial differential equations have an
important place in applied mathematics and physics. Many analytical methods
have been found in literature. Using these methods, partial differential
equations are transformed into ordinary differential equations. These nonlinear
partial differential equations are solved with the help of ordinary
differential equations. In this paper, we implemented an improved tanh function
Method for some exact solutions of Caudrey-Dodd-Gibbon (CDG) Equation and
Dodd-Bullough-Mikhailov Equation.

Anahtar Kelimeler

Caudrey-Dodd-Gibbon (CDG) Equation, Dodd-Bullough-Mikhailov Equation, improved tanh function method, exact solutions

Caudrey-Dodd-Gibbon (CDG) Denklemi ve Dodd-Bullough-Mikhailov Denkleminin Bazı Kesin Çözümleri

Öz

Uygulamalı matematik ve fizikte doğrusal olmayan kısmi
diferansiyel denklemler önemli bir yere sahiptir. Literatürde birçok analitik
yöntem bulunmuştur. Bu yöntemleri kullanarak, kısmi diferansiyel denklemler,
adi diferansiyel denklemlere dönüştürülür. Bu doğrusal olmayan kısmi
diferansiyel denklemler, adi diferansiyel denklemlerin yardımıyla çözülmüştür. Bu
çalışmada, Caudrey-Dodd-Gibbon (CDG) Denklemi ve Dodd-Bullough-Mikhailov
Denkleminin kesin çözümleri için geliştirilmiş tanh fonksiyon metodu
sunulmuştur.

Anahtar Kelimeler

Caudrey-Dodd-Gibbon (CDG) Denklemi, Dodd-Bullough-Mikhailov Denklemi, geliştirilmiş tanh fonksiyon metodu, tam çözümler

Kaynakça

• Debtnath L. 1997. Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, MA.
• Wazwaz A. M. 2002. Partial Differential Equations: Methods and Applications, Balkema, Rotterdam.
• Shang Y. 2007. Backlund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation, Appl.Math.Comput., 187: 1286-1297.
• Bock T. L., Kruskal M. D. 1979. A two-parameter Miura transformation of the Benjamin-Ono equation, Phys. Lett. A 74: 173-176.
• Matveev V. B., Salle M. A. 1991. Darboux Transformations and Solitons, Springer, Berlin.
• Abourabia A. M., El Horbaty M. M. 2006. On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation, Chaos Solitons Fractals, 29: 354-364.
• Malfliet W. 1992. Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60: 650-654. Chuntao Y. 1996. A simple transformation for nonlinear waves, Phys. Lett. A 224: 77-84.
• Cariello F., Tabor M. 1989. Painleve expansions for nonintegrable evolution equations, Physica D 39: 77-94.
• Fan E. 2000. Two new application of the homogeneous balance method, Phys. Lett. A 265: 353-357.
• Clarkson P. A. 1989. New similarity solutions for the modified boussinesq equation, J. Phys. A: Math. Gen. 22: 2355-2367.
• Malfliet W. 1996. The Tanh method Wavw Equation, Physica Scripta, 60: 563-568.
• Fan E. 2000. Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277: 212-218.
• Elwakil S. A., El-labany S. K., Zahran M. A., Sabry R. 2002. Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A 299: 179-188.
• Chen H., Zhang H. 2004. New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos Soliton Fract 19: 71-76.
• Fu Z., Liu, S., Zhao Q. 2001. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A 290: 72-76.
• Shen S., Pan Z. 2003. A note on the Jacobi elliptic function expansion method, Phys. Let. A 308: 143-148.
• Chen H. T., Hong-Qing, Z. 2004. New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation, Chaos Soliton Fract 20: 765-769.
• Chen Y., Wang Q., Li B. 2004. Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations, Z. Naturforsch. A 59: 529-536.
• Chen Y., Yan Z. 2006. The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Soliton Fract 29: 948-964.
• Wang M., Li X., Zhang J. 2008. The -expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A 372: 417-423.Guo S., Zhou Y. 2010. The extended -expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl.Math.Comput. 215: 3214-3221.
• Lü H. L., Liu X. Q., Niu, L. 2010. A generalized -expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput. 215: 3811-3816.
• Li L., Li E., Wang M. 2010. The -expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math-A J. Chin. U 25: 454-462.
• Manafian J. 2016. Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan – expansion Method, Optik 127: 4222-4245.
• Don E. 2001. Schaum's Outline of Theory and Problems of Mathematica, McGraw-Hill.
• Koyunbakan H., Bulut N. 2005. Existence of the transformation operator by the decomposition method. Appl. Anal. 84: 713-719.