Lineer Olmayan Uzay – Zaman Kesirli Diferensiyel Denklem Sistemlerinin Soliter Dalga ve Diğer Çözümleri

Yıl 2018, Cilt: 8 Sayı: 2, 515 - 522, 01.06.2018

Murat Koparan Özkan Güner

Öz

Bu çalışmada, G Gl b l -açılım metodu kullanarak kesir mertebeli Boussinesq denklem sistemleri ve kesirli iki boyutlu Burgers’ denklemlerinin bazı hareketli dalga çözümleri elde edilmiştir. Bu tam çözümler hiperbolik, trigonometrik ve rasyonel fonksiyon çözümlerini içermektedir. Kesirli karmaşık dönüşüm, genellikle, modifiye Riemann-Liouville türevi içeren kesirli kısmi diferansiyel denklemi adi diferensiyel denkleme dönüştürmek için kullanılır. Düşünülen dönüşüm ve metodun, diğer lineer olmayan kesir mertebeli denklemlerin ve sistemlerin çözümünde güvenilir, verimli ve etkili bir yol olduğu gösterilmiştir.

Anahtar Kelimeler

b l -açılım metodu, Tam çözüm, Kesir mertebeli variant Boussinesq denklem sistemi, İki boyutlu kesir mertebeli Burger denklem sistemi

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

Öz

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation FDEs with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

Anahtar Kelimeler

The G&#039, /G -expansion method, exact solution, fractional order system of variant Boussinesq equations, fractional system of two-dimensional Burgers&#039, equations

Kaynakça

• 1. Miller, KS., Ross, B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
• 2. Podlubny, I. 1999. Fractional Differential Equations, Academic Press, California.
• 3. Kilbas, AA., Srivastava, HM., Trujillo, JJ. 2006. Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
• 4. Wang, Q. 2006. Numerical solutions for fractional KdVBurgers equation by Adomian decomposition method. Appl. Math. Comput., 182: 1048-1055.
• 5. Inc, M. 2008. The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl., 345: 476-484.
• 6. Golmankhaneh, AK., Golmankhaneh, Ali K., Dumitru, B. 2011. On nonlinear fractional Klein--Gordon equation. Signal Process., 91: 446-451.
• 7. Xu, H., Cang, J. 2008. Analysis of a time fractional wave-like equation with the homotopy analysis method. Phys. Lett. A., 372: 1250-1255.
• 8. Gepreel, KA., Omran, S. 2012. Exact solutions for nonlinear partial fractional differential equations. Chin. Phys. B., 21: 110204.
• 9. Bekir, A., Guner, O. 2013. Exact solutions of nonlinear fractional differential equations by ^ h G Gl/ -expansion method. Chin. Phys. B., 22: 110202.
• 10. Lu, B. 2012. The first integral method for some time fractional differential equations. J. Math. Anal. Appl., 395: 684-693.
• 11. Eslami, M., Fathi VB., Mirzazadeh, M., Biswas, A. 2014. Application of first integral method to fractional partial differential equations. Indian J. Phys., 88: 177-184.
• 12. Zhang, S., Zhang, HQ. 2011. Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A., 375: 1069-1073.
• 13. Guo, S., Mei, L., Li, Y., Sun, Y. 2012. The improved fractional sub-equation method and its applications to the space--time fractional differential equations in fluid mechanics. Phys. Lett. A., 376: 407-411.
• 14. Bulut, H., Baskonus, HM., Pandir, Y. 2013. The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation. Abstr. Appl. Anal., 2013: 636802.
• 15. Pandir, Y., Gurefe, Y., Misirli, E. 2013. New Exact Solutions of the Time-Fractional Nonlinear Dispersive KdV Equation. Int. J. Model. Optim., 3: 4.
• 16. Bekir, A., Guner, O., Cevikel, AC. 2013. Fractional Complex Transform and exp- Function Methods for Fractional Differential Equations. Abstr. Appl. Anal., 2013: 426462.
• 17. Guner, O., Cevikel, AC. 2014. A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations. Sci., World J. 2014: 489-495.
• 18. Guner, O., Eser, D. 2014. Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods. Adv. Math. Phys., 2014: 456804.
• 19. Wang, M., Li, X., Zhang, J. 2008. The ^ h G Gl/ -expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A., 372: 417- 423.
• 20. Zhang, S., Tong, JL., Wang, W. 2008. A generalized ^ h G Gl/ -expansion method for the mKdV equation with variable coefficients. Phys. Lett. A., 372: 2254-2257.
• 21. Bekir, A. 2008. Application of the ^ h G Gl/ -expansion method for nonlinear evolution equations. Phys. Lett. A., 372: 3400-3406.
• 22. Zayed, EME., Gepreel, KA. 2009. The ^ h G Gl/ -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys., 50: 013502.
• 23. He, JH., Elegan, SK., Li, ZB. 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A., 376: 257-259.
• 24. Aksoy, E., Kaplan, M., Bekir, A. 2016. Exponential rational functionmethod for space-time fractional differential equations. Wave. Random. Complex., 26: 142-151.
• 25. Mohamed, SM., Gepreel, KA. 2013. Numerical solutions for the time fractional variant Bussinesq equation by homotopy analysis method. Sci. Res. Essays., 8: 2163-2170.
• 26. Yan, L. 2015. New travelling wave solutions for coupled fractional variant Boussinesq equation and approximate long water wave equation. Int. J. Numer. Methods Heat Fluid Flow.,25: 33-40.
• 27. Wang, ML. 1995. Solitary wave solutions for variant Boussinesq equations. Phys. Lett.A., 199: 169-172.
• 28. Yan, Z.Y., Zhang, HQ. 1999. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Phys. Lett. A., 252: 291-296.
• 29. Yomba, E. 2005. The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations. Phys. Lett. A., 336: 463-476.
• 30. Lü, D. 2005. Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos, Solitons & Fractals., 24: 1373- 1385.
• 31. Soliman, AA., Abdo, HA. 2009. New Exact Solutions of Nonlinear Variants of the RLW, the PHI-four and Boussinesq Equations Based on Modified Extended Direct Algebraic Method. Int. J. Nonlinear Sci., 7: 274-282.
• 32. Wu, XH., He, JH. 2008. Exp-function method and its application to nonlinear equations. Chaos, Solitons & Fractals., 38: 903-910.
• 33. Zhao, YM., Yang, YJ., Li, W. 2011. Application of the Improved ^ h G Gl/ -expansion method for the Variant Boussinesq Equations. Appl.Math. Sci., 5: 2855-2861.
• 34. Elhanbaly, A., Abdou, MA. 2013. On the solution of fractional space-time nonlinear differential equations. Int. J. Appl. Math.Comput., 5: 47-58.
• 35. Burger, JM. 1948. A mathematical model illustrating the theory of turbulence. Adv. Appl.Mech., 1: 171-199.
• 36. Cole, JD. 1951. On a quasilinear parabolic equations occurring in aerodynamics. Quart.Appl. Math., 9: 225-236.
• 37. Fletcher, CAJ. 1983. Generating exact solutions of the twodimensional Burgers equation. Int. J. Numer. Meth. Fluids., 3: 213-216.
• 38. Fletcher, CAJ. 1983. A comparison of finite element and finite difference solution of the one- and two-dimensional Burgers equations. J. Comput. Phys., 51: 159-188.
• 39. Bahadır, AR. 2003. A fully implicit finite-difference scheme for two-dimensional Burgers’ equations. Appl. Math. Comput., 137: 131-137.
• 40. Wubs, FW. 1992. Goede, E.D. de. An explicit-implicit method for a class of time-dependent partial differential equations. Appl. Numer. Math., 9: 157-181.
• 41. Goyon, O. 1996. Multilevel schemes for solving unsteady equations. Int. J. Numer. Meth.Fluids., 22: 937-959.
• 42. El-Sayed, SM., Kaya, D. 2004. On the numerical solution of the system of two-dimensional Burgers equations by the decomposition method. Appl. Math. Comput., 158: 101-109.
• 43. Biazar, J., Aminikhah, H. 2009. Exact and numerical solutions for non-linear Burger’s equation by VIM. Math. Comp. Model., 49: 1394-1400.
• 44. Zhu, H., Shu, H., Ding, M. 2010. Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method. Comput. Math. Appl., 60: 840-848.