YENİ BİR YARDIMCI DENKLEM KULLANARAK KESİRLİ RLW BURGES DENKLEMI IÇIN TAM ÇÖZÜM BULMA

Yıl 2013, Cilt: 8 Sayı: 1, 1 - 10, 01.02.2013

Hasan Bulut

Öz

Bu çalışmada kesirli lineer olmayan kısmi diferensiyel denklemlerin tam çözümlerinin oluşturulması için Riccati denkleminden farklı bir yardımcı denklem ile yeni bir metod kullanılmıştır. Bu metodun ana fikri, Riccati denkleminden farklı olarak yeni çözümlere sahip yeni bir yardımcı denklemden en iyi şekilde yararlanmaktır. Sonuç olarak, kesirli RLW Burgers denklemi için birçok yeni çözüm elde edilmiştir.

Anahtar Kelimeler

Kesirli Lineer Olmayan Kısmi Diferensiyel Denklemler, Denklemler, Kesirli RLW Burgers Denklemi, Modified Riemann–Liouville Türevi, Lineer Olmayan Kısmi Diferensiyel Denklemler, Riemann–Liouville Türevi,

FINDING EXACT SOLUTION BY USING A NEW AUXILIARY EQUATION FOR FRACTIONAL RLW BURGES EQUATION

Öz

In this study a new method with a different auxiliary equation from the Riccati equation is used for constructing exact solutions of fractional nonlinear partial differential equations. The main idea of this method is to take full advantage of a different auxiliary equation from the Riccati equation which has more new solutions. Finally, more new solutions have been obtained for the fractional RLW Burgers equation.

Anahtar Kelimeler

Fractional Nonlinear Partial Differential Equations, Fractional RLW Burgers Equation, Modified Riemann–Liouville Derivative, Nonlinear Partial Differential Equations, Equations, Riemann–Liouville Derivative,

Kaynakça

• Miller, K.S. and Ross, B., (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
• Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., (2006). Theory and Applications of Fractional Differential Equations, Elsevier, San Diego.
• Podlubny, I., (1999). Fractional Differential Equations, Academic Press, San Diego.
• Jumarie, G., (2006). Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51(9-10), 1367–1376.
• Jumarie, G., (2007). Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function, J. Appl. Math. Comput. 23(1-2), 215–228.
• Jumarie, G., (2009). Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22(3), 378– 385.
• Lu, B., (2012). The first integral method for some time fractional differential equations, J. Math. Anal. Appl. 395, 684–693.
• Song, L.N. and Zhang, H.Q., (2009). Solving the fractional BBM- Burgers equation using the homotopy analysis method, Chaos Solitons Fractals 40,1616–1622.
• Ganji, Z., Ganji, D., Ganji, A.D., and Rostamian, M., (2010). Analytical solution of time-fractional Navier–Stokes equation in polar coordinate by homotopy perturbation method, Numer. Methods Partial Differential Equations 26,117–124.
• Gepreel, K.A., (2011). The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations, Appl. Math. Lett. 24, 1428–1434.
• Gupta, P.K. and Singh, M., (2011). Homotopy perturbation method for fractional Fornberg–Whitham equation, Comput. Math. Appl. 61, 50–254.
• Jumarie, G., (2006). Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order, Appl. Math. Lett. 19, 873–880.
• Zhang, S. and Zhang, H.Q., (2011). Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A 375, 1069–1073.
• Jumarie, G., (2006). Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51, 1367–1376.
• Feng, Z.S. and Roger, K., (2007). Traveling waves to a Burgers– Korteweg–de Vries-type equation with higher-order 10