BAZI KESİRLİ LİNEER OLMAYAN KISMİ DİFERENSİYEL DENKLEMLER İÇİN KUDRYASHOV METODU İLE TAM ÇÖZÜM
Yıl 2013,
Cilt: 8 Sayı: 1, 24 - 63, 01.02.2013
Hasan Bulut
Öz
Bu çalışmada kesirli RLW Burgers ve CRWP denklemlerinin tam çözümleri için Kudryashov metodunu uyguladık. Bu metodu kullanarak, kesirli RLW Burgers ve kesirli CRWP denklemleri için bazı çözümler elde edilmiştir.
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 nonlinearities, J. Math. Anal. Appl. 328, 1435–1450.
- Raslan, K.R., (2008).The first integral method for solving some important nonlinear partial differential equations, Nonlinear. Dynam. 53, 281.
- Lu, B., Zhang, H.Q., and Xie, F.D., (2010). Travelling wave solutions of nonlinear partial equations by using the first integral method, Appl. Math. Comput. 216 1329–1336.
- Taghizadeh, N., Mirzazadeh, M., and Farahrooz, F., (2011). Exact solutions of the nonlinear Schrödinger equation by the first integral method, J. Math. Anal. Appl. 374, 549–553.
- Ganji, Z.Z., Ganji, D.D., and Rostamiyan, Y., (2009). Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled RLW Burgers equation by an analytical technique, Appl. Math. Model. 33, 3107–3113.
- Shateri, M. and Ganji, D.D., (2010). Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled RLW Burgers equation by a new analytical technique,Int. J. Differ. Equ. 2010, Article 954674.
- Song, L.N., Wang, Q., and Zhang, H.Q., (2009). Rational approximation solution of the fractional Sharma–Tasso–Olever equation, J. Comput. Appl. Math. 224,210–218.
- Kudryashov, N.A., (1988). Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech. 52, 361–365.
- Kudryashov, N.A., (1990). Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A 147, 287–291.
- Kudryashov, N.A., (1991). On types of nonlinear nonintegrable equations with exact solutions, Phys. Lett. A 155, 269–275.
- Kudryashov, N.A., (1993). Singular manifold equations and exact solutions for some nonlinear partial differential equations, Phys. Lett. A 182, 356–362.
- Kudryashov, N.A., (2011). On one of methods for finding exact solutions of nonlinear differential equations, arXiv:1108.3288v1[nlin.SI].
EXACT SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS VIA KUDRYASHOV METHOD
Yıl 2013,
Cilt: 8 Sayı: 1, 24 - 63, 01.02.2013
Hasan Bulut
Öz
In this study the Kudryashov method have been implemented for the exact solutions of the fractional RLW Burgers equation and the fractional Clannish Random Walker's Parabolic (CRWP) Equation. Some new solutions of the fractional RLW Burgers equation and the fractional CRWP equation have been obtained by using this method.
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 nonlinearities, J. Math. Anal. Appl. 328, 1435–1450.
- Raslan, K.R., (2008).The first integral method for solving some important nonlinear partial differential equations, Nonlinear. Dynam. 53, 281.
- Lu, B., Zhang, H.Q., and Xie, F.D., (2010). Travelling wave solutions of nonlinear partial equations by using the first integral method, Appl. Math. Comput. 216 1329–1336.
- Taghizadeh, N., Mirzazadeh, M., and Farahrooz, F., (2011). Exact solutions of the nonlinear Schrödinger equation by the first integral method, J. Math. Anal. Appl. 374, 549–553.
- Ganji, Z.Z., Ganji, D.D., and Rostamiyan, Y., (2009). Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled RLW Burgers equation by an analytical technique, Appl. Math. Model. 33, 3107–3113.
- Shateri, M. and Ganji, D.D., (2010). Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled RLW Burgers equation by a new analytical technique,Int. J. Differ. Equ. 2010, Article 954674.
- Song, L.N., Wang, Q., and Zhang, H.Q., (2009). Rational approximation solution of the fractional Sharma–Tasso–Olever equation, J. Comput. Appl. Math. 224,210–218.
- Kudryashov, N.A., (1988). Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech. 52, 361–365.
- Kudryashov, N.A., (1990). Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A 147, 287–291.
- Kudryashov, N.A., (1991). On types of nonlinear nonintegrable equations with exact solutions, Phys. Lett. A 155, 269–275.
- Kudryashov, N.A., (1993). Singular manifold equations and exact solutions for some nonlinear partial differential equations, Phys. Lett. A 182, 356–362.
- Kudryashov, N.A., (2011). On one of methods for finding exact solutions of nonlinear differential equations, arXiv:1108.3288v1[nlin.SI].