The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations
Abstract
The double (G^( ^' )/G,1/G)-expansion method is used to find exact travelling wave solutions to the fractional differantial equations in the sense of Jumarie’s modified Riemann- Liouville derivative. We exploit this method for the combined KdV- negative-order KdV equation (KdV-nKdV) and the Calogero-Bogoyavlinskii-Schiff equation (CBS) of fractional order. We see that these solutions are concise and easy to understand the physical phenomena of the nonlinear partial differential equations. These solutions can be shown in terms of trigonometric, hyperbolic and rational functions.
Keywords
The (G'/G,1/G)-expansion method, the Calogero–Bogoyavlinskii–Schiff equation the combined KdV- negative-order KdV equation
Thanks
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References
- Arafa AAM, Rida SZ, Mohamed H, 2011. Homotopy analysis method for solving biological population model. Communications in Theoretical Physics, 56(5), 797.
- Ekici M, Ayaz F, 2017. Solution of model equation of completely passive natural convection by improved differential transform method. Research on Engineering Structures and Materials, 3(1), 1-10.
- En-Gui F, Hong-Qing Z, 1998. The homogeneous balance method for solving nonlinear soliton equations. Acta Phys Sinica, 47(3), 363.
- Fan E, 2000. Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4), 212-218.
- Fan E, 2002. Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. Journal of Physics A: Mathematical and General, 35(32), 6853.
- 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. Physics Letters A, 376(4), 407-411.
- He JH, Wu XH, 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), 700-708.
- Inan IE, Duran S, Uğurlu Y, 2017. TAN (F(ξ2))-expansion method for traveling wave solutions of AKNS and Burgers-like equations. Optik, 138, 15-20.
- Jumarie G, 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9-10), 1367-1376.
- Li LX, Li EQ, Wang ML, 2010. The (G^( ^' )/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations. Applied Mathematics-A Journal of Chinese Universities, 25(4), 454-462.
- Lu B. 2012. Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Physics Letters A, 376(28-29), 2045-2048.
- Mohyud-Din ST, Saba F, 2017. Extended (G^( ^' )/G)-expansion method for Calogero–Bogoyavlinskii–Schiff equation of fractional order. Journal of Taibah University for Science, 11(6), 1099-1109.
- Odibat Z, Momani S. 2008. A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters, 21(2), 194-199.
- Taşcan F, Bekir A, 2009. Analytic solutions of the (2+ 1)-dimensional nonlinear evolution equations using the sine–cosine method. Applied Mathematics and Computation, 215(8), 3134-3139.
- Wang M, Zhou Y, Li Z, 1996. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters A, 216(1-5), 67-75.
- Wang, M, Zhou Y, 2003. The periodic wave solutions for the Klein–Gordon–Schrödinger equations. Physics Letters A, 318(1-2), 84-92.
- Wang M, Li X, 2005. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Physics Letters A, 343(1-3), 48-54.
- Wang M, Li X, Zhang J, 2007. Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms. Chaos, Solitons & Fractals, 31(3), 594-601.
- Wazwaz AM, 2018. A new integrable equation that combines the KdV equation with the negative-order KdV equation. Mathematical Methods in the Applied Sciences, 41(1), 80-87.
- Yang L, Liu J, Yang K, 2001. Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature. Physics Letters A, 278(5), 267-270.
- Zhang JL, Wang ML, Li XZ, 2006. The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation. Physics Letters A, 357(3), 188-195.
The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations
Abstract
The double (G^( ^' )/G,1/G)-expansion method is used to find exact travelling wave solutions to the fractional differantial equations in the sense of Jumarie’s modified Riemann- Liouville derivative. We exploit this method for the combined KdV- negative-order KdV equation (KdV-nKdV) and the Calogero-Bogoyavlinskii-Schiff equation (CBS) of fractional order. We see that these solutions are concise and easy to understand the physical phenomena of the nonlinear partial differential equations. These solutions can be shown in terms of trigonometric, hyperbolic and rational functions.
Keywords
The double expansion method the Calogero–Bogoyavlinskii–Schiff equation the combined KdV- negative-order KdV equation
References
- Arafa AAM, Rida SZ, Mohamed H, 2011. Homotopy analysis method for solving biological population model. Communications in Theoretical Physics, 56(5), 797.
- Ekici M, Ayaz F, 2017. Solution of model equation of completely passive natural convection by improved differential transform method. Research on Engineering Structures and Materials, 3(1), 1-10.
- En-Gui F, Hong-Qing Z, 1998. The homogeneous balance method for solving nonlinear soliton equations. Acta Phys Sinica, 47(3), 363.
- Fan E, 2000. Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4), 212-218.
- Fan E, 2002. Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. Journal of Physics A: Mathematical and General, 35(32), 6853.
- 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. Physics Letters A, 376(4), 407-411.
- He JH, Wu XH, 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), 700-708.
- Inan IE, Duran S, Uğurlu Y, 2017. TAN (F(ξ2))-expansion method for traveling wave solutions of AKNS and Burgers-like equations. Optik, 138, 15-20.
- Jumarie G, 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9-10), 1367-1376.
- Li LX, Li EQ, Wang ML, 2010. The (G^( ^' )/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations. Applied Mathematics-A Journal of Chinese Universities, 25(4), 454-462.
- Lu B. 2012. Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations. Physics Letters A, 376(28-29), 2045-2048.
- Mohyud-Din ST, Saba F, 2017. Extended (G^( ^' )/G)-expansion method for Calogero–Bogoyavlinskii–Schiff equation of fractional order. Journal of Taibah University for Science, 11(6), 1099-1109.
- Odibat Z, Momani S. 2008. A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters, 21(2), 194-199.
- Taşcan F, Bekir A, 2009. Analytic solutions of the (2+ 1)-dimensional nonlinear evolution equations using the sine–cosine method. Applied Mathematics and Computation, 215(8), 3134-3139.
- Wang M, Zhou Y, Li Z, 1996. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters A, 216(1-5), 67-75.
- Wang, M, Zhou Y, 2003. The periodic wave solutions for the Klein–Gordon–Schrödinger equations. Physics Letters A, 318(1-2), 84-92.
- Wang M, Li X, 2005. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Physics Letters A, 343(1-3), 48-54.
- Wang M, Li X, Zhang J, 2007. Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms. Chaos, Solitons & Fractals, 31(3), 594-601.
- Wazwaz AM, 2018. A new integrable equation that combines the KdV equation with the negative-order KdV equation. Mathematical Methods in the Applied Sciences, 41(1), 80-87.
- Yang L, Liu J, Yang K, 2001. Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature. Physics Letters A, 278(5), 267-270.
- Zhang JL, Wang ML, Li XZ, 2006. The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation. Physics Letters A, 357(3), 188-195.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Matematik / Mathematics |
| Authors | |
| Publication Date | March 1, 2021 |
| Submission Date | July 10, 2020 |
| Acceptance Date | November 14, 2020 |
| Published in Issue | Year 2021 Volume: 11 Issue: 1 |