时空分数阶的广义b-方程组的精确解
Exact Solutions for the Generalized b-Family Equations with Fractional Time and Spatial Derivatives
DOI: 10.12677/AAM.2020.99184, PDF, 下载: 779  浏览: 1,364  科研立项经费支持
作者: 易亚婷, 潘超红*：南华大学数理学院，湖南 衡阳
关键词: 精确解；广义b-方程组；拉普拉斯变换；分数阶微分；Exact Solutions； Generalized b-Family Equations； Laplace Transforms； Sequential Fractional Derivatives
摘要: 在这篇文章中，我们考虑在Riemann−Liouville意义下的时空分数阶广义b-方程组的精确解。我们将通过拉普拉斯变换给出这个方程组带初值条件的解析解。此外，本文将通过一个辅助方程证明这个方程组具有相同的解析解。
Abstract: In this paper, the generalized b-family equations with fractional time and spatial derivatives are considered. The fractional derivative is described in the Riemann−Liouville sense. We present the analytical solutions of the fractional equations with initial con- ditions by the Laplace transform of sequential fractional derivatives. Moreover, the generalized b-family equations with fractional time and spatial derivatives possess com- mon analytical solutions which are solved by considering a simple equation.
文章引用：易亚婷, 潘超红. 时空分数阶的广义b-方程组的精确解[J]. 应用数学进展, 2020, 9(9): 1565-1571. https://doi.org/10.12677/AAM.2020.99184

参考文献

[1] Degasperis, A. and Procesi, M. (1999) Asymptotic Integrability Symmetry and Perturbation Theory. World Scientific, Singapore.
[2] Camassa, R. and Holm, D.D. (1993) An Integrable Shallow Water Equation with Peaked Soli- tons. Physical Review Letters, 71, 1661-1664.
https://doi.org/10.1103/PhysRevLett.71.1661
[3] Zhou, J. and Tian, L.X. (2009) Solitons, Peakons, and Periodic Cuspons of a General- ized Degasperis-Procesi Equation. Mathematical Problems in Engineering, 2009, Article ID: 249361.
https://doi.org/10.1155/2009/249361
[4] Guo, F. (2009) Global Weak Solutions and Wave Breaking Phenomena to the Periodic Degasperis-Procesi Equation with Strong Dispersion. Nonlinear Analysis: Theory, Methods and Applications, 71, 5280-5295.
https://doi.org/10.1016/j.na.2009.04.012
[5] Shen, C.Y., Gao, A. and Tian, L.X. (2010) Optimal Control of the Viscous Generalized Camassa-Holm Equation. Nonlinear Analysis: Real World Applications, 11, 1835-1846.
https://doi.org/10.1016/j.nonrwa.2009.04.003
[6] West, B.J., Bolognab, M. and Grigolini, P. (2003) Physics of Fractal Operators. Springer, New York.
[7] Baleanu, D. and Trujillo, J.J. (2008) On Exact Solutions of a Class of Fractional Euler-Lagrange Equations. Nonlinear Dynamics, 52, 331-335.
https://doi.org/10.1007/s11071-007-9281-7
[8] Srivastava, H.M. and Lashin, A.Y. (2005) Some Applications of the Briot-Bouquet Differential Subordination. Journal of Inequalities in Pure and Applied Mathematics, 6, Article 41.
[9] Odibat, Z.M. (2009) Exact Solitary Solutions for Variants of the KdV Equations with Frac- tional Time Derivatives. Chaos, Solitons and Fractals, 40, 1264-1270.
https://doi.org/10.1016/j.chaos.2007.08.080
[10] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.
[11] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon.
[12] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
[13] Caputo, M. (1967) Linear Models of Dissipation Whose Q Is Almost Frequency Independent, Part II. Geophysical Journal International, 13, 529-539.
https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
[14] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
[15] Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.
[16] Pan, C.H., Liu, Z.R. and Zhang, B.G. (2010) A Simple Method and Its Applications to Non- linear Partial Differential Equations. Applied Mathematics and Computation, 217, 4197-4204.
https://doi.org/10.1016/j.amc.2010.09.070

为你推荐