Lyapunov-Schmidt reduction in the study of bifurcation solutions of nonlinear fractional differential equation. (English) Zbl 1415.35031
Summary: In this article the bifurcation of periodic travelling wave solutions of nonlinear fractional differential equation is studied by using Lyapunov-Schmidt reduction and He’s fractional derivative. The fractional complex transform is used to convert the fractional differential equation into partial differential equation. The reduced equation corresponding to the main problem is found as a system of two nonlinear algebraic equations. The existence of the linear approximation solutions of the nonlinear fractional differential equation is discussed.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35R11 | Fractional partial differential equations |
| 35C07 | Traveling wave solutions |
References:
| [1] | B. V. Loginov, Theory of Branching Nonlinear Equations in The conditions of Invariance Group-Tashkent: Fan, 1985. · Zbl 0593.58028 |
| [2] | B. Lu., The …rst integral method for some time fractional di¤erential equation, J. Math. Anal. Appl., 395(2012), 684-693. · Zbl 1246.35202 |
| [3] | F. J. Liu, Z.B. Li, S. Zhang and H. Y. Liu, He’s fractional derivative for heat conduction in a fFractal medium arising in silkworm cocoon hierarchy, Thermal Science, 19(2015), 4, pp.1155-1159. |
| [4] | M. A. Abdul Hussain, Bifurcation solutions of elastic beams equation with small perturbation, Int. J. Math. Anal., 3(2009), 879-888. · Zbl 1195.74093 |
| [5] | M. A. Abdul Hussain, Two-Mode bifurcation in solution of a perturbed nonlinear fourth order di¤erential equation, Arch. Math., 48(2012), 27-37. · Zbl 1274.34030 |
| [6] | M. A. Abdul Hussain, Nonlinear Ritz approximation for Fredholm functionals, Electron. J. Di¤erential Equations, 2015, No. 294, 11 pp. · Zbl 1330.65121 |
| [7] | M. M. Vainberg and V. A. Trenogin, Theory of Branching Solutions of Nonlinear Equations, M.-Science , 1969. · Zbl 0186.20805 |
| [8] | M. S. Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, Inc., 1977. · Zbl 0368.47001 |
| [9] | S. Zhang and H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A 375 (2011), no. 7, 1069-1073. · Zbl 1242.35217 |
| [10] | S. Guo, L.Q. Mei, Y. Li and Y. F. Sun, The improved fractional sub-equation method and its applications to the space-time fractional di¤erential equations in ‡uid mechanics, Phys. Lett. A 376 (2012), no. 4, 407-411. · Zbl 1255.37022 |
| [11] | Yu. I. Sapronov, Regular Perturbation of Fredholm Maps and Theorem about odd Field, Works Dept. of Math., Voronezh Univ., 1973. V. 10, 82-88. |
| [12] | Yu. I. Sapronov, Finite-dimensional reductions in smooth extremal problems, Usp. Mat. Nauk, 51(1996), 101-132. · Zbl 0888.58010 |
| [13] | Yu. I. Sapronov and V. R. Zachepa, Local Analysis of Fredholm Equation, Voronezh Univ., 2002. · Zbl 0559.47046 |
| [14] | Z. B. Li and J.-H. He, Fractional complex transform for fractional di¤erential equations, Math. Comput. Appl., 15(2010), 970-973. · Zbl 1215.35164 |
| [15] | Z. B. Li and J. H. He, Converting fractional di¤erential equations into partial di¤erential equations, Ther. Sci., 16(2012), 331-334. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
