Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients
Abstract
:1. Introduction
2. Lie Symmetry Analysis of the Time Fractional Partial Differential Equation
3. Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients
- (1)
- If is arbitrary, we obtain the explicit form of infinitesimals
- (2)
- If is a nonzero constant, the infinitesimals areA two-dimensional Lie algebra is spanned by
- (3)
- If (), the expressions of infinitesimals are given asWe obtain a two-dimensional Lie algebra, spanned byWhen , the infinitesimals are given asWhen , the infinitesimals are given as
- (4)
- If (p is a nonzero real number), we obtain the following expressions of infinitesimalsWe have a two-dimensional Lie algebra, which is spanned by
4. Lie Symmetry Analysis of Time Fractional Generalized KdV Equations with Initial and Boundary Value
4.1. Lie Symmetry Analysis of Time Fractional Partial Differential Equations with Initial and Boundary Value
- (i)
- under symmetry V, the manifold S is invariant;
- (ii)
- under symmetry V restricted to manifold S, the initial (boundary) condition is invariant.
4.2. Reduction for Time Fractional Generalized KdV Equations with Initial and Boundary Value
4.3. Explicit Power Series Solutions of Time Fractional Generalized KdV Equations with Initial and Boundary Value
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Chen, C.; Jiang, Y.-L.; Wang, X.-T. Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients. Symmetry 2019, 11, 1281. https://doi.org/10.3390/sym11101281
Chen C, Jiang Y-L, Wang X-T. Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients. Symmetry. 2019; 11(10):1281. https://doi.org/10.3390/sym11101281
Chicago/Turabian StyleChen, Cheng, Yao-Lin Jiang, and Xiao-Tian Wang. 2019. "Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients" Symmetry 11, no. 10: 1281. https://doi.org/10.3390/sym11101281