λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations
Abstract
:1. Introduction
2. λ-Symmetries of Ordinary Differential Equations
2.1. The Basic Concept of λ-symmetries
2.2. Applications of λ-Symmetries
2.2.1. λ-Symmetries Reductions and Integrating Factors without Using Lie Symmetries
2.2.2. λ-Symmetry Reductions and Integrating Factors Using Lie Symmetry
3. μ-Symmetries of Partial Differential Equations
3.1. The Basic Concept of μ-Symmetries
3.2. Applications of μ-Symmetries
3.2.1. An Example of (2 + 1)-Dimensional Equation
3.2.2. An Example of (1+1)-Dimensional Equation
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Bai, Y.-S.; Pei, J.-T.; Ma, W.-X. λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations. Mathematics 2020, 8, 1138. https://doi.org/10.3390/math8071138
Bai Y-S, Pei J-T, Ma W-X. λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations. Mathematics. 2020; 8(7):1138. https://doi.org/10.3390/math8071138
Chicago/Turabian StyleBai, Yu-Shan, Jian-Ting Pei, and Wen-Xiu Ma. 2020. "λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations" Mathematics 8, no. 7: 1138. https://doi.org/10.3390/math8071138
APA StyleBai, Y.-S., Pei, J.-T., & Ma, W.-X. (2020). λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations. Mathematics, 8(7), 1138. https://doi.org/10.3390/math8071138