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Some Connections between Classical and Nonclassical Symmetries of a Partial Differential Equation and Their Applications

1
Arts and Sciences College of Shanghai Maritime University, Shanghai 201306, China
2
Mathematics Department of Inner Mongolia University of Technology, Huhhot 010051, China
3
Mathematics Department of University of British Columbia, Vancouver, BC V6T 1Z2, Canada
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 524; https://doi.org/10.3390/math8040524
Received: 2 March 2020 / Revised: 20 March 2020 / Accepted: 24 March 2020 / Published: 3 April 2020
Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations. View Full-Text
Keywords: classical and nonclassical symmetries; connection; partial differential equation classical and nonclassical symmetries; connection; partial differential equation
MDPI and ACS Style

Temuer, C.; Tong, L.; Bluman, G. Some Connections between Classical and Nonclassical Symmetries of a Partial Differential Equation and Their Applications. Mathematics 2020, 8, 524.

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