Symmetry Operators and Solutions to Differential Equations in Algebra
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The symmetry operator method for finding an analytical form for the solutions to a given PDE system is presented. The main tools for accomplishing this aim include computation and grading of the Lie algebra infinitesimals of symmetry groups admitted by a given PDE system, establishing fusion rules, and then checking the commutator relations after performing the grading. We discuss several examples of the differential systems of mathematical physics based on the splitting of heat and the Euler–Tricomi, generalized Cauchy–Riemann and Dirac equations in non-associative algebras.
KeywordsSymmetry algebra Algebra graduating Polynomial solutions
Mathematics Subject Classification58J70 76M60
I thank Prof. Y. Krasnov for valuable suggestions, useful advice, and for encouraging me to write this paper, to Prof. J. Schiff for helpful comments and for suggesting Theorem 5.3, and Mrs. M. Beller who edited this paper.
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