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Comparison of approximate symmetry methods for differential equations. (English) Zbl 1066.76052
Exact and approximate symmetries of potential Burgers equation are calculated by the usage of two known approximate symmetry methods [V. A. Bajkov, R. K. Gazizov and N. Kh. Ibragimov, Math. USSR, Sb. 64, No. 2, 427–441 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 4(8), 435–450 (1988; Zbl 0683.35004); W. I. Fushchich and W. H. Shtelen, J. Phys A, Math. Gen. 22, 887–890 (1989)] and their modifications. Comparisons are made between those two methods and the exact symmetries of the equation. Approximate solutions corresponding to the approximate symmetries are derived for each case and compared. The non-Newtonian creeping flow equations of a second-grade fluid is considered, and approximate symmetries are computed by different methods together with approximate solutions for each method and their comparisons. Finally, a singular perturbation problem is investigated.
MSC:
76M60Symmetry analysis, Lie group and algebra methods (fluid mechanics)
76A05Non-Newtonian fluids
35Q35PDEs in connection with fluid mechanics
35A30Geometric theory for PDE, characteristics, transformations