On complicated expansions of solutions to ODES
A. D. Bruno
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia
Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.
ordinary differential equation, asymptotic expansion, solution with logarithms, Painlevé equation.
Computational Mathematics and Mathematical Physics, 2018, 58:3, 328–347
A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018), 346–364; Comput. Math. Math. Phys., 58:3 (2018), 328–347
Citation in format AMSBIB
\paper On complicated expansions of solutions to ODES
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
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