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Projector approach to the Butuzov-Nefedov algorithm for asymptotic solution of a class of singularly perturbed problems in a critical case. (English. Russian original) Zbl 1465.34073

Comput. Math. Math. Phys. 60, No. 12, 2007-2018 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 12, 2073-2084 (2020).
In this paper, an asymptotic solution for a \(m\)-dimensional weakly nonlinear singularly perturbed initial value problem \[ \varepsilon^2\frac{dx}{dt}=A(t)x+\varepsilon f(x,t,\varepsilon),\ x=x(t,\varepsilon)\in X,\ t\in[0,T], \] \[ x(0,\varepsilon)=x^0 \] is constructed using the orthogonal projectors of the space \(X\) onto \(\mathrm{ker}\,A(t)\) and \(\mathrm{ker}\,A^T(t)\). It allows to represent expressions of the boundary function method for finding asymptotic terms of any order in an explicit form.
Here, \(\varepsilon\) denotes a small positive parameter, \(A(t)\) is an \(m\times m\) singular matrix with \(k\) zero eigenvalues (\(k < m\)) with equal algebraic and geometric multiplicity for each \(t\in[0,T];\) \(f(x,t,\varepsilon)\) is an \(m\)-dimensional vector function, and \(A(t)\) and \(f(x,t,\varepsilon)\) are assumed to be sufficiently smooth with respect to their arguments.
Theory is illustrated by an example of the construction of a first-order asymptotic approximation with numerical simulation.

MSC:

34E15 Singular perturbations for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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References:

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