Projector approach to the Butuzov-Nefedov algorithm for asymptotic solution of a class of singularly perturbed problems in a critical case
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.
scientific article; zbMATH DE number 7291191
scientific article