Asymptotic expansions of solutions of singularly perturbed equations

V. I. Uskov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2022), pp. 18-32

Résumé

We consider a first-order equation in a Banach space with a small parameter at the derivative and a second-order perturbation of smallness on the right-hand side. A solution to the Cauchy problem is constructed in the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method. The operator A on the right-hand side is degenerate: we consider the case of possessing the property of having a number 0 by a normal eigenvalue and a two-dimensional kernel; core elements have no attached. Formulas for calculating the components of the regular and boundary layer parts of the expansion are determined. A condition for the regularity of degeneration is obtained. The expansion is shown to be asymptotic. An illustrative example is given.

Détail

Keywords: first-order equation in a Banach space, small parameter at the highest derivative, perturbation square on the right-hand side, closed operator, 0-normal eigenvalue, asymptotics, Vasil'eva-Vishik-Lyusternik method.

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