On equations containing derivative of the delta-function

  • Alena V. Shkadzinskaya Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Abstract

The expression u′′ + aδ′u, which is consisted derivative of delta-function as a coefficient, is a formal expression and doesnʼt define operator in L2(R), because a product δ′u is not defined. So according to these reasons the study investigated the family of operators, which are approximated by the following formal expression (L(ε, a, φ)u)(x) = u′′(x) + a(ε) ⋅ (∫Ψε(y)u(y)dy ⋅ φε(x) + ∫φε(y)u(y)dy ⋅ Ψε(x), where φ ∈ D(R); φ(x) ∈ R; ∫φ(x)dx = 1; φε(x) = 1/εφ(x/ε); coefficient a(ε) could be real-valued and not null. The main results of the study were finding the limit in the family in sense of resolvent convergence. As the result, the five different kinds of limits of resolvents in this family had been received which are depended on a behavior of coefficient a(ε) and function φ properties. Therefore the formal expression u′′ + aδ′u could not put in accordance to the operator in L2(R) uniquely. This is the fundamental difference with the case u′′ + aδu expression for which the limit of resolvents doesn’t depend on choosing approximated family.

Author Biography

Alena V. Shkadzinskaya, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

postgraduate student at the department of functional analysis, faculty of mechanics and mathematics

References

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Published
2018-01-25
Keywords: resolvent, resolvent convergence, approximation, fundamental solution
How to Cite
Shkadzinskaya, A. V. (2018). On equations containing derivative of the delta-function. Journal of the Belarusian State University. Mathematics and Informatics, 3, 19-26. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/754
Issue
No 3 (2017): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

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