On a Local Property of One-dimensional Linear Differential Equation of the Second Order
Author(s):
Sofia Stanislavovna Rasova
Institute of Problems in Mechanical Engineering (IPME) RAS
PhD in physics and mathematics,
senior scientific collaborator
s.rasova@gmail.com
Boris Pavlovich Harlamov
Doctor of Siences (physics and mathematics),
head of laboratory of IPME RAS
b.p.harlamov@gmail.com
Abstract:
We consider one-dimensional linear second order differential equation
and the solution of the Dirichlet problem of this equation on an interval
of the unknown function values. Two partial solutions of this problem on a small
symmetric neighborhood of an interior point of this interval are investigated.
The first solution has values 1 on the left edge, and 0 on the right edge of
the neighborhood. The second one has 0 and 1 correspondingly.
It is shown that properties of these partial solutions when the length of
the neighborhood tends to 0 characterize the initial equation completely.
Namely, two initial coefficients of the decomposition of each partial solution
with respect to the neighborhood length are proportional to corresponding
coefficients of this equation. To prove this theorem we use the generalized
Green formula and the generalized semigroup property of the family of the
Dirichlet problem solutions.
Keywords
- differential equation
- diffusion process
- Dirichlet problem
- generalized Green formula
- integral equation
- iteration
- kernel
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