Abstract
The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.
Highlights
Introduction and Formulation of the ProblemThe purpose of this work is to investigate the partial differential equations of the form Lu ≡ ∂3u ∂x3 − ∂u ∂t = (1)in the domain of Ω = {(x, t) : 0 < x < 1, 0 < t ≤ T} with boundary conditions u(x, 0) = μu(x, T), μ = const, 0 ≤ x ≤ 1, (2)α1(t)u(0, t) + α2(t)uxx(0, t) = φ1(t), ux(0, t) = φ2(t), 0 ≤ t ≤ T (3)
The work of L.Cattabriga made it possible to construct a regular solution of boundary value problems with various boundary conditions
As a continuation of this, it is necessary to construct a solution to this problem in an unbounded domain, which the authors are currently working on
Summary
1. Introduction and Formulation of the Problem The purpose of this work is to investigate the partial differential equations of the form in the domain of Ω = {(x, t) : 0 < x < 1, 0 < t ≤ T} with boundary conditions u(x, 0) = μu(x, T), μ = const, 0 ≤ x ≤ 1, (2) In his works, using the asymptotic properties of the Airy function, he proved that for the function U(x − ξ; t − τ), V(x − ξ; t − τ), i.e. for fundamental solutions of Equation (1) the following relation is valid.
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