Workshop on Messico, 2010 B. Tirozzi, S.Yu. Dobrokhotov, E. Nazaikinski “Asymptotic solutions of a 2- dimensional wave equations with degenerate velocity and localized initial conditions”

Messico 2010 μ is a small parameter is the center of perturbation a=(1,0) Position of the Problem

Messico 2010 Operators

Messico 2010 Proposition

Messico 2010 Proposition Let L be the closure of the operator L 0 Consider the differential expression a and its formal adjoint a* Let A 0 be the operator defined by a in the space D 0 and A be its closure in L 2 (R 2 + ), let A be its adjoint operator

Messico 2010 Proposition Then the following identity holds Consider the hyperbolic problem in L 2 (R 2 + ) where L is the self-adjoint operator defined above

Messico 2010 Proposition is a strong solution of the problem if it satisfies the equation and the initial conditions and belong to

Messico 2010 Theorem

Messico 2010 Asymptotic Solution of the Problem Initial Condition Lagrangian manifold Compactification of space, since p 1 goes to infinity in a finite time

Messico 2010 Asymptotic Solution of the Problem function with compact support in Λ, regular in ρ and t, polynomial of degree N in h, fast decaying with its derivatives for ρ → ∞ Error Estimate If K h Λ [φρ(t, h)] satisfies the equation with an error of O(h N ), N > 1 the error of u is

Messico 2010 Asymptotic Solution of the Problem

Messico 2010 Asymptotic Solution of the Problem

Messico 2010 Asymptotic Solution of the Problem

Messico 2010

Asymptotic Solution of the Problem

Messico 2010