The factorization of hermitian matrix functions and its applications to boundary-value problems

1976 ◽  
Vol 27 (6) ◽  
pp. 629-639 ◽  
Author(s):  
A. M. Nikolaichuk ◽  
I. M. Spitkovskii
Keyword(s):  
Boundary Value Problems ◽  
Hermitian Matrix ◽  
Boundary Value ◽  
Matrix Functions

Asymptotic solution of initial boundary-value problems for hyperbolic systems

1967 ◽  
Vol 262 (1129) ◽  
pp. 387-411 ◽  
Keyword(s):  
Asymptotic Solution ◽  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Formal Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Order Partial Differential Equation ◽  
Matrix Functions ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

A theory is developed for the derivation of formal asymptotic solutions for initial boundary-value problems for equations of the form ^ ° S + l / ^ + 'LBu+Cu = f(‘>x ' A)' where A0, Av, B, and C are m xm matrix functions of t and X = (xv ..., xn), u X; A) is an ^-component column vector, and A is a large positive parameter. Our procedure is to consider a formal asymptotic solution of the form 1 00 u(t, X; A) ~ e £ (iA)-i z jt, X). Substitution of this formal solution into the equation yields, for the function s(t, X), a first order partial differential equation which can be solved by the method of characteristics. If the coefficient matrices satisfy certain conditions then we obtain, for the functions z X), linear systems of ordinary differential equations called transport equations along space-time curves called rays . They may be solved explicitly under suitable conditions. A proof is presented of the asymptotic nature of the formal solution when the coefficient matrices and initial data for u are appropriately chosen. The problem of reflexion and refraction at an interface is considered.

Sign in / Sign up

Export Citation Format

Share Document