Asymptotic solution of initial boundary-value problems for hyperbolic systems

Abstract

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.