Identities for Measures of Deviation from Solutions to Parabolic-Hyperbolic Equations

Abstract

Integral identities that are fulfilled for measure of difference between the exact solution of a parabolic-hyperbolic equation and any functions from a corresponding energy class are proved. These identities make it possible to derive two-sided a posteriori estimates for approximate solution to the corresponding Cauchy problem. The left-hand side of such an estimate is a natural measure of deviation from the solution, and the right-hand side depends on the problem data and the approximate solution itself and, therefore, it can be explicitly calculated. These estimates can be used to control the accuracy of approximate solutions and to compare solutions to Cauchy problems with different initial conditions. These estimates also allow one to quantitatively assess the effects occurring due to inaccuracies in the initial data and in the coefficients of the equation.