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

Pub Date : 2024-06-13 DOI:10.1134/s0965542524700313
S. I. Repin
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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.

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抛物线-双曲方程解偏差测量的同一性
证明了抛物线-双曲面方程的精确解与相应能量类别中的任何函数之间的差量的积分等式。通过这些等式,可以推导出相应考希问题近似解的双侧后验估计值。这种估计值的左手边是偏离解的自然度量,右手边取决于问题数据和近似解本身,因此可以明确计算。这些估计值可用于控制近似解的精确度,以及比较具有不同初始条件的 Cauchy 问题的解。通过这些估计值,还可以定量评估由于初始数据和方程系数的不准确而产生的影响。
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