Based on Eq. (12), and after simple transformations, we have Display Formula
$\delta JQPAR=SQPAR\xb7\u03f5QPAR,$(15)
Display Formula$\delta JKPAR=SKPAR\xb7\u03f5KPAR,$(16)
where Display Formula$SQPAR=QPAR(s,t)KPAR(s,t)\u222b0H(s)\lambda (s,z,t)e\u2212Kpar(s,t)zdz,$(17)
Display Formula$SKPAR=QPAR(s,t)KPAR(s,t)\u222b0H(s)[1\u2212KPAR(s,t)z]\lambda (s,z,t)e\u2212Kpar(s,t)zdz,$(18)
Display Formula$\u03f5QPAR=\delta QPAR(s,t)QPAR(s,t),$(19)
Display Formula$\u03f5KPAR=\delta KPAR(s,t)KPAR(s,t),$(20)
where $\u03f5QPAR$ and $\u03f5KPAR$ are the relative errors in $QPAR$ and $KPAR$, respectively, while $SQPAR$ and $SKPAR$ can be interpreted as sensitivities of the heat content $J$ [Eq. (5)] at $t=tf$ to the relative errors in the $QPAR$ and $KPAR$. Note that Eqs. (15)–(20) provide estimate of errors ($\delta JQPAR$ and $\delta JKPAR$) in the heat content $J$ [Eq. (5)] at $t=tf$ due to relative errors in the $QPAR$ ($\u03f5QPAR$) and $KPAR$ ($\u03f5KPAR$) at time $t$ ($t0\u2264t\u2264tf$), therefore, time between initial time $t0$ and verification time $tf$.