A note on the numerical solution of an identification problem for observing two‐phase flow in capillaries

Abstract

In this paper, a right‐hand side identification problem for a parabolic equation with an overdetermined condition on an observation point is considered. A first and second order of accuracy difference schemes are constructed for obtaining approximate solutions of the problem that arises in two‐phase flow in capillaries. Stability estimates and numerical results are also established. Copyright © 2013 John Wiley & Sons, Ltd.

Existence and uniqueness results for an inverse problem for a semilinear equation with final overdetermination

“…Actually, the SIPs for linear equations are of greater interest comparing for the nonlinear ones. In [4,6], well-posedness of SIPs for linear partial differential equations have been studied. Kamynin [8] investigated the existence and uniqueness of the solution of a space-dependent SIP for linear parabolic equation.…”

“…Unfortunately, due to the nonlinearity and variable coefficients in a number of SIPs it is very difficult to obtain their exact solutions. At this point, the numerical methods play a vital role and there are considerably many investigations on the numerical aspects of these problems [4,6,14,15].…”

Existence and uniqueness results for an inverse problem for semilinear parabolic equations

“…Yet these methods need to be investigated theoretically, as well. Recently, there have been many investigations for the approximate solutions of SIPs [7][8][9][10][11].…”

“…In SIPs for identifying the unknown source term an overdetermined condition is given. This condition can be given as local [10,11], nonlocal [13], or integral condition [14]. These conditions affect the nature of problems and according to them various techniques are applied to study SIPs.…”

“…It is also possible to classify SIP with respect to source type. While some papers are into SIPs with space-dependent source [8,9,15], some papers are into the ones with time-dependent source [10,11,16]. In this paper, a time-dependent SIP with an integral overdetermination is considered.…”