The Difference Problem of Obtaining the Parameter of a Parabolic Equation

Abstract

The boundary value problem of determining the parameter p of a parabolic equation υ t Aυ t f t p 0 ≤ t ≤ 1 , υ 0 ϕ, υ 1 ψ in an arbitrary Banach space E with the strongly positive operator A is considered. The first order of accuracy stable difference scheme for the approximate solution of this problem is investigated. The well-posedness of this difference scheme is established. Applying the abstract result, the stability and almost coercive stability estimates for the solution of difference schemes for the approximate solution of differential equations with parameter are obtained.

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

“…Methods of solutions and theoretical aspects for various types of nonlocal BVP for parabolic differential equations are extensively investigated by several researchers (see previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein). Well posedness of nonclassical BVPs for various differential and difference equations were studied in a number of publications (see previous works [1][2][3][4][5][6][7][8] and the references therein).…”

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].…”

On the problem of determining the parameter of an elliptic equation in a Banach space

“…Theory and methods of solutions of inverse problems of determining the parameter of a partial differential equations have been extensively studied by several researchers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein). It is important that several inverse problems of determining the parameter of partial differential equations can be reduced to nonlocal boundary value problems for partial differential equations (see [1][2][3][4][5][6][7][8][9][10][11] and the literature cited therein).…”

“…It is important that several inverse problems of determining the parameter of partial differential equations can be reduced to nonlocal boundary value problems for partial differential equations (see [1][2][3][4][5][6][7][8][9][10][11] and the literature cited therein). Well-posedness of nonlocal boundary value problems for elliptic type differential and difference equations was studied in a number of papers (see [19][20][21][22][23][24][25][26][27][28] and the bibliography therein).…”