On well‐posedness of the nonlocal boundary value problem for parabolic difference equations

Abstract

We consider the nonlocal boundary value problem for difference equations (uk−uk−1)/τ+Auk=φk, 1≤k≤N, Nτ=1, and u0=u[λ/τ]+φ, 0<λ≤1, in an arbitrary Banach space E with the strongly positive operator A. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given

A Note on the Parabolic Differential and Difference Equations

“…Well-posedness of local and nonlocal boundary value problems for abstract parabolic differential and difference equations in Banach spaces have been studied extensively by many researchers (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references therein).…”

Semi-Implicit Difference Scheme for a Two-Dimensional Parabolic Equation With an Integral Boundary Condition

“…In a two-dimensional case one of the first articles in this direction was [5]. Later on, numerical methods for such a problem were analyzed in the articles [1,7,15,18,22]. Particularly, the two-dimensional problems with nonlocal conditions using one-dimensional difference schemes were investigated in [6,10,20,25,31,32].…”

On a first order partial differential equation with the nonlocal boundary condition

“…Such conditions usually are identified as nonlocal boundary conditions and reflect situations when the data on the domain boundary cannot be measured directly, or when the data on the boundary depend on the data inside the domain. The well-posedness of various nonlocal boundary value problems for partial differential and difference equations has been studied extensively by many researchers (see [16,17] and the references given therein). However, such problems were not well investigated in general.…”