Abstract
In the paper, the stability and convergence of difference schemes approximating semilinear parabolic equation with a nonlocal condition are considered. The proof is based on the properties of M-matrices, not requiring the symmetry or diagonal predominance of difference problem. The main presumption is that all the eigenvalues of the corresponding difference problem with nonlocal conditions are positive.
Highlights
Initiations in the research of parabolic equations with nonlocal conditions appeared in papers [3, 19] more than half a century ago
The following three lemmas could be interpreted as the new statement of certain properties of M-matrices, adapted to the systems of difference equations approximating parabolic equations
The results obtained in the paper extend the class of the differential problems with nonlocal conditions when it is possible to prove the stability and convergence of difference schemes in the uniform norm
Summary
Initiations in the research of parabolic equations with nonlocal conditions appeared in papers [3, 19] more than half a century ago. In the paper [33], using the properties of M-matrices, the error of the solution of the difference problem for nonlinear elliptic equation with nonlocal boundary condition was estimated Two aspects of such a method of proof have been noted. The main result of the present paper is that the stability and convergence of difference schemes for a parabolic equation with an integral boundary condition in the uniform norm has been proved using the structure of the spectrum of difference operator and the theory of M-matrices.
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