Stability of a family of weighted finite-difference schemes (Q1037049)

The paper deals with a weighted difference schemes for the heat equation with the main feature that one of the boundary conditions \[ \frac{\partial u}{\partial x}\,(0,t)=\frac{\partial u}{\partial x}\,(1,t)+\alpha u(1,t) \] is nonlocal and depends on a parameter \(\alpha\). Bounds for the weight parameter \(\sigma\) and for the parameter \(\alpha\) are derived that guarantee stability of the solution with respect to the initial data in the mean-square grid norm. Existence and uniqueness theorems are derived.