Characterization of solutions in the discretization of a parabolic equation on an infinite strip

A full discretization scheme is studied for a parabolic equation on a strip which is infinite with respect to the space variable. A criterion is formulated and proven claiming that the discrete solution vanishes at infinity and at each time level if and only if it satisfies a well defined linear two-term recurrence with respect to the discrete space variable. The recurrence involves solution values of lower time levels. The linear relation may be used as a boundary condition. Thus, the problem is reduced to an equivalent one on a finite grid. © 1999 Elsevier Science Ltd. All rights reserved.