Locally One-Dimensional Difference Schemes for Parabolic Equations in Media Possessing Memory

Abstract

Many processes in complex systems are nonlocal and possess long-term memory. Such problems are encountered in the theory of wave propagation in relaxing media [1, p. 86], whose equation of state is distinguished by a noninstantaneous dependence of the pressure p(t) on the density ρ(t); the value of p at a time t is determined by the value of the density ρ at all preceding times; i.e., the medium has memory. Similar problems are also encountered in mechanics of polymers and in the theory of moisture transfer in soil [2]; the same equation arises in the theory of solitary waves [3] and is also called the linearized alternative Korteweg–de Vries equation, or the linearized Benjamin–Bona–Mahony equation. One of such problems was studied in [4]. In the present paper, a locally one-dimensional scheme for parabolic equations with a nonlocal source, where the solution depends on the time t at all preceding times, is considered.