Locally one-dimensional difference schemes for parabolic equations in media possessing memory

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 $\rho(t)$; the value of $p$ at a time $t$ is determined by the value of the density $\rho$ 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.