Periodic step-like contrast structures for a singularly perturbed parabolic equation

A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location

“…Both the time-periodic problems and steady state problems considered in [4] were previously considered by Vasil'eva in [10] and [11]. The algorithm used to find successive terms in the asymptotic expansion of the layer location in the three articles ( [4], [10] and [11]) is identical and corresponds to imposing a C 1 -matching condition on the left and right decompositions of the solution at the layer location.…”

Contrast structures of variable type in quasilinear parabolic equations

“…x,t Ω that satisfies relations (1) and (2) pointwise and also satisfies the periodicity condition u(x, t + T ) = u(x, t) for all (x, t) ∈ Ω.…”

“…In this case, under the additional conditions stated in [1], problem (1), (2) has a classical solution with an internal transition layer (or a step-like contrast structure). Figure 1(a) sketches such a solution in the plane (x, u) for any given value of the variable t.…”

“…Step-like contrast structures for problem (1), (2) were comprehensively investigated in [1], where, along with the existence theorem, the asymptotic expansion of such solutions in the small parameter ε was constructed and the equation…”

“…The assumption, adopted in [1], that the transition line of a step-like contrast structure entirely lies in Ω is quite restrictive. It is actually this condition that guarantees that the solution of problem (1), (2) has the form shown in Fig.…”