Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity

The time-dependent Schrödinger equation with piecewise constant potentials

“…Thus, by Jordan's Lemma, the integrals of exp(−iν (1) x)ψ (1) ν (1) , t and exp(−iν (1) x)ψ (2) −ν (2) , t along a closed, bounded curve in the right-half of the complex κ plane vanish for x < 0. In particular we consider the closed curve L (4) Figure 4. Since the integral along L C vanishes for large C, the fourth and fifth integrals on the right-hand side of (19a) must vanish since the contour L D (4) becomes ∂D (4) as C → ∞.…”

Finite volume schemes for multilayer diffusion

“…Many industrial, environmental and biological problems involve diffusion processes across layered materials. For example, heat conduction in composites [7,18,19], tumour growth across the white and grey matter components of the brain [1,15], contaminant transport across layered soils [14,29] and thermal conduction through skin layers during burning [24] all involve multilayer diffusion processes. Additionally, layered diffusion is of interest to the applied mathematics community as it can be thought of as a simple example of a multiscale problem when the number of layers is large [4,5].…”

“…In the case of infinite contact transfer coefficient, H i → ∞, Type II conditions reduce to Type I. Type I conditions occur in pure diffusion problems [4,10] and in modelling the growth of brain tumours [1,15], Type II conditions are used to model roughness/contact resistance between adjacent layers [4,10], Type III conditions appear in models of concentration diffusion in porous media [22] and dissolved contaminant diffusion in aquitards [14] and Type IV conditions ensure a discontinuity in the solution that is useful in modelling drug release from microcapsules [8,12]. Finite volume/difference schemes for the multilayer diffusion problem described above have been presented by Carr and Turner [4], who implemented a finite volume scheme, and by Hickson et al [11], who implemented a finite difference scheme.…”

“…1) for i ∈ {F, B, C}. Note that equation (3.1) is the stability condition for asymptotic stability, that is, as the number of time steps k → ∞.…”

Initial-to-Interface Maps for the Heat Equation on Composite Domains

“…Using the Fokas method [1,2] such solutions may be constructed. This has been done in the case of the heat equation with n interfaces in infinite, finite, and periodic domains as well as on graphs [3][4][5][6][7]. The method has also been extended to dispersive problems [8,9], and higher order problems [10].…”