Heat equation on a network using the Fokas method

Abstract

The problem of heat conduction on networks of multiply connected rods is solved by providing an explicit solution of the one-dimensional heat equation in each domain. The size and connectivity of the rods is known, but neither temperature nor heat flux are prescribed at the interface. Instead, the physical assumptions of continuity at the interfaces are the only conditions imposed. This work generalizes that of Deconinck, Pelloni, and Sheils [5] for heat conduction on a series of one-dimensional rods connected end-to-end to the case of general configurations.

The Linear KdV Equation with an Interface

“…which are all valid for k ∈ D (r) . Evaluating (19) for t = s, multiplying by e ik 3 s , and integrating from 0 to t one obtains…”

“…Few interface problems allow for an explicit closed-form solution using classical solution methods. Using the Fokas or Unified Transform Method [7,8,9], such solutions may be constructed for both dissipative and dispersive linear interface problems as shown in [3,6,13,16,17,18,19].…”

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

“…Integrating the local relations (4) around the appropriate domain (see Fig. 3) and applying Green's Theorem, we find the global relations (5) and their evaluation at −k (6). In contrast to Section 2, these 2n + 2 global relations are all valid for k ∈ C. In addition to the definitions in Section 2, we define (1) x (x 0 , s) ds,…”

“…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].…”

“…In what follows it is convenient to use the quantity σ j , defined as the positive square root of α j : σ j = √ α j . For a derivation of this system, see [6,13]. If the layer is either at the far left or far right of the wall, Dirichlet, Neumann, or Robin boundary conditions can be imposed on its far left or right boundary, respectively, corresponding to prescribing "outside" temperature, heat flux, or a combination of these.…”