Numerical Identification of External Boundary Conditions for Time Fractional Parabolic Equations on Disjoint Domains

Abstract

We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed.

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

“…Without a loss of generality, we can suppose that the external Dirihlet boundary conditions ( 3) and ( 4) are zero, as well as γ i (y, t) = 0, i = 1, 2. Otherwise, this can be archived by applying linear transformations like those in [33].…”

“…Then, from (33), we express (u 1 ) n+1 j 1 ,N y +1 − (u 1 ) n+1 j 1 ,N y /h y and substitute (31) for s = N y to derive…”

“…We consider corner nodes (x 1 = a 1 , y = c) and (x 1 = a 1 , y = d), where the interface boundary intersects Neumann boundaries. Using (33), j 1 = N 1 , (37), s = N y , (35), j 1 = 0, (37), and s = 0, we derive…”

“…One-dimensional boundary identification and backward heat conduction problems for layer materials are studied in [32]. In our previous works, we constructed a decomposition method to recover external boundary conditions in a time-fractional parabolic problem [33] and a hyperbolic-parabolic problem [34] on disjoint domains under point measurements.…”

Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer

“…Inverse problems for the identification coefficient, source, initial value, boundary conditions, etc., in a time-fractional diffusion equation are investigated in many papers; see, for example, [35][36][37][38][39]. Recently, there has been progress made on solving inverse tempered fractional diffusion problems modeling complex multi-scale problems and anomalous transport phenomena.…”