Abstract
Advection-diffusion equations are widely used in modeling a diverse range of problems. These mathematical models consist in a partial differential equation or system with initial and boundary conditions, which depend on the phenomena being studied. In the modeling, boundary conditions may be neglected and unnecessarily simplified, or even misunderstood, causing a model not to reflect the reality adequately, making qualitative and/or quantitative analyses more difficult. In this work we derive a general linear flux dependent boundary condition for advection-diffusion problems and show that it generates all possible boundary conditions, according to the outward flux on the boundary. This is done through an integral formulation, analyzing the total mass of the system. We illustrate the exposed cases with applications willing to clarify their meanings. Numerical simulations, by means of the Finite Difference Method, are used in order to exemplify the different boundary conditions' impact, making it possible to quantify the flux along the boundary. With qualitative and quantitative analysis, this work can be useful to researchers and students working on mathematical models with advection-diffusion equations.
Highlights
The first application of the diffusion equation was done by Fourier in 1822 [1], when he proposed its use to model heat distribution
The diffusion equation can be combined with advection processes, resulting in advection-diffusion equations, which demand more elaborate boundary conditions, depending on the phenomena being modeled
We show that one unique general boundary condition with linear flux, equation (2.5), generates all other particular cases
Summary
The first application of the diffusion equation was done by Fourier in 1822 [1], when he proposed its use to model heat distribution. General population movement is studied, aiming to compare different models and/or techniques, as in [24, 25, 26, 27] As each of these works has their own particularities, appropriate analyses at the boundaries are necessary in each case, but in most of the cited cases, the authors decided to make simple assumptions in order to obtain more tractable boundary conditions, sometimes due to lack of information about the studied phenomenon. The present work aims to derive and analyze a general boundary condition with linear flux for the advection-diffusion equation. We will consider only two spatial variables (x, y), but all the analyses are analogous to one or three, or even n dimensional problems
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