Presentation of the Problems

Part of the International Series of Numerical Mathematics book series (ISNM, volume 156)

Abstract

In this chapter we introduce the problems we are going to investigate in the next chapters of this part. They are mathematical models for diffusion-convectionreaction processes. We are particularly interested in coupled problems in which a small parameter is present. More precisely, let us consider in the rectangle QT = (a, c) × (0, T), −∞ < a < c < ∞, 0 < T < ∞, the following system of parabolic equations
$$\left\{ {\begin{array}{*{20}c} {u_t + \left( { - \varepsilon u_x + \alpha _1 \left( x \right)u} \right)_x + \beta _1 \left( x \right)u = f\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_{1T} ,} \\ {v_t + \left( { - \mu \left( x \right)v_x + \alpha _2 \left( x \right)v} \right)_x + \beta _2 \left( x \right)v = g\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_T ,} \\ \end{array} } \right.$$
with which we associate initial conditions
$$u\left( {x,0} \right) = u_0 \left( x \right),a \leqslant x \leqslant b;{\mathbf{ }}v\left( {x,0} \right) = v_0 \left( x \right),b \leqslant x \leqslant c,$$
(2)
transmissions conditions at x = b:
$$\left\{ {\begin{array}{*{20}c} {u\left( {b,t} \right) = v\left( {b,t} \right),} \\ { - \varepsilon u_x \left( {b,t} \right) + \alpha _1 \left( b \right)u\left( {b,t} \right) = - \mu \left( b \right)v_x \left( {b,t} \right) + \alpha _2 \left( b \right)v\left( {b,t} \right),{\mathbf{ }}0 \leqslant t \leqslant T,} \\ \end{array} } \right.$$
(3)
as well as one of the following types of boundary conditions:
$$\left\{ {\begin{array}{*{20}c} {u\left( {b,t} \right) = v\left( {b,t} \right),} \\ { - \varepsilon u_x \left( {b,t} \right) + \alpha _1 \left( b \right)u\left( {b,t} \right) = - \mu \left( b \right)v_x \left( {b,t} \right) + \alpha _2 \left( b \right)v\left( {b,t} \right),{\mathbf{ }}0 \leqslant t \leqslant T,} \\ \end{array} } \right.$$
(3)
where $$Q_{1T} = \left( {a,b} \right) \times \left( {0,T} \right),Q_{2T} = \left( {b,c} \right) \times \left( {0,T} \right),b \in \mathbb{R},a < b < c,\gamma$$ , γ is a given nonlinear function and ε is a small parameter, 0 < ε ≪ 1.

Keywords

Heat Transfer Thermal Energy Mass Density Heat Equation Energetic Particle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2007