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On a singulary perturbed, coupled elliptic-elliptic problem. (English) Zbl 1084.34511
Summary: The authors consider the following coupled boundary value problem of elliptic-elliptic type \[ \bigg\{\begin{aligned} -\epsilon u''(x)+\alpha(x)u'(x)+\beta(x)u(x)=f(x),x\in(a,b),\\-(u(x)v'(x))'+ \alpha(x)v'(x)+\beta(x)v(x)=g(x),x\in(b,c),\end{aligned} \tag{E\(_\epsilon\)} \] with homogeneous Dirichlet boundary conditions \[ u(a)=v(c)=0\tag{BC\({}_\epsilon\)} \] and transmition conditions at \(x=b\)
\[ u(b)=v(b),\;\epsilon u'(b)=(uv')(b).\tag{TC\(_\epsilon\)} \]
The behavior of the solution of the problem \((E_\epsilon),\;(BC_\epsilon),\;(TC_\epsilon)\) is studied when the small parameter \(\epsilon\) tends to \(0\).
MSC:
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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