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Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales. (English) Zbl 1278.34064

The problem of the existence of analytic solutions for the linear one-dimensional singularly perturbed boundary value problem \[ -\epsilon^2u''(x)+a_{11}u(x)++a_{12}v(x)=f(x), \]
\[ -\mu^2v''(x)+a_{21}u(x)++a_{22}v(x)=g(x), \]
\[ u(0)=u(1)=0,\quad v(0)=v(1)=0, \] on the interval \((0,1)\) is studied. The functions \(f\), \(g\) and \(a_{ij}\), \(i,j\in\{1,2\}\), are assumed to be analytic on \([0,1]\). The problem is split into four relevant cases in the dependence on the relation between \(\mu/1\) and \(\epsilon/\mu\) for \(0<\epsilon\leq\mu\leq1\).
Using the asymptotic expansion techniques, the existence and regularity of solutions for the problem under consideration are proved.

MSC:

34E05 Asymptotic expansions of solutions to ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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References:

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