Asymptotic Analysis - Volume 10, issue 1

Authors: Abdallah, Naoufel Ben | Mas-Gallic, Sylvie | Raviart, Pierre-Arnaud

Article Type: Research Article

Abstract: In this paper a one-dimensional model for ion extraction from a plasma is investigated. The analysis is achieved thanks to a singular perturbation of the transport equations based on the smallness of both, the Debye length and the thermal voltage. The convergence of the perturbed problems to a limit problem, where a “free surface” appears, is proved (in one dimension the free surface reduces to a single point). The analysis is first done for finite temperature ions. For cold ions, the analysis can be carried out further: a boundary layer lying at the free surface is exhibited and analyzed.

DOI: 10.3233/ASY-1995-10101

Citation: Asymptotic Analysis, vol. 10, no. 1, pp. 1-28, 1995

Authors: van Duijn, C.J. | Floris, F.J.T.

Article Type: Research Article

Abstract: In this paper we consider the singular boundary value problem, −z1/p z″=D(f)/(1+p) for 0<f<1 with z(0)=z(1)=0. Here p>0 and D: [0,1]→R is a given function. This problem arises in a model for two-phase capillary induced flow in porous media. Considering the special case D(f)=fαw (1−f)α0 , with α0 ,αw >−2, we investigate the singular behaviour of the solution z(f) as α0 ,αw↓ −2. We show that the solution then becomes unbounded. We investigate the behaviour of z and z′ in this limit process. The results are incorporated in an algorithm which we use to solve the problem numerically. …The numerical results show significant improvement over standard discretisation techniques near the limit. Non-existence arises for α0 or αw ≤−2. Show more

DOI: 10.3233/ASY-1995-10102

Citation: Asymptotic Analysis, vol. 10, no. 1, pp. 29-48, 1995

Authors: Shibata, Tetsutaro

Article Type: Research Article

Abstract: We consider the non-linear Sturm-Liouville problem with two parameters on general level set Nα : \begin{equation}\left\{\begin{array}{l@{\quad}l}-u''(x)=\mu u(x)-\lambda(u(x)+|u(x)|^{p-1}u(x)),&x\in I=(0,1),\\u(0)=u(1)=0,&\end{array}\right.\end{equation} where p>1, μ,λ∈R are parameters and \begin{equation}N_{\alpha}:=\Biggl\{u\in W_{0}^{1,2}(I){:}\ \int_{0}^{1}(u'(x)^{2}-\mu u(x)^{2})\,\mathrm{d}x=2\alpha,\ \alpha <0{:}\ \mbox{is a normalizing parameter}\Biggr\}.\end{equation} We establish an asymptotic formula of n-th variational eigenvalue λ=λn (α) of (1) as α→−∞: λn (α)=Cμ (−α)(1−p)/2 +o((−α)(1−p)/2 ). Furthermore, we give an asymptotic formula of Cμ as μ→∞.

DOI: 10.3233/ASY-1995-10103

Citation: Asymptotic Analysis, vol. 10, no. 1, pp. 49-61, 1995

Authors: Volkmer, Hans

Article Type: Research Article

Abstract: It is shown that the zero T(a) of the solution y(·,a) of y″+exp (ym −t)=0, y(t,a)→a as t→∞, satisfies T(a)=(1−(m/2))am +(m−1)log a+log (m/2)+3/2(m−1)+Ol((log 3 a)/(am ))  as a→∞ if 1≤m<2. This asymptotic expansion of T(a) is related to a result of Atkinson and Peletier (1986). Moreover, if m=2 then T(a)=log a+s0 +O((log 3 a)/(am ))  as a→∞ where s0 =s=1.6853749… is the solution of the transcendental equation s−1.5=exp (−s).

DOI: 10.3233/ASY-1995-10104

Citation: Asymptotic Analysis, vol. 10, no. 1, pp. 63-75, 1995

Authors: Kersner, R. | Natalini, R. | Tesei, A.

Article Type: Research Article

Abstract: We study the Cauchy problem both for a non-linear diffusion-convection equation and for a first order hyperbolic conservation law. The latter can be regarded as the zero diffusion limit of the former, which is of degenerate parabolic type. Concerning support properties, the behaviour of solutions of both problems exhibits striking analogies when the effect of convection is stronger than diffusion. Hence previously reported phenomena concerning the parabolic problem can be interpreted as being of hyperbolic type.

DOI: 10.3233/ASY-1995-10105

Citation: Asymptotic Analysis, vol. 10, no. 1, pp. 77-93, 1995