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  • 1.
    Allaire, G.
    et al.
    Ecole Polytechnique, Palaiseau Cedex, France.
    Pankratova, Iryna
    Narvik University College, Narvik, Norway.
    Piatnitski, A.
    Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization and concentration for a diffusion equation with large convection in a bounded domain2012Ingår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 262, nr 1, s. 300-330Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.

  • 2.
    Allaire, G.
    et al.
    Ecole Polytechnique, Palaiseau Cedex, France.
    Pankratova, Iryna
    Narvik University College, Narvik, Norway; Ecole Polytechnique, Palaiseau Cedex, France.
    Piatnitski, A.
    Narvik University College, Narvik, Norway; Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer2012Ingår i: SeMA Journal, ISSN 1575-9822, Vol. 58, nr 1, s. 53-95Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we make no assumption on the support of the initial data which may touch the domain’s boundary. This requires the construction of boundary layer correctors in the homogenization process which, surprisingly, play a crucial role in the definition of the leading order term at the limit. Therefore we have to restrict our attention to simple geometries like a rod or a layer for which the definition of boundary layers is easy and explicit.

  • 3.
    Chechkina, Alexandra
    et al.
    Lomonosov Moscow State University, Moscow, Russia.
    Pankratova, Iryna
    Narvik University College, Narvik, Norway.
    Pettersson, Klas
    Narvik University College, Narvik, Norway.
    Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator2015Ingår i: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 93, nr 1-2, s. 141-160Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

  • 4.
    Chiadò Piat, V.
    et al.
    Politecnico di Torino, Torino, Italy.
    Pankratova, Iryna
    Narvik University College, Narvik, Norway.
    Piatnitski, A.
    Narvik University College, Narvik, Norway; P.N. Lebedev Physical Institute RAS, Moscow, Russian Federation.
    Localization effect for a spectral problem in a perforated domain with Fourier boundary conditions2013Ingår i: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 45, nr 3, s. 1302-1327Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of localization of the eigenfunctions. Moreover, the limit behavior of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive estimates for the rate of convergence. 

  • 5.
    Jerez-Hanckes, C.
    et al.
    Universidad Adolfo Ibáñez Diagonal Las Torres, Peñalolén, Santiago, 2700, Chile.
    Pettersson, Irina
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Rybalko, V.
    B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky Ave., Kharkiv, 61103, Ukraine.
    Derivation of cable equation by multiscale analysis for a model of myelinated axons2020Ingår i: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 25, nr 3, s. 815-839Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order ε, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to ε which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model. 

  • 6.
    Jerez-Hanckes, Carlos
    et al.
    Universidad Adolfo Ibáñez, Santiago, Chile.
    Martínez, Isabel A.
    Pontificia Universidad Católica de Chile, Santiago, Chile.
    Pettersson, Irina
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Rybalko, Volodymyr
    Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine.
    Multiscale Analysis of Myelinated Axons2021Ingår i: SEMA SIMAI Springer Series, Springer , 2021, s. 17-35Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    We consider a three-dimensional model for a myelinated neuron, which includes Hodgkin–Huxley ordinary differential equations to represent membrane dynamics at Ranvier nodes (unmyelinated areas). Assuming a periodic microstructure with alternating myelinated and unmyelinated parts, we use homogenization methods to derive a one-dimensional nonlinear cable equation describing the potential propagation along the neuron. Since the resistivity of intracellular and extracellular domains is much smaller than the myelin resistivity, we assume this last one to be a perfect insulator and impose homogeneous Neumann boundary conditions on the myelin boundary. In contrast to the case when the conductivity of the myelin is nonzero, no additional terms appear in the one-dimensional limit equation, and the model geometry affects the limit solution implicitly through an auxiliary cell problem used to compute the effective coefficient. We present numerical examples revealing the forecasted dependence of the effective coefficient on the size of the Ranvier node

  • 7.
    Nazarov, Sergey A.
    et al.
    Institute for Problems in Mechanical Engineering RAS, St. Petersburgh, Russia.
    Pankratova, Iryna L.
    arvik University College, Narvik, Norway; Ecole Polytechnique CNRS, Palaiseau Cedex, France.
    Piatnitski, Andrey L.
    Narvik University College, Narvik, Norway; P. N. Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function2011Ingår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 200, nr 3, s. 747-788Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The paper deals with the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients in the case when the density function (the factor on the spectral parameter) changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum of this problem is discrete and consists of two series, one of them tending towards +∞ and another towards −∞. The asymptotic behaviour of positive and negative eigenvalues and their corresponding eigenfunctions depends crucially on whether the average of the weight function is positive, negative or equal to zero. We construct the asymptotics of eigenpairs in all three cases.

  • 8.
    Panasenko, G.
    et al.
    Université de Saint Étienne, St. Étienne, France.
    Pankratova, Iryna
    Narvik University College, Narvik, Norway.
    Piatnitski, A.
    Narvik University College, Narvik, Norway.
    Homogenization of a convection-diffusion equation in a thin rod structure2010Ingår i: Integral methods in science and engineering: Vol. 1, Boston: Birkhäuser Verlag, 2010, s. 279-290Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    This chapter is devoted to the homogenization of a stationary convection diffusion model problem in a thin rod structure. More precisely, we study the asymptotic behavior of solutions to a boundary value problem for a convection diffusion equation defined in a thin cylinder that is the union of two nonintersecting cylinders with a junction at the origin. We suppose that in each of these cylinders the coefficients are rapidly oscillating functions that are periodic in the axial direction, and that the microstructure period is of the same order as the cylinder diameter. On the lateral boundary of the cylinder we assume the Neumann boundary condition, while at the cylinder bases the Dirichlet boundary conditions are posed.

  • 9.
    Pankratova, Iryna
    et al.
    Department of Technology, Narvik University College, Narvik, Norway.
    Pettersson, Klas
    Department of Technology, Narvik University College, Narvik, Norway.
    Spectral asymptotics for an elliptic operator in a locally periodic perforated domain2015Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, nr 6, s. 1207-1234Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.

  • 10.
    Pankratova, Iryna
    et al.
    Narvik University College, Narvik, Norway; Ecole Polytechnique, Palaiseau Cedex, France.
    Piatnitski, A.
    Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function2011Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, nr 7, s. 3088-3134Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

  • 11.
    Pankratova, Iryna
    et al.
    Narvik University College, Narvik, Norway.
    Piatnitski, Andrey
    Narvik University College, Narvik, Norway; P.N. Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization of convection-diffusion equation in infinite cylinder2011Ingår i: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 6, nr 1, s. 111-126Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.

  • 12.
    Pankratova, Iryna
    et al.
    Narvik University College, Narvik, Norway.
    Piatnitski, Andrey
    Narvik University College, Narvik, Norway.
    On the behaviour at infinity of solutions to stationary convection-diffusion equations in a cylinder2009Ingår i: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 11, nr 4, s. 935-970Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neumann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove the existence of a bounded solution, its stabilization to a constant, and provide necessary and sufficient condition for the uniqueness.

  • 13.
    Pettersson, Irina
    UiT The Arctic University of Norway, Narvik, Norway.
    Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary2017Ingår i: Differential Equations & Applications, ISSN 1847-120X, Vol. 9, nr 3, s. 393-412Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The aim of this paper is to adapt the notion of two-scale convergence in Lp to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients and a p-Laplacian stated in thin cylinders with locally periodic rapidly varying thickness.

  • 14.
    Pettersson, Irina
    et al.
    The Arctic University of Norway, UiT, Norway.
    Piatnitski, Andrey
    The Arctic University of Norway, UiT, Norway; Institute for Information Transmission Problems of Russian Academy of Sciences, Russian Federation.
    Stationary convection-diffusion equation in an infinite cylinder2018Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, nr 7, s. 4456-4487Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

  • 15.
    Viirman, Olov
    et al.
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Pettersson, Irina
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    What to do when there is no formula? Navigating between less and more familiar routines for determining velocity in a calculus task for engineering students.2019Ingår i: Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC / [ed] J. Monaghan, E. Nardi and T. Dreyfus, 2019, s. 167-170Konferensbidrag (Refereegranskat)
  • 16.
    Viirman, Olov
    et al.
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Pettersson, Irina
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Björklund, Johan
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Boustedt, Jonas
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för Industriell utveckling, IT och Samhällsbyggnad, Datavetenskap.
    Programming in mathematics teacher education: A collaborative teaching approach2018Konferensbidrag (Refereegranskat)
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