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• 1. Byström, Johan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Bounds and numerical results for the homogenized degenerated p-Poisson equation2004In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 49, no 2, p. 111-122Article in journal (Refereed)

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.

• 2.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Numerical simulations of glacial rebound using preconditioned iterative solution methods2005In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 50, p. 183-201Article in journal (Refereed)
• 3. Essel, Emmanuel Kwame
Department of Mathematics and Statistics, University of Cape Coast. Department of Mathematics and Statistics, University of Cape Coast. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence2010In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 55, no 4, p. 305-327Article in journal (Refereed)

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe's method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

• 4.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Homogenization of some parabolic operators with several time scales2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 5, p. 431-446Article in journal (Refereed)

The main focus in this paper is on homogenization of a parabolic problem with two local time scales. Under certain assumptions on the coefficient a, there exists a G-limit b, which we characterize by means of multiscale techniques for r>0, r≠1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.

• 5.
Luleå tekniska universitet.
Homogenization of parabolic equations an alternative approach and some corrector-type results1997In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 42, no 5, p. 321-343Article in journal (Refereed)

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold. partial differential equations - homogenization - two-scale convergence - linear parabolic equations - oscillating coefficients in space and time variable - dissimilar speeds of oscillation - admissible test functions - corrector results - compactness result - interpolationThis research was supported by The Swedish Research Council for the Engineering Sciences (TFR), The Swedish National Board for Industrial and Technological Development (NUTEK), and The Country of Jämtland Research Foundation

• 6.
KTH, School of Computer Science and Communication (CSC).
Rational Krylov for Nonlinear Eigenproblems, an Iterative Projection Method2005In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 50, no 6, p. 543-554Article in journal (Refereed)

In recent papers Ruhe [10], [12] suggested rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similar to the Arnoldi method presented in [13], [14] the search space is expanded by the direction from residual inverse iteration. Numercial methods demonstrate that the rational Krylov method can be accelerated considerably by replacing an inner iteration by an explicit solver of projected problems

• 7.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales2018In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, no 5, p. 503-521Article in journal (Refereed)

This paper is devoted to the study of the linear parabolic problem $\varepsilon\partial_tu_{\varepsilon}\left(x,t\right)-\nabla\cdot\left(a\left(x/\varepsilon,t/\varepsilon^3\right)\nabla u_{\varepsilon}\left(x,t\right)\right)=f\left(x,t\right)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $\varepsilon$ in front of the timederivative. First, we have an elliptic homogenized problem although the problem studiedis parabolic. Secondly, we get a parabolic local problem even though the problem has adifferent relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence $\left{u_{\varepsilon}\right}$ different from the standard setting are used, which means that these results are also of independent interest.

• 8. Kjellmert, Bo
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Time-dependent electromagnetic waves in a cavity2009In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 54, no 1, p. 17-45Article in journal (Refereed)

The electromagnetic initial-boundary value problem for a cavity enclosed by perfectly conducting walls is considered. The cavity medium is defined by its permittivity and permeability which vary continuously in space. The electromagnetic field comes from a source in the cavity. The field is described by a magnetic vector potential A satisfying a wave equation with initial-boundary conditions. This description through A is rigorously shown to give a unique solution of the problem and is the starting point for numerical computations. A Chebyshev collocation solver has been implemented for a cubic cavity, and it has been compared to a standard finite element solver. The results obtained are consistent while the collocation solver performs substantially faster. Some time histories and spectra are computed.

• 9. Kuliev, K.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied mathematics.
An extension of Rothe´s method to noncylindrical domains2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 5, p. 365-389Article in journal (Refereed)

In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

• 10.
University of West Bohemia.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
An extension of Rothe's method to noncylindrical domains2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 5, p. 365-389Article in journal (Refereed)

In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied

• 11.
Luleå tekniska universitet.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A study of bending waves in infinite and anisotropic plates1997In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 42, no 3, p. 213-232Article in journal (Refereed)

In this paper we present a unified approach to obtain integral representation formulas for describing the propagation of bending waves in infinite plates. The general anisotropic case is included and both new and well-known formulas are obtained in special cases (e.g. the classical Boussinesq formula). The formulas we have derived have been compared with experimental data and the coincidence is very good in all cases

• 12. Lukkassen, Dag
Centre of Mathematics, Lund University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On some iterated means arising in homogenization theory2004In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 49, no 4, p. 343-356Article in journal (Refereed)

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.

• 13.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
Homogenization of monotone parabolic problems with several temporal scales2012In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 57, no 3, p. 191-214Article in journal (Refereed)

In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the “rapid” self-similar case), respectively.

• 14.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
On general two-scale convergence and its application to the characterization of G-limits2007In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 52, no 4, p. 285-302Article in journal (Refereed)

We characterize some G-limits using two-scale techniques and investigate a method to detect deviations from the arithmetic mean in the obtained G-limit, where no peirodicity assumptions are involved. We also prove some results on the properties of generalized two-scale convergence.

• 15.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations1997In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 42, no 4, p. 245-257Article in journal (Refereed)

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.

• 16.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Bounds and estimates on the effective properties for nonlinear composites2000In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 45, no 6, p. 419-437Article in journal (Refereed)

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior.

• 17.
Luleå tekniska universitet.
Homogenization of the Maxwell Equations: Case I. Linear Theory2001In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 46, no 1, p. 29-51Article in journal (Refereed)

The Maxwell equations in a heterogeneous medium are studied. Nguetseng's method of two-scale convergence is applied to homogenize and prove corrector results for the Maxwell equations with inhomogeneous initial conditions. Compactness results, of two-scale type, needed for the homogenization of the Maxwell equations are proved

• 18.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.
Department of Mathematics, College of Sciences, Yasouj University, Yasouj, Iran. Örebro University, School of Science and Technology.
A free boundary problem for a predator-prey model with nonlinear prey-taxix2018In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 63, no 2, p. 125-147Article in journal (Refereed)

This paper deals with a reaction-diffusion system modeling a free boundary problem of the predator-prey type with prey-taxis over a one-dimensional habitat. The free boundary represents the spreading front of the predator species. The global existence and uniqueness of classical solutions to this system are established by the contraction mapping principle. With an eye on the biological interpretations, numerical simulations are provided which give a real insight into the behavior of the free boundary and the stability of the solutions.

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