In this paper, we consider a diffusion approximation of various kinetic equations using a modifie... more In this paper, we consider a diffusion approximation of various kinetic equations using a modified Chapman‐Enskog method called the compressed asymptotic procedure. In each case the results of computational experiments are presented, and the effects of different kind of correctors, applied for improving the accuracy of the approximation, are shown.
Mathematical Methods in The Applied Sciences, 2007
The paper is devoted to boundary value problems for transport equations with non divergence free ... more The paper is devoted to boundary value problems for transport equations with non divergence free external field. The crucial role is played by integration along characteristics and associated Green's formula for which we provide a new proof which generalizes and clarifies previous versions. The paper is concluded with an application of general theory to the Spencer-Lewis equation.
Journal of Mathematical Analysis and Applications, 2004
In the paper we shall present a survey of recent results on substochastic semigroups and provide ... more In the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty. These methods are applied to the fragmentation equation with mass loss, yielding sufficient conditions for the existence of conservative and shattering solutions. Our results provide a mathematical framework that clarifies the discussion of [Phys. Rev. A 43 (1991) 656, Phys. Rev. A 41 (1990) 5755, J. Phys. A 24 (1991) 3967] on shattering fragmentation in such models showing, among others, that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss, contrary to the conjecture of [Phys. Rev. A 43 (1991) 656, Phys. Rev A 41 (1990) 5755]. 2004 Elsevier Inc. All rights reserved.
The paper deals with an analysis of well-posedness of the Boltzmann-like semiconductor equation w... more The paper deals with an analysis of well-posedness of the Boltzmann-like semiconductor equation with unbounded collision frequency, introduced recently by Majorana and Milazzo. 17 The equation is derived by writing the balance of the electrons lost and gained at each energy level due to scattering on the crystalline lattice of the semiconductor. As the total amount of electrons is expected to be constant, the process can be viewed as a Markov process, and from the functional analytic point of view it fits into the general theory of substochastic semigroups. 5,26 In this paper we present two methods of solving the evolution equation describing this process: one is a generalization of the approach of Reuter and Ledermann 23 to solving differential equations governing Markov processes with denumerably many states, while the other is based on the Kato-Voigt perturbation technique for substochastic semigroups. The combination of these two techniques is a powerful tool yielding strong results on the existence and uniqueness of conservative solutions. It is also shown how the solution method employed in Ref. 17 fits into the theory developed in this paper.
Zeitschrift Fur Analysis Und Ihre Anwendungen, 2005
J. Banasiak and M. Lachowicz proved in [Math. Models Methods Appl. Sci. 12 (2002), 755 -775] that... more J. Banasiak and M. Lachowicz proved in [Math. Models Methods Appl. Sci. 12 (2002), 755 -775] that, under certain conditions on the coefficients, the dynamics generated by birth-and-death type systems with proliferation was chaotic. In this paper we extend this result to systems with parameter-dependent coefficients and present an application to a linear Boltzmann equation describing inelastic collisions.
In the last few years there has been a renewed interest in infinite systems of differential equat... more In the last few years there has been a renewed interest in infinite systems of differential equations, similar to the classical birth-and-death system of population dynamics, due to their rôle in modelling the evolution of drug resistance in cancer cells. In [J. Banasiak, M. Lachowicz, Topological chaos for birth-and-death models with proliferation, Math. Models Methods Appl. Sci. 12 755] such systems were shown to generate a chaotic dynamics under, however, very restrictive assumptions on the growth of coefficients. In this paper, using recently developed concept of subspace chaos [J. Banasiak, M. Moszyń ski, A generalization of Desch-Schappacher-Webb criteria for topological chaos with applications, Discrete Contin. Dyn. Syst. -A 12 959], we show that for a linear growth of the coefficients, which are more acceptable from biological point of view, the dynamics of these systems is chaotic in some subspaces of the original state space.
Mathematical Methods in The Applied Sciences, 1994
In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons wi... more In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons with a small positive parameter ϵ related to the mean free path, based upon the Chapman–Enskog procedure of the kinetic theory. We prove that if proper initial conditions derived by considering initial layer solutions are used, the diffusion equation gives the uniform approximation to the neutron density function with the O(ϵ2) accuracy.
Journal of Applied Mathematics and Stochastic Analysis, 1998
In the paper we analyze singularly perturbed telegraph systems applying the newly developed compr... more In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.
Comptes Rendus De L Academie Des Sciences Serie Ii Fascicule B-mecanique, 2001
In recent years it was observed that chaotic behaviour can occur in some infinitedimensional line... more In recent years it was observed that chaotic behaviour can occur in some infinitedimensional linear systems. An example of this type, related to a kinetic model (death process), has been previously reported. In this paper we generalize these earlier results to the case of variable coefficients, showing that the property of being chaotic can be in a certain sense stable. On the other hand the 'opposite' birth process cannot be chaotic. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
We present three different asymptotic expansion procedures for a linear kinetic equation, of whic... more We present three different asymptotic expansion procedures for a linear kinetic equation, of which only two lead to a diffusion equation. We show that the diffusion equation gives a first-order approximation to the kinetic equation if it is supplemented by an initial condition derived from the initial layer considerations. As an example, we take a one-speed neutron transport equation in
Published 23 Communicated by N. Bellomo 24 In the present paper we study the macroscopic limits o... more Published 23 Communicated by N. Bellomo 24 In the present paper we study the macroscopic limits of a kinetic model for interacting 25 entities (individuals, organisms, cells). The kinetic model is one-dimensional and the 26 entities are characterized by their position and orientation (+/−) with swarming inter-27 action controlled by a sensitivity parameter. The macroscopic limits of the model are 28 considered for solutions close either to the diffusive (isotropic) or to the aligned (swarm-29 ing) equilibrium states for various values of that parameter. In the former case the 30 classical linear diffusion equation results whereas in the latter a traveling wave solution 31
In this paper, we consider a diffusion approximation of various kinetic equations using a modifie... more In this paper, we consider a diffusion approximation of various kinetic equations using a modified Chapman‐Enskog method called the compressed asymptotic procedure. In each case the results of computational experiments are presented, and the effects of different kind of correctors, applied for improving the accuracy of the approximation, are shown.
Mathematical Methods in The Applied Sciences, 2007
The paper is devoted to boundary value problems for transport equations with non divergence free ... more The paper is devoted to boundary value problems for transport equations with non divergence free external field. The crucial role is played by integration along characteristics and associated Green's formula for which we provide a new proof which generalizes and clarifies previous versions. The paper is concluded with an application of general theory to the Spencer-Lewis equation.
Journal of Mathematical Analysis and Applications, 2004
In the paper we shall present a survey of recent results on substochastic semigroups and provide ... more In the paper we shall present a survey of recent results on substochastic semigroups and provide new methods for determining their honesty. These methods are applied to the fragmentation equation with mass loss, yielding sufficient conditions for the existence of conservative and shattering solutions. Our results provide a mathematical framework that clarifies the discussion of [Phys. Rev. A 43 (1991) 656, Phys. Rev. A 41 (1990) 5755, J. Phys. A 24 (1991) 3967] on shattering fragmentation in such models showing, among others, that there occurs an unexpected mass loss associated with shattering which is not accounted for by the discrete and continuous mass loss, contrary to the conjecture of [Phys. Rev. A 43 (1991) 656, Phys. Rev A 41 (1990) 5755]. 2004 Elsevier Inc. All rights reserved.
The paper deals with an analysis of well-posedness of the Boltzmann-like semiconductor equation w... more The paper deals with an analysis of well-posedness of the Boltzmann-like semiconductor equation with unbounded collision frequency, introduced recently by Majorana and Milazzo. 17 The equation is derived by writing the balance of the electrons lost and gained at each energy level due to scattering on the crystalline lattice of the semiconductor. As the total amount of electrons is expected to be constant, the process can be viewed as a Markov process, and from the functional analytic point of view it fits into the general theory of substochastic semigroups. 5,26 In this paper we present two methods of solving the evolution equation describing this process: one is a generalization of the approach of Reuter and Ledermann 23 to solving differential equations governing Markov processes with denumerably many states, while the other is based on the Kato-Voigt perturbation technique for substochastic semigroups. The combination of these two techniques is a powerful tool yielding strong results on the existence and uniqueness of conservative solutions. It is also shown how the solution method employed in Ref. 17 fits into the theory developed in this paper.
Zeitschrift Fur Analysis Und Ihre Anwendungen, 2005
J. Banasiak and M. Lachowicz proved in [Math. Models Methods Appl. Sci. 12 (2002), 755 -775] that... more J. Banasiak and M. Lachowicz proved in [Math. Models Methods Appl. Sci. 12 (2002), 755 -775] that, under certain conditions on the coefficients, the dynamics generated by birth-and-death type systems with proliferation was chaotic. In this paper we extend this result to systems with parameter-dependent coefficients and present an application to a linear Boltzmann equation describing inelastic collisions.
In the last few years there has been a renewed interest in infinite systems of differential equat... more In the last few years there has been a renewed interest in infinite systems of differential equations, similar to the classical birth-and-death system of population dynamics, due to their rôle in modelling the evolution of drug resistance in cancer cells. In [J. Banasiak, M. Lachowicz, Topological chaos for birth-and-death models with proliferation, Math. Models Methods Appl. Sci. 12 755] such systems were shown to generate a chaotic dynamics under, however, very restrictive assumptions on the growth of coefficients. In this paper, using recently developed concept of subspace chaos [J. Banasiak, M. Moszyń ski, A generalization of Desch-Schappacher-Webb criteria for topological chaos with applications, Discrete Contin. Dyn. Syst. -A 12 959], we show that for a linear growth of the coefficients, which are more acceptable from biological point of view, the dynamics of these systems is chaotic in some subspaces of the original state space.
Mathematical Methods in The Applied Sciences, 1994
In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons wi... more In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons with a small positive parameter ϵ related to the mean free path, based upon the Chapman–Enskog procedure of the kinetic theory. We prove that if proper initial conditions derived by considering initial layer solutions are used, the diffusion equation gives the uniform approximation to the neutron density function with the O(ϵ2) accuracy.
Journal of Applied Mathematics and Stochastic Analysis, 1998
In the paper we analyze singularly perturbed telegraph systems applying the newly developed compr... more In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.
Comptes Rendus De L Academie Des Sciences Serie Ii Fascicule B-mecanique, 2001
In recent years it was observed that chaotic behaviour can occur in some infinitedimensional line... more In recent years it was observed that chaotic behaviour can occur in some infinitedimensional linear systems. An example of this type, related to a kinetic model (death process), has been previously reported. In this paper we generalize these earlier results to the case of variable coefficients, showing that the property of being chaotic can be in a certain sense stable. On the other hand the 'opposite' birth process cannot be chaotic. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
We present three different asymptotic expansion procedures for a linear kinetic equation, of whic... more We present three different asymptotic expansion procedures for a linear kinetic equation, of which only two lead to a diffusion equation. We show that the diffusion equation gives a first-order approximation to the kinetic equation if it is supplemented by an initial condition derived from the initial layer considerations. As an example, we take a one-speed neutron transport equation in
Published 23 Communicated by N. Bellomo 24 In the present paper we study the macroscopic limits o... more Published 23 Communicated by N. Bellomo 24 In the present paper we study the macroscopic limits of a kinetic model for interacting 25 entities (individuals, organisms, cells). The kinetic model is one-dimensional and the 26 entities are characterized by their position and orientation (+/−) with swarming inter-27 action controlled by a sensitivity parameter. The macroscopic limits of the model are 28 considered for solutions close either to the diffusive (isotropic) or to the aligned (swarm-29 ing) equilibrium states for various values of that parameter. In the former case the 30 classical linear diffusion equation results whereas in the latter a traveling wave solution 31
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