Showing papers in "Japan Journal of Industrial and Applied Mathematics in 2001"
TL;DR: In this paper, a new type of competition-diffusion system with a small parameter is proposed, and it is shown that any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms.
Abstract: A new type of competition-diffusion system with a small parameter is proposed. By singular limit analysis, it is shown that any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms. This result exhibits the relation between an ecological population model and water-ice solidification problems.
64 citations
TL;DR: In this paper, the stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains.
Abstract: Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal “energy” is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.
54 citations
TL;DR: In this article, the spatio-temporal dynamics of three competitive species is considered and the community is described by a system of partial differential equations of Lotka-Volterra type.
Abstract: The spatio-temporal dynamics of three competitive species is considered. Mathematically, the community is described by a system of partial differential equations of Lotka-Volterra type. The properties of the system are investigated both numerically and analytically. We show that for finite initial conditions the dynamics of the system is typically reduced to a succession of travelling diffusive waves, some of which demonstrate rather an unusual behaviour. Particularly, a locally unstable equilibrium can become stable in the wake of a diffusive front. After propagation of the waves, the domain is invaded by irregular spatiotemporal population oscillations that can be classified as spatio-temporal chaos.
51 citations
TL;DR: In this work, the order of accuracy of the method for general ODEs is investigated, and it is shown that the attainable order is 2s, like conventional Runge-Kutta methods.
Abstract: In this paper, we propose a functional fittings-stage Runge-Kutta method which is based on the exact integration of the set of the linearly independent functions φi(t), (i = 1,...,s). The method is exact when the solution of the ODE can be expressed as the linear combination of φi(t), although the method has an error for general ODE. In this work we investigate the order of accuracy of the method for general ODEs, and show that the order of accuracy of the method is at leasts, if the functions φi(t) are sufficiently smooth and the method is non-confluent. Furthermore, it is shown that the attainable order of the method is 2s, like conventional Runge-Kutta methods. Two- and three-stage methods including embedded one of this type are developed.
50 citations
TL;DR: In this paper, the authors reformulated Kermack's and McKendrick's variable susceptibility model for infectious diseases as a nonlinear age-dependent population dynamics model, and proved an existence and uniqueness result for the endemic steady state.
Abstract: In this paper, we reformulate Kermack’s and McKendrick’s variable susceptibility model for infectious diseases as a nonlinear age-dependent population dynamics model, then we prove an existence and uniqueness result for the endemic steady state. Subsequently we discuss the local stability of the endemic steady state. Finally we show that Pease’s evolutionary epidemic model can be seen as a special case of the variable susceptibility model and discuss possible extensions.
45 citations
TL;DR: In this paper, the authors constructed a local invariant manifold near the onset of self-replication and derived the nonlinear ODE on it, and studied the manner of splitting by analyzing the resulting ODE, and answer the question "2n-splitting or edge splitting?" starting from a single pulse.
Abstract: Since early 90’s, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k afterk-th splitting, not 2n-splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question “2n-splitting or edge-splitting?” starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.
38 citations
TL;DR: In this paper, a general method to find exact traveling and standing wave solutions of reaction-diffusion systems and nonlinear wave equations is proposed, which is applied to several well-known reaction diffusion systems such as the Gray-Scott model, a simplification of the Noyes-Field model for the Belousov-Zhabotinskii reaction, and two and three component models for quadratic solitons.
Abstract: We propose a general method to find exact travelling and standing wave solutions of reaction-diffusion systems and nonlinear wave equations. The method is applied to several well-known reaction-diffusion systems such as a competition-diffusion system of Lotka-Volterra type, the Gray-Scott model, a simplification of the Noyes-Field model for the Belousov-Zhabotinskii reaction, and two- and three-component models for quadratic solitons. We also find exact solutions of several generalized nonlinear dispersive equations occurring in mathematical physics.
37 citations
TL;DR: In this paper, a mathematical model for the dynamic behavior of hippocampus is described by the skew product transformation in terms of chaotic dynamics and contracting dynamics, where fractal objects are generated.
Abstract: We construct a mathematical model for the dynamic behavior of hippocampus. The model is described by the skew product transformation in terms of chaotic dynamics and contracting dynamics. In the contracting subspace, fractal objects are generated. We show that such fractal objects are characterized by a code of a temporal sequence generated by chaotic dynamics.
35 citations
Abstract: A technique based on the abstract Cauchy-Kovalevskaya theorem is used to derive uniform estimates of solutions of BBGKY hierarchy, which improves Lanford’s theorem on the Boltzmann-Grad limit and simplifies its proof, in part. It is also applied to the Euler limit of the Boltzmann hierarchy.
34 citations
TL;DR: In this paper, an introductory review of the mathematics for investigating the interfacial motion in crystal growth problems is presented, where the main theme is how the kinetic anisotropy determines the growth form of the crystal and how the curvature effect works on it.
Abstract: We will present an introductory review of the mathematics for investigating the interfacial motion in crystal growth problems. Anisotropy is quite an important factor in such problems. There are two types of anisotropy — the kinetic anisotropy and the one of curvature effect. The main theme of this article is how the kinetic anisotropy determines the growth form of the crystal and how the curvature effect works on it.
28 citations
TL;DR: In this article, the authors proposed a solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed of the interface. And they also gave a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet.
Abstract: Anisotropic curvature flow equations with singular interfacial energy are important for good understanding of motion of phase-boundaries. If the energy and the interfacial surface were smooth, then the speed of the interface would be equal to the gradient of the energy. However, this is not so simple in the case of non-smooth crystalline energy. But it’s well-known that a unique gradient characterization of the velocity is possible if the interface is a curve in the two-dimensional space. In this paper we propose a notion of solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed. We also give a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet, a flat portion of the interface.
TL;DR: In this paper, the existence and non-existence of stable nonconstant steady states in reaction-diffusion systems on graphs was studied. But the authors focused on the stability of the PDEs.
Abstract: Diffusion process on a network of thin media can be described by a system of parabolic PDEs on a graph which interact with each other through connecting nodes. The aim of this paper is to study the existence and non-existence of stable nonconstant steady states in reaction-diffusion systems on graphs. It is shown that there are five types of graphs on which any stable steady state is necessarily constant. It is also shown that if the graph is of none of these five types, then a stable nonconstant steady state can exist by takingf appropriately.
TL;DR: In this paper, a universal theory describing the deformation and atomization processes of liquid droplets and columns is proposed on the basis of the first principle of fluid dynamics, which provides a formulation for breakup phenomena when two droplets collide.
Abstract: A universal theory describing the deformation and atomization processes of liquid droplets and columns is proposed on the basis of the first principle of fluid dynamics. Based on the proposed theory, previously reported empirical models such as the TAB model and the OPT model can be derived along with their arbitrary constants. Moreover, this theory provides a formulation for breakup phenomena when two droplets collide. It is also shown on the basis of this theory that the atomization processes of liquid droplets are mathematically similar to biological cell proliferation. This is because actual living cells mainly consist of liquid and because both systems are dominated by three essential forces, that is, internal convection, surface tension, and the internal pressure gradient due to energy input. Finally, it will be shown that the present theory offers a qualitative explanation of the unlacing processes of biological molecules such as the base pairs of purines and pyrimidines surrounded with water molecules, that is, the chemical reaction processes related to the hydrogen bonds.
TL;DR: In this paper, an operator that extends functions in finite element spaces to the exact domain is constructed and some estimates in the boundary skin are presented, which is successfully used to prove uniform solvability in approximate domains for problems subject to slip boundary conditions and so on.
Abstract: Uniform solvability of finite element solutions in approximate domains is studied. An operator that extends functions in finite element spaces to the exact domain is constructed and some estimates in the boundary skin are presented. The extension operator is successfully used to prove uniform solvability in approximate domains for problems subject to slip boundary conditions and so on.
TL;DR: In this article, an iterative procedure using finite element method without the Lagrange multiplier is proposed for three-dimensional eddy current problems, which is based on an iteration procedure derived from a perturbation problem of the magnetostatic problem.
Abstract: An iterative procedure using finite element method without the Lagrange multiplier is proposed for three-dimensional eddy current problems, which is based on an iterative procedure derived from a perturbation problem of the magnetostatic problem. To consider the continuity of an excitation current density, a correction method is also proposed. Numerical results show that the BiConjugate Gradient (BiCG) method is applicable to the complex symmetric linear systems arising in the iterative procedure, and that approximate physical quantities are suitable.
TL;DR: Hierarchy structure, or lattice structure among invariant subspaces supporting a skelton of associative dynamics, is elucidated in chaotic neural networks with ability of dynamical association.
Abstract: We analyse symmetrical structure among invariant subspaces in chaotic neural networks with ability of dynamical association. In particular, we elucidate hierarchical structure, or lattice structure among invariant subspaces supporting a skelton of associative dynamics. We’d like to dedicate this paper to the late Professor Masaya Yamaguti from whom we learned much.
TL;DR: It is logically prove that the time-space patterns of rule 180 are those cut off from theTime-space pattern of rule 90, and then prove the conjecture of Braga et al.
Abstract: Cellular automata have interested many researchers because of the unexpected complex time-space patterns that are generated by very simple rules of cellular automata. Especially Wolfram [5] [6] classified the cellular automata by computer simulations, and his work has inspired the research on cellular automata since then.
TL;DR: In this article, the finite element method (FEM) and the charge simulation method (CSM) were combined inside a planar exterior domain of a bounded domain, and a result of mathematical analysis for this FEM-CSM combined method was reported.
Abstract: Consider the Poisson equation −Δu= ƒ in a planar exterior domain of a bounded domainO. Assume thatƒ = 0 in the outside of a disc with sufficiently large diameter. The solutionu is assumed to be bounded at infinity. Discretizing the problem, we employ the finite element method (FEM, in short) inside the disc, and the charge simulation method (CSM, in short) outside the disc. A result of mathematical analysis for this FEM-CSM combined method is reported.
TL;DR: In this article, the authors present new results on discrete compactness of the edge elements for more general edge elements using an approach slightly different from that employed by Boffi to obtain results on the same subject.
Abstract: The Nedelec edge elements are now widely used for numerical analysis of various electromagnetic problems. However, it has not been easy to show their mathematical validity since the formulations associated with the edge elements are usually based on some mixed variational principles on special function spaces. In particular case of the simplest Nedelec simplex elements, the present author formerly showed the discrete compactness which plays essential roles in theoretical analysis of such elements. Here we present some new results on such a property for more general edge elements using an approach slightly different from that employed by Boffi to obtain results on the same subject.
TL;DR: In this article, the motion law of vortices in the limit ase → 0 of the Ginzburg-Landau equation was studied in a planar contractible domain with Neumann boundary condition.
Abstract: This paper deals with the motion law of vortices in the limit ase → 0 of the Ginzburg-Landau equationu
t
= Δu+ (1/e2)(1 − ¦u¦2),u = (u1,u2)T in a planar contractible domain with Neumann boundary condition, where the vortices are meant by zeros of a solution. As e → 0, applying the argument by Jerrard-Soner to the Neumann case yields an ordinary differential equation, called a limit equation, describing the dynamics of the vortices. We show that the limit equation can be written by using the Green function with Dirichlet condition and the Robin function of it. With this nice form we discuss the dynamics of a single or two vortices together with equilibrium states of the limit equation. In addition for the disk domain an explicit form of the equation is proposed and the dynamics for multi-vortices is investigated.
TL;DR: In this paper, the bending of a beam under relatively strong tension on an elastic foundation has been studied, and 9 different Green's functions have been proposed, all of which are positive-valued and have a hierarchical structure.
Abstract: Green’s functions to 2-point simple type self-adjoint boundary value problems for bending of a beam under relatively strong tension on an elastic foundation are studied. We have 9 different Green’s functions. All are positive-valued and have a suitable hierarchical structure.
TL;DR: In this paper, a free boundary problem which arises from the physical model called "peeling" was analyzed numerically and the fixed domain method was applied to obtain an equation which describes the motion of the free boundary.
Abstract: A free boundary problem which arises from the physical model called “peeling” will be analyzed numerically. To obtain an equation which describes the motion of the free boundary, the fixed domain method is applied. By using the equation, numerical computations are carried out. Our numerical computations suggest that the peeling speed plays an important role in the existence of global solutions.
TL;DR: A difference equation is constructed which preserves any time evolution pattern of the rule 90 Elementary cellular automaton and it is demonstrated that such difference equations can be obtained for any elementary cellular automata.
Abstract: We construct a difference equation which preserves any time evolution pattern of the rule 90 elementary cellular automaton. We also demonstrate that such difference equations can be obtained for any elementary cellular automata.
TL;DR: In this paper, the existence of travelling waves for autocatalytic reaction-diffusion systems of a non-diffusive reactant and a diffusive auto-atalyst with quadratic and cubic autocatalyses was proved.
Abstract: This paper proves the existence of travelling waves for autocatalytic reaction-diffusion systems of a non-diffusive reactant and a diffusive autocatalyst where quadratic and cubic autocatalyses occur concurrently with the ratio 1 :k. We give the estimate of the minimal speed of travelling waves which is consistent with the result obtaind by S. Focant and Th. Gallay for the systems where a reactant and an autocatalyst are both diffusive. We further discuss the value of the parameterk which assures the validity of the heuristic argument employed by J. Murray and others.
TL;DR: In this paper, a personal survey of Lagrangian chaos, the chaotic motion of fluid particles, in two-dimensional time-periodic flows and in three-dimensional steady flows is given.
Abstract: After describing the significance of the mixing of fluids by laminar flows, we give a personal survey of Lagrangian chaos, the chaotic motion of fluid particles, in two-dimensional timeperiodic flows and in three-dimensional steady flows. Next, a few studies of the chaotic mixing, mixing of fluids using Lagrangian chaos, are introduced. Finally, some results of the study of the mixing in a two-dimensional time-periodic flow based on a transport matrix are shown.
TL;DR: In this article, the limiting behavior of Seen's spiral flows for viscous incompressible fluid is investigated and the width of the interior layer is proved to be O(R−1/2), where R is the Reynolds number.
Abstract: Oseen’s spiral flows for viscous incompressible fluid are considered. Their limiting behavior as the Reynolds number tends to infinity is rigorously analyzed and the width of the interior layer is proved to be of O(R−1/2), whereR is the Reynolds number.
TL;DR: In this article, the existence of equilibrium internal layers intersecting the boundary of two dimensional bounded smooth domains was shown for the Allen-Cahn equation with balanced nonlinearity, and stability analysis was carried out for the layers and local shape of the boundary was classified according to the stability of the layers.
Abstract: For the Allen-Cahn equation with balanced nonlinearity, we show the existence of equilibrium internal layers intersecting the boundary of two dimensional bounded smooth domains. Stability analysis is carried out for the layers and local shape of the boundary is classified according to the stability of the layers. Numerical simulations are exhibited which indicate that bifurcations occur as the domain boundary is perturbed.
TL;DR: In this paper, it is shown that the boundary value problems can be reduced to a canonical form after suitable change of variables, which can be used to study the properties of radial solutions of semilinear elliptic equations in a systematic way.
Abstract: Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is shown that the boundary value problems can be reduced to a canonical form after suitable change of variables. Through the canonical form, we can study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations. We also clarify the implication of the Kelvin transformation and the Rellich-Pohozaev identity, and give their generalized forms.
TL;DR: Using the wavelet transform, this paper decomposes time series into frequency components and connects weak stationarity and prediction methods of original time series to those of each frequency component, accompanied with numeric results.
Abstract: Time series are conventionally analyzed only in the time domain or only in the frequency domain, and few analyses make use of information in both domains simultaneously. On the other hand, time series analysis based on the wavelet transform has been concentrated on the irregularity detection or the analysis of stochastic processes constructed by the wavelet transform. The wavelet transform is applied to stationarity analysis and predictions in the present paper. Using the wavelet transform, we can decompose time series into frequency components. Consequently, we can extract local information with respect to frequency. We observe the time series in both the time domain and the frequency domain simultaneously. And we connect weak stationarity and prediction methods of original time series to those of each frequency component, accompanied with numeric results.
TL;DR: In this paper, the authors estimate the term structure of credit spreads from the possibility of future defaults of firms using a discrete-state Markov chain model, constructed as a model which can be used to estimate the baseline transition matrix of the credit-rating class, recovery amount, and risk adjusting factors from yield spreads for every rating.
Abstract: For the purposes of this article, first, we will estimate the term structure of credit spreads results from the possibility of future defaults of firms. It is assumed that credit risk is specified as a discrete-state Markov chain, constructed as a model which can be used to estimate the baseline transition matrix of the credit-rating class, recovery amount, and risk-adjusting factors from yield spreads for every rating. This enables us to compute the implied term structure from market data. Next, we will provide a valuation model for downgrade protection.