arXiv subjects

Kaname Matsue

Publications and source records attributed to Kaname Matsue.

At least 19 recordsLinked to original sources

arXiv2026

A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions

This paper presents a novel framework for enclosing solutions of Poisson's equation based on generalized sub- and super-solutions constructed using fundamental solutions. The conventional definition of sub- and super-solutions based on variational inequalities often fails for natural function classes such as piecewise linear functions and encounters theoretical difficulties in non-convex polygonal domains, where H^2 regularity is lost because of corner singularities. To overcome these limitations, we introduce the concept of ``Green-representable solutions'' utilizing test functions constructed from fundamental solutions. This framework enables a new formulation of sub- and super-solutions that permits rigorous pointwise evaluation. For one-dimensional problems, we derive explicit constructions of the test functions. For two-dimensional polygonal domains, we employ the Method of Fundamental Solutions to generate test functions. The approach is validated through numerical experiments in both settings, including non-convex polygons. The results demonstrate that the proposed method yields strict and accurate pointwise enclosures of the true solution, even for problems with discontinuous source terms or geometric singularities.

math.NA↗

arXiv2025

A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs

A simple criterion of the existence of (type-I) blow-up solutions for nonautonomous ODEs is provided. In a previous study [Matsue, SIADS, 24(2025), 415-456], geometric criteria for characterizing blow-up solutions for nonautonomous ODEs are provided by means of dynamics at infinity. The basic idea towards the present aim is to correspond such criteria to leading-term equations associated with blow-up ansatz characterizing multiple-order asymptotic expansions, which originated from the corresponding study developed in the framework of autonomous ODEs. Restricting our attention to constant coefficients of leading terms of blow-ups, results involving the simple criterion of blow-up characterizations in autonomous ODEs can be mimicked to nonautonomous ODEs.

math.DS↗

arXiv2024

Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity

We describe blow-up behavior for ODEs by means of dynamics at infinity with complex asymptotic behavior in autonomous systems, as well as in nonautonomous systems. Based on preceding studies, a variant of closed embeddings of phase spaces and the time-scale transformation determined by the structure of vector fields at infinity reduce our description of blow-ups to unravel the shadowing property of (pre)compact trajectories on the horizon, the geometric object expressing the infinity, with the specific convergence rates. Geometrically, this description is organized by asymptotic phase of invariant sets on the horizon. Blow-up solutions in nonautonomous systems can be described in a similar way. As a corollary, normally, or partially hyperbolic invariant manifolds on the horizon possessing asymptotic phase are shown to induce blow-ups.

math.DS↗

arXiv2024

Characterization of Maximizers in A Non-Convex Geometric Optimization Problem With Application to Optical Wireless Power Transfer Systems

This research studies a non-convex geometric optimization problem arising from the field of optical wireless power transfer. In the considered optimization problem, the cost function is a sum of negatively and fractionally powered distances from given points arbitrarily located in a plane to another point belonging to a different plane. Therefore, it is a strongly nonlinear and non-convex programming, hence posing a challenge on the characterization of its optimizer set, especially its set of global optimizers. To tackle this challenge, the bifurcation theory is employed to investigate the continuation and bifurcation structures of the Hessian matrix of the cost function. As such, two main results are derived. First, there is a critical distance between the two considered planes such that beyond which a unique global optimizer exists. Second, the exact number of maximizers is locally derived by the number of bifurcation branches determined via one-dimensional isotropic subgroups of a Lie group acting on $\mathbb{R}^2$, when the inter-plane distance is smaller than the above-mentioned critical distance. Consequently, numerical simulations and computations of bifurcation points are carried out for various configurations of the given points, whose results confirm the derived theoretical outcomes.

math.OC↗

arXiv2022

Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. I -- Method and Justification

In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of blow-up solutions for a class of vector fields, we extract algebraic objects determining all possible orders in the asymptotic expansions. Examples for calculating concrete multi-order asymptotic expansions of blow-up solutions are finally collected.

math.CA↗

arXiv2022

Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. II -- Dynamical Correspondence

In this paper, we provide a natural correspondence of eigenstructures of Jacobian matrices associated with equilibria for appropriately transformed two systems describing finite-time blow-ups for ODEs with quasi-homogeneity in an asymptotic sense. As a corollary, we see that asymptotic expansions of blow-ups proposed in Part I themselves provide a criterion of the existence of blow-ups with an intrinsic gap structure of stability information among two systems. Examples provided in Part I are revisited to show the above correspondence.

math.CA↗

arXiv2022

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times, and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

math.DS↗

arXiv2021

A mathematical treatment of bump structure for the particle laden flows with particle features

In this paper, we consider the particle laden flows on a inclined plane under the effect of the gravity. It is observed from preceding experimental works that the particle-rich ridge is generated near the contact line. The bump structure observed in particle-rich ridge is studied in terms of Lax's shock waves in the mathematical theory of conservation laws. In the present study, the effect of particles with nontrivial radii on morphology of particle laden flows is explicitly considered, and dependence of radius and concentration of particles on the bump structure is extracted.

physics.flu-dyn↗

arXiv2020

A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations

In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: $u_{t}=u^{p}(u_{xx}+u)-δu$ ($δ= 0$ or $1$) for $ξ\equiv x - ct \to - \infty$ with $c>0$. We give a refined one of them, which was not obtain in the preceding work [Ichida-Sakamoto, 2020], by an appropriate asymptotic study and properties of the Lambert $W$ function.

math.DS↗

arXiv2020

A crossover between open quantum random walks to quantum walks

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters $M\in \mathbb{N}$ controlling a decoherence effect; if $M=1$, the walk coincides with an open quantum random walk, while $M=\infty$, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on $\mathbb{Z}$ for $M=\infty$ and $M=1$ and we observe intermediate behavior through numerical simulations for varied positive values $M$. In the case for $M=2$, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato's perturbation theory for linear operator. We futher analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.

quant-ph↗

arXiv2019

Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of $u_t = u_{xx} + e^{u^m}$ with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.

math.NA↗

arXiv2018

Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates.

math.DS↗

arXiv2018

Rigorous numerics of tubular, conic, star-shaped neighborhoods of slow manifolds for fast-slow systems

We provide a rigorous numerical computation method to validate tubular neighborhoods of normally hyperbolic slow manifolds with the explicit radii for the fast-slow system \begin{equation*} \begin{cases} x' = f(x,y,ε), and y' =εg(x,y,ε). & \end{cases} \end{equation*} Our main focus is the validation of the continuous family of eigenpairs $\{λ_i(y;ε), u_i(y;ε)\}_{i=1}^n$ of $f_x(h_ε(y),y,ε)$ over the slow manifold $S_ε= \{x = h_ε(y)\}$ admitting the graph representation. In order to obtain such a family, we apply the interval Newton-like method with rigorous numerics. The validated family of eigenvectors generates a vector bundle over $S_ε$ determining normally hyperbolic eigendirections rigorously. The generated vector bundle enables us to construct a tubular neighborhood centered at slow manifolds with explicit radii. Combining rate conditions for providing smoothness of center-(un)stable manifolds, we can validate smooth tubular neighborhoods with diffeomorphic family of affine change of coordinates, as well as several extensions such as conic and star-shaped neighborhoods. Our procedure provides a systematic construction of smooth neighborhoods of slow manifolds in an explicit range $[0,ε_0]$ of $ε$ with rigorous numerics.

math.DS↗

arXiv2017

Quantum Search on Simplicial Complexes

In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we newly define a quantum walk on simplicial complex which is an alternative of preceding studies by authors. We show that the quantum search on the specific simplicial complex corresponding to the triangulation of $n$-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability $1+o(1)$ with in a time $O(\sqrt{N})$, where $N$ is the number of simplices with the dimension of marked simplex.

math-ph↗

arXiv2017

Resonant-tunneling in discrete-time quantum walk

We show that discrete-time quantum walks on the line, $\mathbb{Z}$, behave as "the quantum tunneling". In particular, quantum walkers can tunnel through a double-well with the transmission probability $1$ under a mild condition. This is a property of quantum walks which cannot be seen on classical random walks, and is different from both linear spreadings and localizations.

quant-ph↗

arXiv2017

Rigorous numerics of finite-time singularities in dynamical systems - methodology and applications

This paper aims at providing rigorous numerical computation procedure for finite-time singularities in dynamical systems. Combination of time-scale desingularization as well as Lyapunov functions validation on stable manifolds of invariant sets for desingularized vector fields with standard integration procedure for ordinary differential equations give us validated trajectories of dynamical systems involving finite-time singularities. Our focus includes finite-time extinction, traveling wave solutions with half-line or compact support, and singular canards in fast-slow systems, including rigorous validations of enclosures of extinction, finite-passage times or size of supports for compactons. Such validated solutions lead to a plenty of composite wave solutions for degenerate parabolic equations, for example, with concrete information of profiles and evolutions. The present procedure also provides a universal aspect of finite-time singularities with rigorous numerics, combining with rigorous numerics of blow-up solutions in preceding works.

math.NA↗

arXiv2017

Toward a mathematical analysis for a model of suspension flowing down an inclined plane

We consider the Riemann problem of the dilute approximation equations with spatiotemporally dependent volume fractions from the full model of suspension, in which the particles settle to the solid substrate and the clear liquid film flows over the sediment [Murisic et al., J. Fluid. Mech. {\bf{717}}, 203--231 (2013)]. We present a method to find shock waves, rarefaction waves for the Riemann problem of this system. Our method is mainly based on [Smoller, Springer-Verlag, New York, second edition, (1994)].

math.AP↗

arXiv2017

Numerical validation of blow-up solutions with quasi-homogeneous compactifications

We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing a new compactification, which shall be called a quasi-parabolic compactification. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up times as well as rigorous profile enclosures of blow-up solutions.

math.NA↗