9 citations · 45 across the 19 of their papers we have counts for
19 papers
Computer-Assisted Proofs in Dynamical Systems: A Case Study of a Heteroclinic Orbit in the Shimizu--Morioka System
Olivier Hénot, Akitoshi Takayasu
The radii polynomial approach is an a posteriori validation method based on the contraction of a quasi-Newton operator. We apply this strategy to give a computer-assisted proof of…
Computer-assisted proofs for finding the monodromy of Picard-Fuchs differential equations for a family of K3 toric hypersurfaces
Toshimasa Ishige, Akitoshi Takayasu
In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are…
Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs
Matthieu Cadiot, Jonathan Jaquette, Jean-Philippe Lessard +1
In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First,…
A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory
Gabriel William Duchesne, Jean-Philippe Lessard, Akitoshi Takayasu
In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular…
Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. II -- Dynamical Correspondence
Hisatoshi Kodani, Kaname Matsue, Hiroyuki Ochiai +1
In this paper, we provide a natural correspondence of eigenstructures of Jacobian matrices associated with equilibria for appropriately transformed two systems describing finite-ti…
Monodromy Approach to Pair Production of Charged Black Holes and Electric Fields
Chiang-Mei Chen, Toshimasa Ishige, Sang Pyo Kim +2
To find the pair production, absorption cross section and quasi-normal modes in background fields, we advance the monodromy method that makes use of the regular singular points of…
Complex moment-based methods for differential eigenvalue problems
Akira Imakura, Keiichi Morikuni, Akitoshi Takayasu
This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on…
Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh$\unicode{x2013}$Ritz procedure for generalized Hermitian eigenvalue problems
Akira Imakura, Keiichi Morikuni, Akitoshi Takayasu
We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses compl…
Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity
Jonathan Jaquette, Jean-Philippe Lessard, Akitoshi Takayasu
In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial…
Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature
Jean-Philippe Lessard, Kaname Matsue, Akitoshi Takayasu
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 s…
Global dynamics in nonconservative nonlinear Schrödinger equations
Jonathan Jaquette, Jean-Philippe Lessard, Akitoshi Takayasu
In this paper, we study the global dynamics of a class of nonlinear Schrödinger equations using perturbative and non-perturbative methods. We prove the semi-global existence of sol…
Rigorous numerics for nonlinear heat equations in the complex plane of time
Akitoshi Takayasu, Jean-Philippe Lessard, Jonathan Jaquette +1
In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. Using a solution…
Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems
Akira Imakura, Keiichi Morikuni, Akitoshi Takayasu
We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolv…
Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity
Kaname Matsue, Akitoshi Takayasu
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 di…
Rigorous numerical computations for 1D advection equations with variable coefficients
Akitoshi Takayasu, Suro Yoon, Yasunori Endo
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial dif…
Numerical validation of blow-up solutions with quasi-homogeneous compactifications
Kaname Matsue, Akitoshi Takayasu
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 q…
Accurate method of verified computing for solutions of semilinear heat equations
Akitoshi Takayasu, Makoto Mizuguchi, Takayuki Kubo +1
We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqu…
Numerical validation of blow-up solutions of ordinary differential equations
Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki +3
This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on c…
Verified computations for hyperbolic 3-manifolds
Neil Hoffman, Kazuhiro Ichihara, Masahide Kashiwagi +3
For a given cusped 3-manifold admitting an ideal triangulation, we describe a method to rigorously prove that either or a filling of admits a complete hyperbolic struct…