Estimate of the Hausdorff Dimension of a Self-Similar Set due to Weak Contractions

As for the remarkable study on the estimate of the Hausdorff dimension of a self-similar set due to weak contractions (Kitada A. et al. Chaos, Solitons & Fractals 13 (2002) 363-366), we present a mathematically simplified form which will be more applicable to various phenomena.Comment: 5 page

Numerical verification method for positiveness of solutions to elliptic equations

In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic equations. We provide a sufficient condition for a solution to an elliptic equation to be positive in the domain of the equation, which can be checked numerically without requiring a complicated computation. We present some numerical examples.Comment: 16 pages and 2 figure

Estimation of the Sobolev embedding constant on domains with minimally smooth boundary

In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains.Comment: 22 pages, 5 figure

Numerical method for deriving sharp inclusion of the Sobolev embedding constant on bounded convex domain

In this paper we proposed a verified numerical method for deriving a sharp inclusion of the Sobolev embedding constant from H^1_0 to L^p on bounded convex domain in R^2. We estimated the embedding constant by computing the corresponding extremal function using verified numerical computation. Some concrete numerical inclusions of the constant on a square domain were presented.Comment: 8 pages, 1 figur

Verified numerical computation for semilinear elliptic problems with lack of Lipschitz continuity of the first derivative

In this paper, we propose a numerical method for verifying solutions to the semilinear elliptic equation -{\Delta}u=f(u) with homogeneous Dirichlet boundary condition. In particular, we consider the case in which the Fr\'echet derivative of f is not Lipschitz continuous. A numerical example for a concrete nonlinearity is presented.Comment: 20 pages, 2 figures. arXiv admin note: text overlap with arXiv:1606.0381

Verified computations for hyperbolic 3-manifolds

For a given cusped 3-manifold $M$ admitting an ideal triangulation, we describe a method to rigorously prove that either $M$ or a filling of $M$ admits a complete hyperbolic structure via verified computer calculations. Central to our method are an implementation of interval arithmetic and Krawczyk's Test. These techniques represent an improvement over existing algorithms as they are faster, while accounting for error accumulation in a more direct and user friendly way.Comment: 27 pages, 3 figures. Version 2 has minor changes, mostly attributed to a small simplification of the code associated to this paper and the correction of typographical error

A new formulation for the numerical proof of the existence of solutions to elliptic problems

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {\mathcal L}^{-1} {\mathcal F}(\hat{u}) + {\mathcal L}^{-1} {\mathcal G}(w)$, where ${\mathcal L}$ is a linearized operator, ${\mathcal F}(\hat{u})$ is a residual, and ${\mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $\| {\mathcal L}^{-1} {\mathcal F}(\hat{u}) \|$ and $\| {\mathcal L}^{-1}{\mathcal G}(w) \|$ play major roles in the verification procedures. In this paper, using a similar concept as the `Schur complement' for matrix problems, we represent the inverse operator ${\mathcal L}^{-1}$ as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as ${\mathcal L}^{-1}$ are presented in the appendix.Comment: 16 page, 1 figur

Numerical validation of blow-up solutions of ordinary differential equations

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 compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.Comment: Accepted version, to appear in Journal of Computational and Applied Mathematic

Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.Comment: 8 table

Numerical verification for asymmetric solutions of the H\'enon equation on the unit square

The H\'enon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We numerically prove the existence of asymmetric solutions of the H\'enon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of solutions with three peaks. The bifurcation curves of such solutions are shown for a square domain.Comment: 13 pages, 1 figur