A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

@article{Mizuguchi2017AMO,
  title={A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory},
  author={Makoto Mizuguchi and Akitoshi Takayasu and Takayuki Kubo and Shin'ichi Oishi},
  journal={SIAM J. Numer. Anal.},
  year={2017},
  volume={55},
  pages={980-1001}
}
This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived… 

Figures and Tables from this paper

Accurate method of verified computing for solutions of semilinear heat equations

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact

Rigorous numerics for nonlinear heat equations in the complex plane of time

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 map operator, we

Rigorous numerical computations for 1D advection equations with variable coefficients

TLDR
The provided method is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions as well as a foundational approach of verified numerical computations for hyperbolic PDEs.

Pointwise a Posteriori Error Bounds for Blow-Up in the Semilinear Heat Equation

TLDR
Conditional a posteriori error bounds are derived in the first order in time, implicit-explicit (IMEX), conforming finite element method in space discretization of the problem.

Validated forward integration scheme for parabolic PDEs via Chebyshev series

A Rigorous Implicit $$C^1$$ C 1 Chebyshev Integrator for Delay Equations

We present a new approach to validated numerical integration for systems of delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with

Rigorous FEM for 1D Burgers equation

We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and

Numerical verification for positive solutions of Allen–Cahn equation using sub- and super-solution method

This paper describes a numerical verification method for positive solutions of the Allen–Cahn equation on the basis of the suband super-solution method. Our application range extends to

Evolutional Equations

References

SHOWING 1-10 OF 46 REFERENCES

A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem

In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a

A numerical approach to the proof of existence of solutions for elliptic problems

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the

On the finite element method for parabolic equations, I; approximation of holomorphic semi-groups1)

In the present paper and a few papers to follow, we shall make an operator theoretical study of the finite element method applied to the initial boundary value problems for partial differential

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

TLDR
This work considers the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions and derives an a posterioru estimate of the norm for the infinite-dimensional operator.

A numerical approach to the proof of existence of solutions for elliptic problems II

This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder’s fixed point theorem, of the existence of weak solutions for

Numerical verification methods for solutions of semilinear elliptic boundary value problems

This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here “numerical

Constructive A Priori Error Estimates for a Full Discrete Approximation of the Heat Equation

TLDR
The constructive a priori error estimates for a full discrete numerical solution of the heat equation based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space are considered.

Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the NewtonKantorovich theorem, our

On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems

TLDR
A new technique for obtaining the estimates of the inverse operator by using the finite dimensional approximation and error estimates enables us to obtain very sharp bounds compared with a priori estimates.