NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

DOI10.1081/NFA-100105107zbMath1106.65315OpenAlexW2066629027MaRDI QIDQ2769532

Mitsuhiro T. Nakao

Publication date: 5 February 2002

Published in: Numerical Functional Analysis and Optimization (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1081/nfa-100105107

Mathematics Subject Classification ID

Algorithms with automatic result verification (65G20)

Related Items (52)

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations ⋮ Microscopic patterns in the 2D phase-field-crystal model ⋮ A general method for computer-assisted proofs of periodic solutions in delay differential problems ⋮ Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach ⋮ Some remarks on the behaviour of the finite element solution in nonsmooth domains ⋮ A posteriori error bounds for two point boundary value problems: a Green's function approach ⋮ A numerical verification method for two-coupled elliptic partial differential equations ⋮ An application of Taylor models to the Nakao method on ODEs ⋮ Numerical verification method of solutions for elliptic equations and its application to the Rayleigh-Bénard problem ⋮ Numerical verification method for infinite dimensional eigenvalue problems ⋮ Introduction to rigorous numerics in dynamics: General functional analytic setup and an example that forces chaos ⋮ Rigorous continuation of bifurcation points in the diblock copolymer equation ⋮ Some improvements of invertibility verifications for second-order linear elliptic operators ⋮ Rigorous computation of non-uniform patterns for the 2-dimensional Gray-Scott reaction-diffusion equation ⋮ Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains ⋮ Saddle-type blow-up solutions with computer-assisted proofs: validation and extraction of global nature ⋮ Computer assisted proofs of bifurcating solutions for nonlinear heat convection problems ⋮ Constructive proofs for localised radial solutions of semilinear elliptic systems on Rd ⋮ The error term approximation in interval method of Crank-Nicolson type ⋮ Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator ⋮ Periodic orbits in Hořava-Lifshitz cosmologies ⋮ Interval versions of central-difference method for solving the Poisson equation in proper and directed interval arithmetic ⋮ Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation ⋮ Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation ⋮ Error analysis of Galerkin's method for semilinear equations ⋮ Efficient and generic algorithm for rigorous integration forward in time of dPDEs. I ⋮ A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator ⋮ Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications ⋮ Computer-assisted proofs for radially symmetric solutions of PDEs ⋮ Computational fixed-point theory for differential delay equations with multiple time lags ⋮ An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces ⋮ Numerical Verification Methods for Solutions of the Free Boundary Problem ⋮ A Numerical Verification Method for a System of FitzHugh-Nagumo Type ⋮ Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions ⋮ Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields ⋮ A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems ⋮ A numerical verification method for solutions of singularly perturbed problems with nonlinearity ⋮ Spontaneous periodic orbits in the Navier-Stokes flow ⋮ Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach ⋮ Verified error bounds for multiple roots of systems of nonlinear equations ⋮ Sharp numerical inclusion of the best constant for embedding \(H_0^1(\Omega) \hookrightarrow L^p(\Omega)\) on bounded convex domain ⋮ A numerical proof algorithm for the non-existence of solutions to elliptic boundary value problems ⋮ On theL²a PrioriError Estimates to the Finite Element Solution of Elliptic Problems with Singular Adjoint Operator ⋮ A symbolic-numeric validation algorithm for linear ODEs with Newton-Picard method ⋮ Computer-assisted proofs in PDE: a survey ⋮ Global dynamics in nonconservative nonlinear Schrödinger equations ⋮ Numerical verification of positiveness for solutions to semilinear elliptic problems ⋮ Inclusion method of optimal constant with quadratic convergence for \(H_0^1\)-projection error estimates and its applications ⋮ Rigorous validation of stochastic transition paths ⋮ Some considerations of the invertibility verifications for linear elliptic operators ⋮ A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem ⋮ Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Scientific computing with automatic result verification
• Numerical verifications of solutions for nonlinear elliptic equations
• Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method
• Computer-assisted existence proofs for two-point boundary value problems
• Solving nonlinear parabolic problems with result verification. I: One- space dimensional case
• Bounds for eigenvalues of second-order elliptic differential operators
• A numerical verification method for the existence of weak solutions for nonlinear boundary value problems
• Explicit \(H_ 2\)-estimates and pointwise bounds for solutions of second- order elliptic boundary value problems
• Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems
• The construction of a priori bounds for the solution of a two point boundary value problem with finite elements. I
• On the best constant in the error bound for the \(H_0^1\)-projection into piecewise polynomial spaces
• Numerical verification of solutions for variational inequalities
• A numerically based existence theorem for the Navier-Stokes equations
• A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations
• Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains
• On computational proofs of the existence of solutions to nonlinear parabolic problems
• Numerical verifications of solutions for nonlinear parabolic equations in one-space dimensional case
• Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points
• Numerical verification of solutions for nonlinear hyperbolic equations
• Numerical verification of solutions for nonlinear elliptic problems using an \(L^\infty\) residual method
• Numerical verification of solutions for elasto-plastic torsion problems
• Numerical verification of solutions for a simplified Signorini problem
• A numerical method to verify the elliptic eigenvalue problems including a uniqueness property
• Discovery of closed orbits of dynamical systems with the use of computers
• Numerical verification of existence and inclusion of solutions for nonlinear operator equations
• Existence and enclosure results for continua of solutions of parameter- dependent nonlinear boundary value problems
• Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element
• A numerically rigorous proof of curve veering in an eigenvalue problem for differential equations
• A numerical verification method of solutions for the Navier-Stokes equations
• Numerical verifications for eigenvalues of second-order elliptic operators
• Galerkin's procedure for nonlinear periodic systems
• Aufgaben monotoner Art
• A numerical approach to the proof of existence of solutions for elliptic problems II
• A numerical approach to the proof of existence of solutions for elliptic problems
• Computable Error Bounds in the Finite-Element Method
• A Posteriori Error Bounds for Two-Point Boundary Value Problems
• Interval Methods for Systems of Equations
• Computable a Posteriori $L_\infty $-Error Bounds for the Approximate Solution of Two-Point Boundary Value Problems
• A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem
• An Approach to The Numerical Verification of Solutions for Nonlinear Elliptic Problems With Local Uniqueness
• Chaos in the Lorenz equations: A computer assisted proof. Part II: Details
• Numerical verifications of solutions for elliptic equations with strong nonlinearity
• Computer-assisted enclosure methods for elliptic differential equations

This page was built for publication: NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS