Numerical verification methods for solutions of ordinary and partial differential equations

Numerical Functional Analysis and Optimization

Volumn 22, Issue 3-4, 2001, Pages 321-356

  Nakao, M T ^a  

^a KYUSHU UNIVERSITY   (Japan)

Author keywords

Computer assisted proof in analysis; Nonlinear differential equation; Numerical verification

Indexed keywords

BOUNDARY VALUE PROBLEMS; EIGENVALUES AND EIGENFUNCTIONS; NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

COMPUTER ASSISTED PROOF IN ANALYSIS; NUMERICAL VERIFICATION METHODS;

NUMERICAL METHODS;

EID: 0035326665     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1081/NFA-100105107     Document Type: Article

References (92)

1. The determination of guaranteed bounds to eigenvalues with the use of variational methods II
2. A numerically rigorous proof of curve veering in an eigenvalue problem for differential equations
3. On the eigenvalue problem of the rectangular clamped plate
4. Functional analysis and periodic solutions for one dimensional problems, contrib.
5. Aufgaben monotoner art
6. The determination of guaranteed bounds to eigenvalues with the use of variational methods I
7. A numerically based existence theorem for the Navier-Stokes equations
8. Generalization of bifurcation theorems aiming for the study of nonlinear forced vibrations described by PDEs
9. A posterior error bounds for two-point boundary value problems
10. Computable error bounds in the finite-element method
11. Enclosing the solutions of ordinary initial and boundary value problems
12. ∞-error bounds for the approximate solution of two-point boundary value problems
13. Numerical verifications of solutions for nonlinear parabolic equations in one-space dimensional case
14. Numerical verifications of solutions for nonlinear hyperbolic equations in one-space dimensional case
15. Numerical verification method for solutions of the perturbed Gelfand equation
16. Numerical existence and uniqueness proof for solutions of semilinear parabolic equations
17. Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations
18. Chaos in the Lorenz equations: A computer assisted proof
19. An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness
20. A numerical method to verify the elliptic eigenvalue problems including a uniqueness property
21. An enclosure method of eigenvalues for the elliptic operator linearlized at an exact solution of nonlinear problems
22. A numerical approach to the proof of existence of solutions for elliptic problems
23. A computational verification method of existence of solutions for nonlinear elliptic equations
24. A numerical approach to the proof of existence of solutions for elliptic problems II
25. Solving nonlinear parabolic problems with result verification: Part I
26. A numerical verification method for the existence of weak solutions for nonlinear boundary value problems
27. -1 residual iteration
28. ∞ error estimates in the finite element method with application to nonlinear elliptic problems
29. Numerical verifications of solutions for elliptic equations with strong nonlinearity
30. Numerical verifications of solutions for nonlinear hyperbolic equations
31. On computational proofs of the existence of solutions to nonlinear parabolic problems
32. A simplified method of numerical verification for nonlinear elliptic equations
33. Guaranteed error bounds for finite element solutions of the stokes problem
34. ∞ residual method
35. 1-projection into piecewise polynomial spaces
36. Numerical verification of the solution curve for some parameterized nonlinear elliptic problem
37. Numerical verifications of eigenvalues of second-order elliptic operators
38. A posteriori and constructive a priori error bounds for finite element solutions of stokes equations
39. Numerical verification of solutions for elasto-plastic torsion problems
40. A numerical verification of bifurcated solutions for the heat convection problems
41. Computer assisted enclosure of the optimal constant in the error estimation of a finite element projection
42. Berechenbare fehlerschranken für die finite elemente methode
43. The construction of a priori bounds for the solution of a two point boundary value problem with finite elements I
44. Bifurcation problems for equations of fluid dynamics and computer assisted proof
45. Two topics in nonlinear system analysis through fixed point theorems
46. Numerical verification of existence and inclusion of solutions for nonlinear operator equations
47. Numerical analysis with guaranteed error bounds of connecting orbits in continuous dynamics
48. Method of numerical inclusion for solutions of fixed point type equations
49. Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method
50. Computer-assisted existence proofs for two-point boundary value problems
51. Bounds for eigenvalues of second-order elliptic differential operators
52. Existence proofs in combination with error bounds for approximate solutions of weakly nonlinear second-order elliptic boundary value problems
53. 2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems
54. Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems
55. Enclosures for weak solutions of nonlinear elliptic boundary value problems
56. Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems
57. Enclosures for two-point boundary value problems near bifurcation points
58. Guaranteed numerical bounds for eigenvalues
59. Computer assisted enclosure methods for elliptic differential equations
60. Interval methods for initial value problems in ODE's
61. Solving algebraic problems with high accuracy, a new approach to scientific computation
62. Verification methods for dense and sparse systems of equations
63. Numerical verification of solutions for variational inequalities
64. Numerical verifications of solutions for simplified Signorini problems
65. A method for producing verified results for two-point boundary value problems
66. Operator inequalities and applications
67. Discovery of closed orbits of dynamical systems with the use of computers
68. Validated solution of initial value problems for ODE
69. Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points
70. Numerical verification of simple bifurcation points
71. Galerkin's procedure for nonlinear periodic systems
72. The Newton method and its application to boundary value problems with nonlinear boundary conditions
73. Numerical verifications of solutions for nonlinear elliptic equations
74. Verified computation of solutions for nondifferentiable elliptic equations related to MHD equilibria
75. A numerical verification method of solutions for the Navier-Stokes equations
76. Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains
77. Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite element
78. A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed point theorem
79. Validated computation of polynomial zeros by Durand-Kerner method