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A posteriori error bounds for two point boundary value problems: A green's function approach
1. | Program in Applied Mathematics, The University of Arizona, Tucson, Arizona 85721, United States |
References:
[1] |
D. Ambrosi, G. Arioli and H. Koch, A homoclinic solution for excitation waves on a contractile substratum,, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1533.
doi: 10.1137/12087654X. |
[2] |
G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation,, Nonlinear Analysis: Theory, 113 (2015), 51.
doi: 10.1016/j.na.2014.09.023. |
[3] |
U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, Society for Industrial and Applied Mathematics (SIAM, (3600).
doi: 10.1137/1.9781611971392. |
[4] |
P.-O. Åsén, Stability of Plane Couette Flow and Pipe Poiseuille Flow,, PhD thesis, (2007). Google Scholar |
[5] |
B. A. Coomes, H. Koçak and K. J. Palmer, Rigorous computational shadowing of orbits of ordinary differential equations,, Numerische Mathematik, 69 (1995), 401.
doi: 10.1007/s002110050100. |
[6] |
Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis: An Introduction,, Graduate studies in mathematics, (2004).
doi: 10.1090/gsm/066. |
[7] |
C. Fefferman and R. de la Llave, Relativistic stability of matter-I,, I. Rev. Mat. Iberoamericana, 2 (1986), 119.
doi: 10.4171/RMI/30. |
[8] |
O. Fogelklou, W. Tucker and G. Kreiss, A computer-assisted proof of the existence of traveling wave solutions to the scalar euler equations with artificial viscosity,, Nonlinear Differential Equations and Applications NoDEA, 19 (2012), 97.
doi: 10.1007/s00030-011-0120-7. |
[9] |
O. Fogelklou, W. Tucker, G. Kreiss and M. Siklosi, A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition,, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1227.
doi: 10.1016/j.cnsns.2010.07.008. |
[10] |
G. Folland, Real Analysis: Modern Techniques and Their Applications,, A Wiley-Interscience Publication. John Wiley & Sons, (1984).
|
[11] |
Z. Galias and P. Zgliczyski, Computer assisted proof of chaos in the Lorenz equations,, Physica D: Nonlinear Phenomena, 115 (1998), 165.
doi: 10.1016/S0167-2789(97)00233-9. |
[12] |
M. Göhlen, M. Plum and J. Schröder, A programmed algorithm for existence proofs for two-point boundary value problems,, Computing, 44 (1990), 91.
doi: 10.1007/BF02241862. |
[13] |
A. Hungria, J.-P. Lessard and J. D. M. James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach,, Math. Comp., 85 (2016), 1427.
doi: 10.1090/mcom/3046. |
[14] |
L. Jaulin, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics,, no. v. 1 in Applied Interval Analysis: With Examples in Parameter and State Estimation, (2001).
doi: 10.1007/978-1-4471-0249-6. |
[15] |
L. Kantorovich and G. Akilov, Functional Analysis in Normed Spaces,, International series of monographs in pure and applied mathematics, (1964).
|
[16] |
R. B. Kearfott, Interval computations: Introduction, uses, and resources,, Euromath Bulletin, 2 (1996), 95.
|
[17] |
G. Kedem, A posteriori error bounds for two-point boundary value problems,, SIAM Journal on Numerical Analysis, 18 (1981), 431.
doi: 10.1137/0718028. |
[18] |
H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study,, SIAM Review, 38 (1996), 565.
doi: 10.1137/S0036144595284180. |
[19] |
J. Lee and L. Greengard, A fast adaptive numerical method for stiff two-point boundary value problems,, SIAM Journal on Scientific Computing, 18 (1997), 403.
doi: 10.1137/S1064827594272797. |
[20] |
J.-P. Lessard, J. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields,, Journal of Dynamics and Differential Equations, 26 (2014), 267.
doi: 10.1007/s10884-014-9367-0. |
[21] |
J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series,, SIAM Journal on Numerical Analysis, 52 (2014), 1.
doi: 10.1137/13090883X. |
[22] |
M. A. McCarthy and R. A. Tapia, Computable a posteriori $L_\infty$-error bounds for the approximate solution of two-point boundary value problems,, SIAM Journal on Numerical Analysis, 12 (1975), 919.
doi: 10.1137/0712068. |
[23] |
R. Moore, Methods and Applications of Interval Analysis,, Studies in Applied and Numerical Mathematics, (1979).
|
[24] |
R. Moore, R. Kearfott and M. Cloud, Introduction to Interval Analysis,, Cambridge University Press, (2009).
doi: 10.1137/1.9780898717716. |
[25] |
K. Nagatou, N. Yamamoto and M. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness,, Numerical Functional Analysis and Optimization, 20 (1999), 543.
doi: 10.1080/01630569908816910. |
[26] |
M. T. Nakao, Solving nonlinear parabolic problems with result verification. Part I: One-space dimensional case,, Journal of Computational and Applied Mathematics, 38 (1991), 323.
doi: 10.1016/0377-0427(91)90179-N. |
[27] |
M. T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems,, Journal of Mathematical Analysis and Applications, 164 (1992), 489.
doi: 10.1016/0022-247X(92)90129-2. |
[28] |
M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations,, Numerical Functional Analysis and Optimization, 22 (2001), 321.
doi: 10.1081/NFA-100105107. |
[29] |
N. Nedialkov, K. Jackson and G. Corliss, Validated solutions of initial value problems for ordinary differential equations,, Applied Mathematics and Computation, 105 (1999), 21.
doi: 10.1016/S0096-3003(98)10083-8. |
[30] |
M. Plum, Computer-assisted existence proofs for two-point boundary value problems,, Computing, 46 (1991), 19.
doi: 10.1007/BF02239009. |
[31] |
M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance,, Jahresbericht der Deutschen Mathematiker Vereinigung, 110 (2008), 19.
|
[32] |
S. Rump, INTLAB - INTerval LABoratory,, in Developments in Reliable Computing (ed. T. Csendes), (1999), 77.
doi: 10.1007/978-94-017-1247-7_7. |
[33] |
A. Takayasu, S. Oishi and T. Kubo, Numerical existence theorem for solutions of two-point boundary value problems of nonlinear differential equations,, Nonlinear Theory and Its Applications, 1 (2010), 105.
doi: 10.1587/nolta.1.105. |
[34] |
W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations,, Princeton University Press, (2011).
|
[35] |
W. Tucker, The Lorenz attractor exists,, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 1197.
doi: 10.1016/S0764-4442(99)80439-X. |
[36] |
M. Urabe, Galerkin's procedure for nonlinear periodic systems,, Archive for Rational Mechanics and Analysis, 20 (1965), 120.
doi: 10.1007/BF00284614. |
[37] |
J. van den Berg, J. Mireles-James, J. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation,, SIAM Journal on Mathematical Analysis, 43 (2011), 1557.
doi: 10.1137/100812008. |
[38] |
J. B. van den Berg, C. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 423.
doi: 10.1137/140987973. |
[39] |
Y. Watanabe, M. Plum and M. T. Nakao, A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow,, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 89 (2009), 5.
doi: 10.1002/zamm.200700158. |
[40] |
N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem,, SIAM Journal on Numerical Analysis, 35 (1998), 2004.
doi: 10.1137/S0036142996304498. |
[41] |
P. Zgliczynski, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243.
doi: 10.1088/0951-7715/10/1/016. |
show all references
References:
[1] |
D. Ambrosi, G. Arioli and H. Koch, A homoclinic solution for excitation waves on a contractile substratum,, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1533.
doi: 10.1137/12087654X. |
[2] |
G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation,, Nonlinear Analysis: Theory, 113 (2015), 51.
doi: 10.1016/j.na.2014.09.023. |
[3] |
U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations,, Society for Industrial and Applied Mathematics (SIAM, (3600).
doi: 10.1137/1.9781611971392. |
[4] |
P.-O. Åsén, Stability of Plane Couette Flow and Pipe Poiseuille Flow,, PhD thesis, (2007). Google Scholar |
[5] |
B. A. Coomes, H. Koçak and K. J. Palmer, Rigorous computational shadowing of orbits of ordinary differential equations,, Numerische Mathematik, 69 (1995), 401.
doi: 10.1007/s002110050100. |
[6] |
Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis: An Introduction,, Graduate studies in mathematics, (2004).
doi: 10.1090/gsm/066. |
[7] |
C. Fefferman and R. de la Llave, Relativistic stability of matter-I,, I. Rev. Mat. Iberoamericana, 2 (1986), 119.
doi: 10.4171/RMI/30. |
[8] |
O. Fogelklou, W. Tucker and G. Kreiss, A computer-assisted proof of the existence of traveling wave solutions to the scalar euler equations with artificial viscosity,, Nonlinear Differential Equations and Applications NoDEA, 19 (2012), 97.
doi: 10.1007/s00030-011-0120-7. |
[9] |
O. Fogelklou, W. Tucker, G. Kreiss and M. Siklosi, A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition,, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1227.
doi: 10.1016/j.cnsns.2010.07.008. |
[10] |
G. Folland, Real Analysis: Modern Techniques and Their Applications,, A Wiley-Interscience Publication. John Wiley & Sons, (1984).
|
[11] |
Z. Galias and P. Zgliczyski, Computer assisted proof of chaos in the Lorenz equations,, Physica D: Nonlinear Phenomena, 115 (1998), 165.
doi: 10.1016/S0167-2789(97)00233-9. |
[12] |
M. Göhlen, M. Plum and J. Schröder, A programmed algorithm for existence proofs for two-point boundary value problems,, Computing, 44 (1990), 91.
doi: 10.1007/BF02241862. |
[13] |
A. Hungria, J.-P. Lessard and J. D. M. James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach,, Math. Comp., 85 (2016), 1427.
doi: 10.1090/mcom/3046. |
[14] |
L. Jaulin, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics,, no. v. 1 in Applied Interval Analysis: With Examples in Parameter and State Estimation, (2001).
doi: 10.1007/978-1-4471-0249-6. |
[15] |
L. Kantorovich and G. Akilov, Functional Analysis in Normed Spaces,, International series of monographs in pure and applied mathematics, (1964).
|
[16] |
R. B. Kearfott, Interval computations: Introduction, uses, and resources,, Euromath Bulletin, 2 (1996), 95.
|
[17] |
G. Kedem, A posteriori error bounds for two-point boundary value problems,, SIAM Journal on Numerical Analysis, 18 (1981), 431.
doi: 10.1137/0718028. |
[18] |
H. Koch, A. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study,, SIAM Review, 38 (1996), 565.
doi: 10.1137/S0036144595284180. |
[19] |
J. Lee and L. Greengard, A fast adaptive numerical method for stiff two-point boundary value problems,, SIAM Journal on Scientific Computing, 18 (1997), 403.
doi: 10.1137/S1064827594272797. |
[20] |
J.-P. Lessard, J. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields,, Journal of Dynamics and Differential Equations, 26 (2014), 267.
doi: 10.1007/s10884-014-9367-0. |
[21] |
J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series,, SIAM Journal on Numerical Analysis, 52 (2014), 1.
doi: 10.1137/13090883X. |
[22] |
M. A. McCarthy and R. A. Tapia, Computable a posteriori $L_\infty$-error bounds for the approximate solution of two-point boundary value problems,, SIAM Journal on Numerical Analysis, 12 (1975), 919.
doi: 10.1137/0712068. |
[23] |
R. Moore, Methods and Applications of Interval Analysis,, Studies in Applied and Numerical Mathematics, (1979).
|
[24] |
R. Moore, R. Kearfott and M. Cloud, Introduction to Interval Analysis,, Cambridge University Press, (2009).
doi: 10.1137/1.9780898717716. |
[25] |
K. Nagatou, N. Yamamoto and M. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness,, Numerical Functional Analysis and Optimization, 20 (1999), 543.
doi: 10.1080/01630569908816910. |
[26] |
M. T. Nakao, Solving nonlinear parabolic problems with result verification. Part I: One-space dimensional case,, Journal of Computational and Applied Mathematics, 38 (1991), 323.
doi: 10.1016/0377-0427(91)90179-N. |
[27] |
M. T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems,, Journal of Mathematical Analysis and Applications, 164 (1992), 489.
doi: 10.1016/0022-247X(92)90129-2. |
[28] |
M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations,, Numerical Functional Analysis and Optimization, 22 (2001), 321.
doi: 10.1081/NFA-100105107. |
[29] |
N. Nedialkov, K. Jackson and G. Corliss, Validated solutions of initial value problems for ordinary differential equations,, Applied Mathematics and Computation, 105 (1999), 21.
doi: 10.1016/S0096-3003(98)10083-8. |
[30] |
M. Plum, Computer-assisted existence proofs for two-point boundary value problems,, Computing, 46 (1991), 19.
doi: 10.1007/BF02239009. |
[31] |
M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance,, Jahresbericht der Deutschen Mathematiker Vereinigung, 110 (2008), 19.
|
[32] |
S. Rump, INTLAB - INTerval LABoratory,, in Developments in Reliable Computing (ed. T. Csendes), (1999), 77.
doi: 10.1007/978-94-017-1247-7_7. |
[33] |
A. Takayasu, S. Oishi and T. Kubo, Numerical existence theorem for solutions of two-point boundary value problems of nonlinear differential equations,, Nonlinear Theory and Its Applications, 1 (2010), 105.
doi: 10.1587/nolta.1.105. |
[34] |
W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations,, Princeton University Press, (2011).
|
[35] |
W. Tucker, The Lorenz attractor exists,, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 1197.
doi: 10.1016/S0764-4442(99)80439-X. |
[36] |
M. Urabe, Galerkin's procedure for nonlinear periodic systems,, Archive for Rational Mechanics and Analysis, 20 (1965), 120.
doi: 10.1007/BF00284614. |
[37] |
J. van den Berg, J. Mireles-James, J. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation,, SIAM Journal on Mathematical Analysis, 43 (2011), 1557.
doi: 10.1137/100812008. |
[38] |
J. B. van den Berg, C. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem,, SIAM Journal on Applied Dynamical Systems, 14 (2015), 423.
doi: 10.1137/140987973. |
[39] |
Y. Watanabe, M. Plum and M. T. Nakao, A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow,, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 89 (2009), 5.
doi: 10.1002/zamm.200700158. |
[40] |
N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem,, SIAM Journal on Numerical Analysis, 35 (1998), 2004.
doi: 10.1137/S0036142996304498. |
[41] |
P. Zgliczynski, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243.
doi: 10.1088/0951-7715/10/1/016. |
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