A posteriori error bounds for two point boundary value problems: A green's function approach

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June 2015, 2(2): 143-164. doi: 10.3934/jcd.2015001

A posteriori error bounds for two point boundary value problems: A green's function approach

Jeremiah Birrell ^1,

1.	Program in Applied Mathematics, The University of Arizona, Tucson, Arizona 85721, United States

Received January 2015 Revised August 2015 Published May 2016

Abstract
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We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

Keywords: a posteriori error analysis, computer assisted proofs, singular perturbations., periodic orbits, Two point boundary value problems.

Mathematics Subject Classification: Primary: 37C27, 65G20, 34B15; Secondary: 34B27, 65L10, 65L1.

Citation: Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

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), 1533-1542. 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, Methods & Applications, 113 (2015), 51-70. 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 Market Street, Floor 6, Philadelphia, PA 19104), 1998. doi: 10.1137/1.9781611971392.
[4]	P.-O. Åsén, Stability of Plane Couette Flow and Pipe Poiseuille Flow, PhD thesis, KTH, Numerical Analysis and Computer Science, NADA, 2007, QC 20100825.
[5]	B. A. Coomes, H. Koçak and K. J. Palmer, Rigorous computational shadowing of orbits of ordinary differential equations, Numerische Mathematik, 69 (1995), 401-421. doi: 10.1007/s002110050100.
[6]	Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis: An Introduction, Graduate studies in mathematics, American Mathematical Society, 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), 119-213. 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), 97-131. 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), 1227-1243. 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, Inc., New York, 1984, URL http://books.google.com/books?id=wI4fAwAAQBAJ.
[11]	Z. Galias and P. Zgliczyski, Computer assisted proof of chaos in the Lorenz equations, Physica D: Nonlinear Phenomena, 115 (1998), 165-188. 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), 91-132. 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), 1427-1459, URL http://archimede.mat.ulaval.ca/jplessard/Publications_files/final_draft.pdf. 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, Robust Control and Robotics, Springer London, 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, Pergamon Press; [distributed in the Western Hemisphere by Macmillan, New York], 1964, URL http://books.google.com/books?id=g_tQAAAAMAAJ.
[16]	R. B. Kearfott, Interval computations: Introduction, uses, and resources, Euromath Bulletin, 2 (1996), 95-112.
[17]	G. Kedem, A posteriori error bounds for two-point boundary value problems, SIAM Journal on Numerical Analysis, 18 (1981), 431-448. 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), 565-604. 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), 403-429. 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), 267-313. 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), 1-22. 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), 919-937. doi: 10.1137/0712068.
[23]	R. Moore, Methods and Applications of Interval Analysis, Studies in Applied and Numerical Mathematics, Society for Industrial and Applied Mathematics, 1979, URL http://books.google.com/books?id=WYjD2-R2zMgC.
[24]	R. Moore, R. Kearfott and M. Cloud, Introduction to Interval Analysis, Cambridge University Press, 2009, URL http://books.google.com/books?id=kd8FmmN7sAoC. 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), 543-565. 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), 323-334. 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), 489-507. 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), 321-356. 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), 21-68. doi: 10.1016/S0096-3003(98)10083-8.
[30]	M. Plum, Computer-assisted existence proofs for two-point boundary value problems, Computing, 46 (1991), 19-34. 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), 19-54.
[32]	S. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77-104, URL http://www.ti3.tuhh.de/rump/. 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, IEICE, 1 (2010), 105-118. doi: 10.1587/nolta.1.105.
[34]	W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, 2011, URL http://books.google.com/books?id=IEN56sqHtR8C.
[35]	W. Tucker, The Lorenz attractor exists, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 1197-1202. 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), 120-152. 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), 1557-1594. 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), 423-447. 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), 5-18. 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), 2004-2013. doi: 10.1137/S0036142996304498.
[41]	P. Zgliczynski, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252. 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), 1533-1542. 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, Methods & Applications, 113 (2015), 51-70. 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 Market Street, Floor 6, Philadelphia, PA 19104), 1998. doi: 10.1137/1.9781611971392.
[4]	P.-O. Åsén, Stability of Plane Couette Flow and Pipe Poiseuille Flow, PhD thesis, KTH, Numerical Analysis and Computer Science, NADA, 2007, QC 20100825.
[5]	B. A. Coomes, H. Koçak and K. J. Palmer, Rigorous computational shadowing of orbits of ordinary differential equations, Numerische Mathematik, 69 (1995), 401-421. doi: 10.1007/s002110050100.
[6]	Y. Eidelman, V. Milman and A. Tsolomitis, Functional Analysis: An Introduction, Graduate studies in mathematics, American Mathematical Society, 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), 119-213. 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), 97-131. 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), 1227-1243. 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, Inc., New York, 1984, URL http://books.google.com/books?id=wI4fAwAAQBAJ.
[11]	Z. Galias and P. Zgliczyski, Computer assisted proof of chaos in the Lorenz equations, Physica D: Nonlinear Phenomena, 115 (1998), 165-188. 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), 91-132. 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), 1427-1459, URL http://archimede.mat.ulaval.ca/jplessard/Publications_files/final_draft.pdf. 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, Robust Control and Robotics, Springer London, 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, Pergamon Press; [distributed in the Western Hemisphere by Macmillan, New York], 1964, URL http://books.google.com/books?id=g_tQAAAAMAAJ.
[16]	R. B. Kearfott, Interval computations: Introduction, uses, and resources, Euromath Bulletin, 2 (1996), 95-112.
[17]	G. Kedem, A posteriori error bounds for two-point boundary value problems, SIAM Journal on Numerical Analysis, 18 (1981), 431-448. 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), 565-604. 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), 403-429. 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), 267-313. 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), 1-22. 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), 919-937. doi: 10.1137/0712068.
[23]	R. Moore, Methods and Applications of Interval Analysis, Studies in Applied and Numerical Mathematics, Society for Industrial and Applied Mathematics, 1979, URL http://books.google.com/books?id=WYjD2-R2zMgC.
[24]	R. Moore, R. Kearfott and M. Cloud, Introduction to Interval Analysis, Cambridge University Press, 2009, URL http://books.google.com/books?id=kd8FmmN7sAoC. 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), 543-565. 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), 323-334. 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), 489-507. 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), 321-356. 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), 21-68. doi: 10.1016/S0096-3003(98)10083-8.
[30]	M. Plum, Computer-assisted existence proofs for two-point boundary value problems, Computing, 46 (1991), 19-34. 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), 19-54.
[32]	S. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77-104, URL http://www.ti3.tuhh.de/rump/. 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, IEICE, 1 (2010), 105-118. doi: 10.1587/nolta.1.105.
[34]	W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, 2011, URL http://books.google.com/books?id=IEN56sqHtR8C.
[35]	W. Tucker, The Lorenz attractor exists, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 328 (1999), 1197-1202. 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), 120-152. 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), 1557-1594. 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), 423-447. 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), 5-18. 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), 2004-2013. doi: 10.1137/S0036142996304498.
[41]	P. Zgliczynski, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252. doi: 10.1088/0951-7715/10/1/016.

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