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Integrable algorithm

April 12, 2024

Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.[1]

Background edit

The theory of integrable systems has advanced with the connection between numerical analysis. For example, the discovery of solitons came from the numerical experiments to the KdV equation by Norman Zabusky and Martin David Kruskal.[2] Today, various relations between numerical analysis and integrable systems have been found (Toda lattice and numerical linear algebra,[3][4] discrete soliton equations and series acceleration[5][6]), and studies to apply integrable systems to numerical computation are rapidly advancing.[7][8]

Integrable difference schemes edit

Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions".[9][10][11][12][13]

At the same time, Mark J. Ablowitz and others have not only made discrete soliton equations with discrete Lax pair but also compared numerical results between integrable difference schemes and ordinary methods.[14][15][16][17][18] As a result of their experiments, they have found that the accuracy can be improved with integrable difference schemes at some cases.[19][20][21][22]

References edit

  1. ^ Nakamura, Y. (2004). A new approach to numerical algorithms in terms of integrable systems. International Conference on Informatics Research for Development of Knowledge Society Infrastructure. IEEE. pp. 194–205. doi:10.1109/icks.2004.1313425. ISBN 0-7695-2150-9.
  2. ^ Zabusky, N. J.; Kruskal, M. D. (1965-08-09). "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States". Physical Review Letters. 15 (6). American Physical Society (APS): 240–243. Bibcode:1965PhRvL..15..240Z. doi:10.1103/physrevlett.15.240. ISSN 0031-9007.
  3. ^ Sogo, Kiyoshi (1993-04-15). "Toda Molecule Equation and Quotient-Difference Method". Journal of the Physical Society of Japan. 62 (4). Physical Society of Japan: 1081–1084. Bibcode:1993JPSJ...62.1081S. doi:10.1143/jpsj.62.1081. ISSN 0031-9015.
  4. ^ Iwasaki, Masashi; Nakamura, Yoshimasa (2006). "Accurate computation of singular values in terms of shifted integrable schemes". Japan Journal of Industrial and Applied Mathematics. 23 (3). Springer Science and Business Media LLC: 239–259. doi:10.1007/bf03167593. ISSN 0916-7005. S2CID 121824363.
  5. ^ Papageorgiou, V.; Grammaticos, B.; Ramani, A. (1993). "Integrable lattices and convergence acceleration algorithms". Physics Letters A. 179 (2). Elsevier BV: 111–115. Bibcode:1993PhLA..179..111P. doi:10.1016/0375-9601(93)90658-m. ISSN 0375-9601.
  6. ^ Chang, Xiang-Ke; He, Yi; Hu, Xing-Biao; Li, Shi-Hao (2017-07-01). "A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia's sequence transformation via pfaffians". Numerical Algorithms. 78 (1). Springer Science and Business Media LLC: 87–106. doi:10.1007/s11075-017-0368-z. ISSN 1017-1398. S2CID 4974630.
  7. ^ Nakamura, Yoshimasa (2001). "Algorithms associated with arithmetic, geometric and harmonic means and integrable systems". Journal of Computational and Applied Mathematics. 131 (1–2). Elsevier BV: 161–174. Bibcode:2001JCoAM.131..161N. doi:10.1016/s0377-0427(00)00316-2. ISSN 0377-0427.
  8. ^ Chu, Moody T. (2008-04-25). "Linear algebra algorithms as dynamical systems". Acta Numerica. 17. Cambridge University Press (CUP): 1–86. doi:10.1017/s0962492906340019. ISSN 0962-4929. S2CID 8746366.
  9. ^ Hirota, Ryogo (1977-10-15). "Nonlinear Partial Difference Equations. I. A Difference Analogue of the Korteweg-de Vries Equation". Journal of the Physical Society of Japan. 43 (4). Physical Society of Japan: 1424–1433. Bibcode:1977JPSJ...43.1424H. doi:10.1143/jpsj.43.1424. ISSN 0031-9015.
  10. ^ Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations. II. Discrete-Time Toda Equation". Journal of the Physical Society of Japan. 43 (6). Physical Society of Japan: 2074–2078. Bibcode:1977JPSJ...43.2074H. doi:10.1143/jpsj.43.2074. ISSN 0031-9015.
  11. ^ Hirota, Ryogo (1977-12-15). "Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation". Journal of the Physical Society of Japan. 43 (6). Physical Society of Japan: 2079–2086. Bibcode:1977JPSJ...43.2079H. doi:10.1143/jpsj.43.2079. ISSN 0031-9015.
  12. ^ Hirota, Ryogo (1978-07-15). "Nonlinear Partial Difference Equations. IV. Bäcklund Transformation for the Discrete-Time Toda Equation". Journal of the Physical Society of Japan. 45 (1). Physical Society of Japan: 321–332. Bibcode:1978JPSJ...45..321H. doi:10.1143/jpsj.45.321. ISSN 0031-9015.
  13. ^ Hirota, Ryogo (1979-01-15). "Nonlinear Partial Difference Equations. V. Nonlinear Equations Reducible to Linear Equations". Journal of the Physical Society of Japan. 46 (1). Physical Society of Japan: 312–319. Bibcode:1979JPSJ...46..312H. doi:10.1143/jpsj.46.312. ISSN 0031-9015.
  14. ^ Ablowitz, M. J.; Ladik, J. F. (1975). "Nonlinear differential−difference equations". Journal of Mathematical Physics. 16 (3). AIP Publishing: 598–603. Bibcode:1975JMP....16..598A. doi:10.1063/1.522558. ISSN 0022-2488.
  15. ^ Ablowitz, M. J.; Ladik, J. F. (1976). "Nonlinear differential–difference equations and Fourier analysis". Journal of Mathematical Physics. 17 (6). AIP Publishing: 1011–1018. Bibcode:1976JMP....17.1011A. doi:10.1063/1.523009. ISSN 0022-2488.
  16. ^ Ablowitz, M. J.; Ladik, J. F. (1976). "A Nonlinear Difference Scheme and Inverse Scattering". Studies in Applied Mathematics. 55 (3). Wiley: 213–229. doi:10.1002/sapm1976553213. ISSN 0022-2526.
  17. ^ Ablowitz, M. J.; Ladik, J. F. (1977). "On the Solution of a Class of Nonlinear Partial Difference Equations". Studies in Applied Mathematics. 57 (1). Wiley: 1–12. doi:10.1002/sapm19775711. ISSN 0022-2526.
  18. ^ Ablowitz, Mark J.; Segur, Harvey (1981). Solitons and the Inverse Scattering Transform. Philadelphia: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611970883. ISBN 978-0-89871-174-5.
  19. ^ Taha, Thiab R; Ablowitz, Mark J (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical". Journal of Computational Physics. 55 (2). Elsevier BV: 192–202. Bibcode:1984JCoPh..55..192T. doi:10.1016/0021-9991(84)90002-0. ISSN 0021-9991.
  20. ^ Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation". Journal of Computational Physics. 55 (2). Elsevier BV: 203–230. Bibcode:1984JCoPh..55..203T. doi:10.1016/0021-9991(84)90003-2. ISSN 0021-9991.
  21. ^ Taha, Thiab R; Ablowitz, Mark I (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation". Journal of Computational Physics. 55 (2). Elsevier BV: 231–253. Bibcode:1984JCoPh..55..231T. doi:10.1016/0021-9991(84)90004-4. ISSN 0021-9991.
  22. ^ Taha, Thiab R; Ablowitz, Mark J (1988). "Analytical and numerical aspects of certain nonlinear evolution equations IV. Numerical, modified Korteweg-de Vries equation". Journal of Computational Physics. 77 (2). Elsevier BV: 540–548. Bibcode:1988JCoPh..77..540T. doi:10.1016/0021-9991(88)90184-2. ISSN 0021-9991.

See also edit

April 12, 2024
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This article relies excessively on references to primary sources Please improve this article by adding secondary or tertiary sources Find sources Integrable algorithm news newspapers books scholar JSTOR April 2020 Learn how and when to remove this template message Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems 1 Contents 1 Background 2 Integrable difference schemes 3 References 4 See alsoBackground editThe theory of integrable systems has advanced with the connection between numerical analysis For example the discovery of solitons came from the numerical experiments to the KdV equation by Norman Zabusky and Martin David Kruskal 2 Today various relations between numerical analysis and integrable systems have been found Toda lattice and numerical linear algebra 3 4 discrete soliton equations and series acceleration 5 6 and studies to apply integrable systems to numerical computation are rapidly advancing 7 8 Integrable difference schemes editGenerally it is hard to accurately compute the solutions of nonlinear differential equations due to its non linearity In order to overcome this difficulty R Hirota has made discrete versions of integrable systems with the viewpoint of Preserve mathematical structures of integrable systems in the discrete versions 9 10 11 12 13 At the same time Mark J Ablowitz and others have not only made discrete soliton equations with discrete Lax pair but also compared numerical results between integrable difference schemes and ordinary methods 14 15 16 17 18 As a result of their experiments they have found that the accuracy can be improved with integrable difference schemes at some cases 19 20 21 22 References edit Nakamura Y 2004 A new approach to numerical algorithms in terms of integrable systems International Conference on Informatics Research for Development of Knowledge Society Infrastructure IEEE pp 194 205 doi 10 1109 icks 2004 1313425 ISBN 0 7695 2150 9 Zabusky N J Kruskal M D 1965 08 09 Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States Physical Review Letters 15 6 American Physical Society APS 240 243 Bibcode 1965PhRvL 15 240Z doi 10 1103 physrevlett 15 240 ISSN 0031 9007 Sogo Kiyoshi 1993 04 15 Toda Molecule Equation and Quotient Difference Method Journal of the Physical Society of Japan 62 4 Physical Society of Japan 1081 1084 Bibcode 1993JPSJ 62 1081S doi 10 1143 jpsj 62 1081 ISSN 0031 9015 Iwasaki Masashi Nakamura Yoshimasa 2006 Accurate computation of singular values in terms of shifted integrable schemes Japan Journal of Industrial and Applied Mathematics 23 3 Springer Science and Business Media LLC 239 259 doi 10 1007 bf03167593 ISSN 0916 7005 S2CID 121824363 Papageorgiou V Grammaticos B Ramani A 1993 Integrable lattices and convergence acceleration algorithms Physics Letters A 179 2 Elsevier BV 111 115 Bibcode 1993PhLA 179 111P doi 10 1016 0375 9601 93 90658 m ISSN 0375 9601 Chang Xiang Ke He Yi Hu Xing Biao Li Shi Hao 2017 07 01 A new integrable convergence acceleration algorithm for computing Brezinski Durbin Redivo Zaglia s sequence transformation via pfaffians Numerical Algorithms 78 1 Springer Science and Business Media LLC 87 106 doi 10 1007 s11075 017 0368 z ISSN 1017 1398 S2CID 4974630 Nakamura Yoshimasa 2001 Algorithms associated with arithmetic geometric and harmonic means and integrable systems Journal of Computational and Applied Mathematics 131 1 2 Elsevier BV 161 174 Bibcode 2001JCoAM 131 161N doi 10 1016 s0377 0427 00 00316 2 ISSN 0377 0427 Chu Moody T 2008 04 25 Linear algebra algorithms as dynamical systems Acta Numerica 17 Cambridge University Press CUP 1 86 doi 10 1017 s0962492906340019 ISSN 0962 4929 S2CID 8746366 Hirota Ryogo 1977 10 15 Nonlinear Partial Difference Equations I A Difference Analogue of the Korteweg de Vries Equation Journal of the Physical Society of Japan 43 4 Physical Society of Japan 1424 1433 Bibcode 1977JPSJ 43 1424H doi 10 1143 jpsj 43 1424 ISSN 0031 9015 Hirota Ryogo 1977 12 15 Nonlinear Partial Difference Equations II Discrete Time Toda Equation Journal of the Physical Society of Japan 43 6 Physical Society of Japan 2074 2078 Bibcode 1977JPSJ 43 2074H doi 10 1143 jpsj 43 2074 ISSN 0031 9015 Hirota Ryogo 1977 12 15 Nonlinear Partial Difference Equations III Discrete Sine Gordon Equation Journal of the Physical Society of Japan 43 6 Physical Society of Japan 2079 2086 Bibcode 1977JPSJ 43 2079H doi 10 1143 jpsj 43 2079 ISSN 0031 9015 Hirota Ryogo 1978 07 15 Nonlinear Partial Difference Equations IV Backlund Transformation for the Discrete Time Toda Equation Journal of the Physical Society of Japan 45 1 Physical Society of Japan 321 332 Bibcode 1978JPSJ 45 321H doi 10 1143 jpsj 45 321 ISSN 0031 9015 Hirota Ryogo 1979 01 15 Nonlinear Partial Difference Equations V Nonlinear Equations Reducible to Linear Equations Journal of the Physical Society of Japan 46 1 Physical Society of Japan 312 319 Bibcode 1979JPSJ 46 312H doi 10 1143 jpsj 46 312 ISSN 0031 9015 Ablowitz M J Ladik J F 1975 Nonlinear differential difference equations Journal of Mathematical Physics 16 3 AIP Publishing 598 603 Bibcode 1975JMP 16 598A doi 10 1063 1 522558 ISSN 0022 2488 Ablowitz M J Ladik J F 1976 Nonlinear differential difference equations and Fourier analysis Journal of Mathematical Physics 17 6 AIP Publishing 1011 1018 Bibcode 1976JMP 17 1011A doi 10 1063 1 523009 ISSN 0022 2488 Ablowitz M J Ladik J F 1976 A Nonlinear Difference Scheme and Inverse Scattering Studies in Applied Mathematics 55 3 Wiley 213 229 doi 10 1002 sapm1976553213 ISSN 0022 2526 Ablowitz M J Ladik J F 1977 On the Solution of a Class of Nonlinear Partial Difference Equations Studies in Applied Mathematics 57 1 Wiley 1 12 doi 10 1002 sapm19775711 ISSN 0022 2526 Ablowitz Mark J Segur Harvey 1981 Solitons and the Inverse Scattering Transform Philadelphia Society for Industrial and Applied Mathematics doi 10 1137 1 9781611970883 ISBN 978 0 89871 174 5 Taha Thiab R Ablowitz Mark J 1984 Analytical and numerical aspects of certain nonlinear evolution equations I Analytical Journal of Computational Physics 55 2 Elsevier BV 192 202 Bibcode 1984JCoPh 55 192T doi 10 1016 0021 9991 84 90002 0 ISSN 0021 9991 Taha Thiab R Ablowitz Mark I 1984 Analytical and numerical aspects of certain nonlinear evolution equations II Numerical nonlinear Schrodinger equation Journal of Computational Physics 55 2 Elsevier BV 203 230 Bibcode 1984JCoPh 55 203T doi 10 1016 0021 9991 84 90003 2 ISSN 0021 9991 Taha Thiab R Ablowitz Mark I 1984 Analytical and numerical aspects of certain nonlinear evolution equations III Numerical Korteweg de Vries equation Journal of Computational Physics 55 2 Elsevier BV 231 253 Bibcode 1984JCoPh 55 231T doi 10 1016 0021 9991 84 90004 4 ISSN 0021 9991 Taha Thiab R Ablowitz Mark J 1988 Analytical and numerical aspects of certain nonlinear evolution equations IV Numerical modified Korteweg de Vries equation Journal of Computational Physics 77 2 Elsevier BV 540 548 Bibcode 1988JCoPh 77 540T doi 10 1016 0021 9991 88 90184 2 ISSN 0021 9991 See also editSoliton Integrable system Retrieved from https en wikipedia org w index php title Integrable algorithm amp oldid 1191087478, wikipedia, wiki, book, books, library,

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