Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Difference Equations by Differential Equation Methods
  • Cited by 35
Publisher:
Cambridge University Press
Online publication date:
August 2014
Print publication year:
2014
Online ISBN:
9781139016988
DOI:
https://doi.org/10.1017/CBO9781139016988
Subjects:
Mathematics, Computational Science, Differential and Integral Equations, Dynamical Systems and Control Theory, Numerical Analysis and Computational Science
Series:
Cambridge Monographs on Applied and Computational Mathematics (27)
Information

Book description

Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
Abramowitz, M. and Stegun, I. A. 1965. Handbook of mathematical functions. New York: Dover.
Adler, V. E., Bobenko, A. I., and Suris, Y. B. 2003. Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 233, 513–543.
Adler, V. E., Bobenko, A. I., and Suris, Y. B. 2009. Discrete nonlinear hyperbolic equations: classification of integrable cases. Funktsional. Anal. i Prilozhen., 43, 3–21.
Adler, V. E., Bobenko, A. I., and Suris, Y. B. 2012. Classification of integrable discrete equations of octahedron type. Int. Math. Res. Not., 1822–1889.
Bluman, G. W. and Anco, S. C. 2002. Symmetry and integration methods for differential equations. New York: Springer.
Bobenko, A. I. and Suris, Y. B. 2008. Discrete differential geometry: integrable structure. Providence, RI: American Mathematical Society.
Brading, K. A. 2002. Which symmetry? Noether, Weyl, and conservation of electric charge. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys., 33, 3–22.
Bridgman, T., Hereman, W., Quispel, G. R. W., and van der Kamp, P. H. 2013. Symbolic computation of Lax pairs of partial difference equations using consistency around the cube. Found. Comput. Math., 13, 517–544.
Budd, C. J. and Piggott, M. D. 2003. Geometric integration and its applications. In Handbook of numerical analysis, Vol. XI pp. 35–139. Amsterdam: North-Holland.
Casimiro, A. C. and Rodrigo, C. 2012. First variation formula and conservation laws in several independent discrete variables. J. Geom. Phys., 62, 61–86.
Christiansen, S. H. and Halvorsen, T. G. 2011. Discretizing the Maxwell–Klein–Gordon equation by the lattice gauge theory formalism. IMA J. Numer. Anal., 31, 1–24.
Dorodnitsyn, V. 2001. Noether-type theorems for difference equations. Appl. Numer. Math., 39, 307–321.
Dorodnitsyn, V. 2011. Applications of Lie groups to difference equations. Boca Raton, FL: CRC Press.
Duistermaat, J. J. 2010. Discrete integrable systems: QRT maps and elliptic surfaces. New York: Springer.
Elaydi, S. 2005. An introduction to difference equations, 3rd edn. New York: Springer.
Erdmann, K. and Wildon, M. J. 2006. Introduction to Lie algebras. Springer Undergraduate Mathematics Series. London: Springer.
Fuchs, J. and Schweigert, C. 1997. Symmetries, Lie algebras and representations. Cambridge: Cambridge University Press.
Gaeta, G. 1993. Lie-point symmetries of discrete versus continuous dynamical systems. Phys. Lett. A, 178, 376–384.
Gao, M., Kato, Y., and Ito, M. 2004. Some invariants for kth-order Lyness equation. Appl. Math. Lett., 17, 1183–1189.
Garifullin, R. N. and Yamilov, R. I. 2012. Generalized symmetry Classification of discrete equations of a class depending on twelve parameters. J. Phys. A, 45, 345205, 23pp.
Grammaticos, B., Kosmann-Schwarzbach, Y., and Tamizhmani, T. (eds). 2004. Discrete integrable systems. Berlin: Springer.
Grammaticos, B., Halburd, R. G., Ramani, A., and Viallet, C.-M. 2009. How to detect the integrability of discrete systems. J. Phys. A, 42, 454002, 30pp.
Grant, T. J. and Hydon, P. E. 2013. Characteristics of conservation laws for difference equations. Found. Comput. Math., 13, 667–692.
Gregor, J. 1998. The Cauchy problem for partial difference equations. Acta Appl. Math., 53, 247–263.
Hairer, E., Lubich, C., and Wanner, G. 2006. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd edn. Berlin: Springer.
Hardy, G. H. and Wright, E. M. 1979. An introduction to the theory of numbers, 5th edn. Oxford: Clarendon Press.
Hirota, R. 1977. Nonlinear partial difference equations. I. A difference analogue of the Korteweg–de Vries equation. J. Phys. Soc. Japan, 43, 1424–1433.
Hydon, P. E. 2000a. Symmetries and first integrals of ordinary difference equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456, 2835–2855.
Hydon, P. E. 2000b. Symmetry methods for differential equations: a beginner's guide. Cambridge: Cambridge University Press.
Hydon, P. E. and Mansfield, E. L. 2004. A variational complex for difference equations. Found. Comput. Math., 4, 187–217.
Hydon, P. E. and Mansfield, E. L. 2011. Extensions of Noether's second theorem: from continuous to discrete systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467, 3206–3221.
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P., and Zanna, A. 2000. Lie-group methods. Acta Numer., 9, 215–365.
Kelley, W. G. and Peterson, A. C. 2001. difference equations: an introduction with applications, 2nd edn. San Diego, CA: Harcourt/Academic Press.
Kim, P. and Olver, P. J. 2004. Geometric integration via multi-space. Regul. Chaotic Dyn., 9, 213–226.
Kosmann–Schwarzbach, Y. 2011. The Noether theorems: Invariance and conservation laws in the twentieth century, Translated, revised and augmented from the 2006 French edition by B. E., Schwarzbach. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer.
Leimkuhler, B. and Reich, S. 2004. Simulating Hamiltonian dynamics. Cambridge: Cambridge University Press.
Levi, D. and Yamilov, R. I. 2011. Generalized symmetry integrability test for discrete equations on the square lattice. J. Phys. A, 44, 145207, 22pp.
Levi, D., Vinet, L., and Winternitz, P. 1997. Lie group formalism for difference equations. J. Phys. A, 30, 633–649.
Maeda, S. 1987. The similarity method for difference equations. IMA J. Appl. Math., 38, 129–134.
Mansfield, E. L. and Hydon, P. E. 2008. difference forms. Found. Comput. Math., 8, 427–467.
Mansfield, E. L. and Szanto, A. 2003. Elimination theory for differential difference polynomials. In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 191–198. New York: ACM.
Marsden, J. E. and West, M. 2001. Discrete mechanics and variational integrators. Acta Numer., 10, 357–514.
McLachlan, R. and Quispel, R. 2001. Six lectures on the geometric integration of ODEs. In Foundations of computational mathematics (Oxford, 1999), pp. 155–210. Cambridge: Cambridge University Press.
Mickens, R. E. 1990. difference equations: theory and applications, 2nd edn. New York: Van Nostrand Reinhold.
Mikhailov, A. V. and Xenitidis, P. 2013. Second order integrability conditions for difference equations: an integrable equation. Lett. Math. Phys., DOI: 10.1007/s11005-013-0668-8.
Mikhailov, A. V., Wang, J. P., and Xenitidis, P. 2011a. Cosymmetries and Nijenhuis recursion operators for difference equations. Nonlinearity, 24, 2079–2097.
Mikhailov, A. V., Wang, J. P., and Xenitidis, P. 2011b. Recursion operators, conservation laws, and integrability conditions for difference equations. Theor. Math. Phys., 167, 421–443.
Olver, P. J. 1993. Applications of Lie groups to differential equations, 2nd edn. New York: Springer.
Olver, P. J. 1995. Equivalence, invariants, and symmetry. Cambridge: Cambridge University Press.
Ovsiannikov, L. V. 1982. Group analysis of differential equations. Translated from the Russian by Y., Chapovsky, Translation edited by W. F., Ames. New York: Academic Press.
Papageorgiou, V. G., Nijhoff, F. W., and Capel, H. W. 1990. Integrable mappings and nonlinear integrable lattice equations. Phys. Lett. A, 147, 106–114.
Quispel, G. R. W. and Sahadevan, R. 1993. Lie symmetries and the integration of difference equations. Phys. Lett. A, 184, 64–70.
Quispel, G. R. W., Nijhoff, F. W., Capel, H. W., and van der Linden, J. 1984. Linear integral equations and nonlinear difference-difference equations. Phys. A, 125, 344–380.
Quispel, G. R. W., Roberts, J. A. G., and Thompson, C. J. 1988. Integrable mappings and soliton equations. Phys. Lett. A, 126, 419–421.
Rasin, A. G. 2010. Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method. J. Phys. A, 43, 235201, 11pp.
Rasin, A. G. and Schiff, J. 2013. The Gardner method for symmetries. J. Phys. A, 46, 155202, 15pp.
Rasin, O. G. and Hydon, P. E. 2007a. Conservation laws for integrable difference equations. J. Phys. A, 40, 12763–12773.
Rasin, O. G. and Hydon, P. E. 2007b. Symmetries of integrable difference equations on the quad-graph. Stud. Appl. Math., 119, 253–269.
Stephani, H. 1989. differential equations: their solution using symmetries. Cambridge: Cambridge University Press.
van der Kamp, P. H. 2009. Initial value problems for lattice equations. J. Phys. A, 42, 404019, 16pp.
Wilf, H. S. 2006. generatingfunctionology, 3rd edn. Wellesley, MA: A. K. Peters. The second edition may be downloaded (subject to conditions) from the author's website.
Xenitidis, P. and Nijhoff, F. 2012. Symmetries and conservation laws of lattice Boussinesq equations. Phys. Lett. A, 376, 2394–2401.
Zhang, D.-J., Cheng, J.-W., and Sun, Y.-Y. 2013. Deriving conservation laws for ABS lattice equations from Lax pairs. J. Phys. A, 46, 265202, 19pp.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.