Hydon PE, 2014, Difference Equations by Differential Equation Methods, Cambridge University Press


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.


'Following his successful Cambridge text on symmetry methods for differential equations, Hydon has written a superb introduction to the modern theory of difference equations, concentrating on explicit solution techniques, Lie symmetry methods, Noether's Theorem relating symmetries and conservation laws, and integrability properties. Hydon is an exceptionally clear expositor, and his new book adapts well-known geometric and algebraic constructions for differential equations to the more challenging discrete realm, copiously illustrated by explicit examples and exercises. It will prove to be an essential textbook and reference volume, introducing the next generation of researchers, practitioners, and students to this increasingly important and active area of contemporary applied mathematics.'
Peter Olver, University of Minnesota

Errata (PDF)


Hydon PE, 2000, Symmetry Methods for Differential Equations: A Beginner's Guide, Cambridge University Press


Symmetry is the key to solving differential equations. There are many well-known techniques for obtaining exact solutions, but most of them are special cases of a few powerful symmetry methods. Furthermore, these methods can be applied to differential equations of an unfamiliar type; they do not rely on special 'tricks'. Instead, a given differential equation is forced to reveal its symmetries, which are then used to construct exact solutions. This book is a straightforward introduction to the subject, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily. The book contains methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.


'Hydon's book stands out as perhaps the best introductory level text currently available ... Hydon's book is extremely well-written, and a welcome addition to the literature on Lie's methods. The author has clearly devoted a lot of effort to pedagogical details, and the exposition is designed to effortlessly bring the beginning student up to speed in basic applications of the method.'
Peter Olver, ZAMM

'Throughout the text numerous examples are worked out in detail and the exercises have been well chosen. This is the most readable text on this material I have seen and I would recommend the book for self-study (as an introduction).'
Mark Fels, MathSciNet

'This new book by Peter Hydon ... is eminently suitable for advanced undergraduates and beginning postgraduate students ... Overall I thoroughly recommend this book and believe that it will be a useful textbook for introducing students to symmetry methods for differential equations.'
Peter Clarkson, Journal of Nonlinear Mathematical Physics

'. a nice introduction to symmetry methods for ordinary and partial differential equations written with passion by a specialist ... in a lucid and precise manner. The presentation is vivid and informal, without a traditional "theorem-proof-corollary" format which quite often frightens non-mathematicians interested mostly in applications of theoretical results rather than in their justification. Despite this unusual choice of style, the exposition is not lacking in neatness and rigour, and all the main details which are necessary for the understanding of the material are provided.'
Svitlana Rogovchenko, Zentralblatt MATH

Errata (PDF)

Papers and Preprints

(in PostScript (ps) and/or PDF (pdf) format - very large files are zipped)

59. Bastankhah M, Zunder JK, Hydon PE, Deebank C, Placidi, M, 2024, Modelling turbulence in axisymmetric wakes: an application to wind turbine wakes, J. Fluid Mech., in press. The accepted manuscript is available here: arXiv link

58. White LC, Hydon PE, 2024, Moving Frames: Difference and Differential-Difference Lagrangians, Sigma, 20:006. The published version is available without charge here: Direct link

57. Hydon PE, 2023, Partial Euler operators and the efficient inversion of Div, Eur. J. Appl. Math., 34:1046-1066. The published version is available without charge here: Direct link

56. Peng L, Hydon PE, 2022, Transformations, symmetries and Noether theorems for differential-difference equations, Proc. Roy. Soc. Lond. A, 478: 20210944. pdf

55. Frasca-Caccia G, Hydon PE, 2022, A new technique for preserving conservation laws, Found. Comput. Math. 22: 477-506. The published version is available without charge here: Direct link

54. Frasca-Caccia G, Hydon PE, 2021, Numerical preservation of multiple local conservation laws, Appl. Math. Comput. 403: 126203. doi:10.1016/j.amc.2021.126203. pdf The published version is available here: Direct link

53. Frasca-Caccia G, Hydon PE, 2020, Simple bespoke preservation of two conservation laws, IMA J. Numer. Anal., 40: 1294-1329. doi:10.1093/imanum/dry087. Direct link

52. Frasca-Caccia G, Hydon PE, 2019, Locally conservative finite difference schemes for the Modified KdV equation, J. Comput. Dyn. 6: 307-323. pdf

51. Mansfield EL, Rojo-Echeburua A, Hydon PE, Peng L, 2019, Moving Frames and Noether's Finite Difference Conservation Laws I, Trans. Math. Appl., 3: tnz004. doi:10.1093/imatrm/tnz004. Direct link

50. Grant TJ, Hydon PE, 2013, Characteristics of conservation laws for difference equations, Found. Comp. Math., 13: 667-692. pdf

49. Fisher DJ, Gray RJ, Hydon PE, 2013, Automorphisms of real Lie algebras of dimension five or less, J. Phys. A: Math. Theor., 46: 225204. pdf

48. Delahaies S, Hydon PE, 2011, Multisymplectic formulation of near-local Hamiltonian balanced models, Proc. Roy. Soc. Lond. A, 467: 3424-3442. ps pdf

47. Hydon PE, Mansfield EL, 2011, Extensions of Noether's Second Theorem: from continuous to discrete systems, Proc. Roy. Soc. Lond. A, 467: 3206-3221. ps pdf

46. Tsuda A, Laine-Pearson FE, Hydon PE, 2011, Why chaotic mixing of particles is inevitable in the deep lung, J. Theor. Biol., 286: 57-66. pdf

45. Laine-Pearson FE, Hydon PE, 2010, Alternating flow in a moving corner, Eur. J. Fluids/B, 29: 278-284. ps pdf

44. Hydon PE, Viallet C-M, 2010, Asymmetric integrable quad-graph equations, (Invited paper for special issue on Continuous and Discrete Integrable Systems) Applicable Analysis 89: 493-506. ps pdf

43. Bridges TJ, Hydon PE, Lawson JK, 2010, Multisymplectic structures and the variational bicomplex, Math. Proc. Camb. Phil. Soc. 148: 159-178. ps pdf

42. Laine-Pearson FE, Hydon PE, 2009, Inertial particle motion in recirculating Stokes flow, Stud. Appl. Math. 122: 139-152. ps pdf (The published version is available at the Wiley Online Library.)

41. Mansfield EL, Hydon PE, 2008, Difference forms, Found. Comp. Math. 8: 427-467. ps pdf

40. Laine-Pearson FE, Hydon PE, 2008, The carousel effect in alveolar models, ASME J. Biomech. Engng. 130: 021016. ps pdf

39. Rasin OG, Hydon PE, 2007, Conservation laws for integrable difference equations, J. Phys. A: Math. Theor. 40: 12763-12773. ps pdf

38. Cotter CJ, Holm DD, Hydon PE, 2007, Multisymplectic formulation of fluid dynamics using the inverse map, Proc. Roy. Soc. Lond. A 463: 2671-2687. ps pdf

37. Rasin OG, Hydon PE, 2007, Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119: 253-269. ps pdf (The definitive version is available at the Wiley Online Library.)

36. Sahadevan R, Rasin OG, Hydon PE, 2007, Integrability conditions for nonautonomous quad-graph equations, J. Math. Anal. Appl. 331: 712-726. ps pdf

35. Rasin OG, Hydon PE, 2006, Conservation laws for NQC-type difference equations, J. Phys. A: Math. Gen. 39: 14055-14066. ps pdf

34. Laine-Pearson FE, Hydon PE, 2006, Particle transport in a moving corner, J. Fluid Mech. 559: 379-390. ps pdf

33. Rasin, OG, Hydon PE, 2005, Conservation laws of discrete KdV equation, SIGMA 1 (026): 1-6. ps pdf

32. Hydon PE, 2005, An introduction to symmetry methods in the solution of differential equations that occur in chemistry and chemical biology, Int. J. Quantum Chem. 106: 266-277. ps pdf

31. Hydon PE, 2005, Symmetry analysis of initial-value problems, J. Math. Anal. Appl. 309:103-116. ps pdf

30. Hydon PE, 2005, Multisymplectic conservation laws for differential and differential-difference equations, Proc. Roy. Soc. Lond. A 461: 1627-1637. ps pdf

29. Bridges, TJ, Hydon PE, Reich S, 2005, Vorticity and symplecticity in Lagrangian fluid dynamics, J. Phys. A: Math. Gen. 38: 1403-1418. ps pdf

28. Hydon PE, 2004, Self-invariant contact symmetries, J. Nonlinear Math. Phys. 11: 233-242. ps pdf

27. Hydon PE, Mansfield EL, 2004, A variational complex for difference equations, Found. Comp. Math. 4: 187-217. ps pdf

26. Laine-Pearson FE, Hydon PE, 2003, Classification of matrices for discrete symmetries of ordinary differential equations, Stud. Appl. Math. 111: 269-299. ps pdf (The definitive version is available at the Wiley Online Library.)

25. Tsuda A, Rogers RA, Hydon PE, Butler JP, 2002, Chaotic mixing deep in the lung, Proc. Nat. Acad. Sci. 99: 10173-10178. ps pdf

24. Gammack D, Hydon PE, 2001, Flow in pipes with non-uniform curvature and torsion, J. Fluid Mech. 433: 357-382. ps pdf

23. Hydon PE, 2001, Conservation laws of partial difference equations with two independent variables, J. Phys. A: Math. Gen. 34: 10347-10355. ps pdf

22. Mansfield EL, Hydon PE, 2001, Towards approximations of difference equations which preserve first integrals, Proc. of the 2001 International Symposium on Symbolic and Algebraic Computation (ISSAC 2001), ed. B. Mourrain, ACM, New York, pp 217-222. ps pdf

21. Mansfield EL, Hydon PE, 2001, On a variational complex for difference equations, The Geometrical Study of Differential Equations, eds Joshua A. Leslie and Thierry P. Robart, American Mathematical Society, Providence, RI, 121-129. ps pdf

20. Hydon PE, 2001, Discrete symmetries of differential equations, The Geometrical Study of Differential Equations, eds Joshua A. Leslie and Thierry P. Robart, American Mathematical Society, Providence, RI, 61-70. ps pdf

19. Hydon PE, Mansfield EL, 2000, A variational complex for difference equations, University of Kent Tech. Rep. UKC/IMS/00/32.

18. Hydon PE, 2000, How to construct the discrete symmetries of partial differential equations, Eur. J. Appl. Math. 11: 515-527. ps pdf

17. Hydon PE, 2000, Symmetries and first integrals of ordinary difference equations, Proc. Roy. Soc. Lond. A 456: 2835-2855. ps pdf

16. Hydon PE, 1999, How to use Lie symmetries to find discrete symmetries, Modern Group Analysis VII, eds N. H. Ibragimov, K. R. Naqvi, E. Straume, MARS Publishers, Trondheim, 141-147. ps pdf

15. Hydon PE, 1998, Construction of the discrete symmetries of field equations, Univ. Surrey Tech. Rep. 98/18/Ma.

14. Hydon PE, 1998, How to find discrete contact symmetries, J. Nonlin. Math. Phys. 5: 405-416. ps pdf

13. Hydon PE, 1998, Discrete point symmetries of ordinary differential equations, Proc. Roy. Soc. Lond. A 454: 1961-1972. ps pdf

12. Gammack D, Hydon PE, 1997, Steady flow in pipes of non-uniform curvature and torsion, Univ. Surrey Tech. Rep. 97/19/Ma.

11. Hydon PE, 1996, Chaotic transport in a pipe of uniform curvature, Zeit. Ang. Math. Mech. 292: 215-216.

10. Hydon PE, 1996, A direct method for linear equations in branching networks, Med. Biomed. Engng. Comp. 34: 389-393.

9. Hydon PE, Pedley TJ, 1996, Dispersion in oscillatory channel flow with coupled transverse wall motion, Eur. J. Mech. B 15: 143-156.

8. Hydon PE, 1996, Conformal symmetries of first order ordinary differential equations, J. Phys. A: Math. Gen. 29: 385-392.

7. Pedley TJ, Corieri P, Kamm RD, Grotberg JB, Hydon PE, Schroter RC, 1994, Gas flow and mixing in the airways, Critical Care Medicine 22: S24-S36.

6. Hydon PE, 1994, Resonant and chaotic advection in a curved pipe, Chaos, Solitons and Fractals 4: 941-954.

5. Hydon PE, 1994, Resonant advection by oscillatory flow in a curved pipe, Physica D 76: 44-54.

4. Hydon PE, Pedley TJ, 1993, Axial dispersion in a channel with oscillating walls, J. Fluid Mech. 249: 535-555.

3. Smye SW, Hydon PE, Will E, 1993, An analysis of the single pool urea kinetic model and estimation of errors, Phys. Med. Biol. 38: 115-122.

2. Hydon PE, Marshall JE, Walton K, 1992, All-pass feedback systems, Fifth IMA International Conference on the Mathematics of Control Theory, ed. NK Nichols, DH Owens, 371-380, Oxford University Press.

1. Hydon PE, Higenbottam T, Pedley TJ, 1990, Modelling the pulmonary circulation in health and disease, Biomechanical Transport Processes, ed. F Mosora, C Caro, E Krause, H Schmid-Schonbein, Ch. Baquey, R Pelissier, 135-142, Plenum Press, New York.