Publications

Books

  1. Numerical Analysis of Dynamical Systems, A.M. Stuart and A.R. Humphries, 685 + xxii pages, Cambridge University Press, Hardback 1996, Paperback 1998.

Academic Journal Papers

  1. Spurious Solutions of Numerical Methods for Initial Value Problems, A.R. Humphries. IMA J. Num. Anal., 13 (1993), 263-290.
  2. Runge-Kutta Methods for Dissipative and Gradient Dynamical Systems, A.R. Humphries and A.M. Stuart. SIAM J. Num. Anal., 31 (1994), 1452-1485.
  3. Model Problems in Numerical Stability Theory for Initial Value Problems, A.M. Stuart and A.R. Humphries. SIAM Review, 36 (1994), 226-257.
  4. The Essential Stability of Local Error Control for Dynamical Systems, A.M. Stuart and A.R. Humphries. SIAM J. Num. Anal., 32 (1995) 1940-1971.
  5. Adaptive Methods for semi-linear elliptic equations with critical exponents and interior singularities, C.J. Budd and A.R. Humphries. Appl Numer. Math. 26 (1997) 227-240.
  6. Weak Finite Dimensional Approximations of Semi-Linear Elliptic PDEs With Near Critical Exponents, C.J. Budd and A.R. Humphries. J. Asymp. Anal. 17 (1998) 185-220.
  7. The finite element approximation of Semi-linear Elliptic PDEs with Critical Sobolev Exponents in the cube, C.J. Budd, A.R. Humphries and A.J. Wathen. SIAM J. Sci. Comp. 20, (1999) 1875-1904.
  8. Phase Space Error Control for Dynamical Systems, D.J. Higham, A.R. Humphries and R.J. Wain, SIAM J. Sci. Comp. 21, (2000) 2275-2294.
  9. Mosaic solutions and Entropy for Spatially Discrete Cahn-Hilliard Equations, K.A. Abell, A.R. Humphries and E.S. Van Vleck, IMA J. Appl. Math. 65 (2000) 219-255.
  10. Mosaic solutions and Entropy for Spatially Discrete Coupled Phase Transition Equations, K.A. Abell, A.R. Humphries and E.S. Van Vleck, Physica D, 155 (2001), 274-310.
  11. Numerical and analytical estimates of existence regions for Semi-linear Elliptic equations with Critical Sobolev Exponents in cuboid and cylindrical domains, C.J. Budd and A.R. Humphries. J. Comput. Appl. Math. 151 (2003), 59-84..
  12. Phase Space Stability Error Control with Variable Time-stepping Runge-Kutta Methods for Dynamical Systems, A.R. Humphries and R. Vigneswaran, Appl. Num. Anal. Comp. Math. 1 (2004) 469-488.
  13. Computation of Mixed Type Functional Differential Boundary Value Problems, K.A. Abell, C.E. Elmer, A.R. Humphries and E.S. Van Vleck. SIAM J. Appl. Dyn. Sys., 4 (2005) 755-781.
  14. Electromagnetic two-body problem: recurrent dynamics in the presence of state-dependent delay, J. De Luca, N. Guglielmi, A.R. Humphries and A. Politi. J. Phys. A, 43 (2010) 205103 (22 pages).
  15. Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion, A.R. Humphries, B.E. Moore, and E.S. van Vleck. SIAM J. Appl. Math, 71 (2011) 1374-1400.
  16. Dynamics of a Delay Differential Equation With Multiple State Dependent Delays, A.R. Humphries, O.A. DeMasi, F.M.G. Magpantay and F. Upham. Disc. Cont. Dynam. Syst. series A, 32 (2012) 2701 - 2727.
  17. Finite Element Boundary Value Integration of Wheeler-Feynman Electrodynamics, J. De Luca, A.R. Humphries, and S.B. Rodrigues, J. Comput. Appl. Math, 236 (2012), 3319-3337.
  18. Boundary-Value Problem Formulations for Computing Invariant Manifolds and Connecting Orbits in the Circular Restricted Three Body Problem, R.C. Calleja, E.J. Doedel, A.R. Humphries, A. Lemus, B.E. Oldeman, Celest. Mech. & Dynam. Astron., 114 (2012) 77 - 106.
  19. Synchronization in ecological systems by weak dispersal coupling with time delay, E. Wall, F. Guichard and A.R. Humphries, Theoretical Ecology, (2013). doi:10.1007/s12080-013-0176-6

Proceedings

  1. Approximation of Dissipative Partial Differential Equations over Long Time Intervals, 180-207 in Numerical Analysis, 1993, Proceedings of the 15th Dundee Numerical Analysis Conference, June-July 1993", (eds. DF Griffiths and GA Watson), Longman, 1994, AR Humphries, DA Jones and A.M. Stuart.
  2. Finite Element Method Solutions for Semi-linear Elliptic PDEs with Critical Sobolev Exponents on Non-spherical 3D Domains, A.R. Humphries, C.J. Budd and Wathen, 418--425 in LT Magalhães, C Rocha and L Sanchez, editors, Equadiff 95 Proceedings of the International Conference on Differential Equations, World Scientific, 1998.
  3. Deterministic and Random Dynamical Systems: Theory and Numerics, A.R. Humphries and A.M. Stuart. In Modern Methods in Scientific Computing and Applications, NATO Science Series II, volume 75, Kluwer Academic Publishers, 2002.

Research Reports

  1. A.R. Humphries and N. Christodoulou. Phase Space Error Control for Dynamical Systems, II. University of Sussex, Centre for Mathematical Analysis and its Applications Research Report no CMAIA00-13.
  2. A.R. Humphries and R.P. Wilds. Traveling wave solutions of lattice differential equations through an embedding approach, 2006, (23 pages).

Some Recent (and Not so Recent) Seminars

  1. Dynamics of Differential Equations with Multiple State Dependent Delays at Workshop on New Developments in Dynamical Systems Arising from the Biosciences at the MBI, Ohio, 25th March 2011
  2. State Dependent Delay and Advanced-Delay Differential Equations, Soup and Science, McGill University, Friday 16th January, 2009 (pdf),
  3. Model Problems and Truncation of Advanced-Retarded FDEs arising in Lattice Travelling Wave Problems, One Day Workshop on Analysis and Computation of Lattice, Delay and Functional Differential Equations, McGill University, Monday 25th April, 2005 (pdf),
  4. Numerics of Dynamics, Symposium on "Numerical Algorithms for Differential Equations and Dynamical Systems", CMS/CAIMS Summer 2004 Meeting, June 13-15, Dalhousie University. (pdf)
  5. Dynamics of Adaptive Time-Stepping ODE Solvers, University of Kansas 2002. pdf (2.8MB)
  6. Audio and overheads for the talk "Travelling waves in lattice differential equations" which I gave at the Fields Institute in December 2001 can be found here.
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