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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Chen, Xiaopeng Roberts, Anthony J. and Duan, Jinqiao 2015. Centre manifolds for stochastic evolution equations. Journal of Difference Equations and Applications, Vol. 21, Issue. 7, p. 606.


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    2002. Complex Wave Dynamics on Thin Films.


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    Martel, Carlos and Vega, José M 1996. Finite size effects near the onset of the oscillatory instability. Nonlinearity, Vol. 9, Issue. 5, p. 1129.


    Roberts, A.J. 1996. Low-dimensional models of thin film fluid dynamics. Physics Letters A, Vol. 212, Issue. 1-2, p. 63.


    Watt, S. D. and Roberts, A. J. 1996. The construction of zonal models of dispersion in channels via matched centre manifolds. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 38, Issue. 01, p. 101.


    Xu, Chao and Roberts, A.J. 1996. On the low-dimensional modelling of Stratonovich stochastic differential equations. Physica A: Statistical Mechanics and its Applications, Vol. 225, Issue. 1, p. 62.


    Watt, Simon D. and Roberts, Anthony J. 1995. The Accurate Dynamic Modelling of Contaminant Dispersion in Channels. SIAM Journal on Applied Mathematics, Vol. 55, Issue. 4, p. 1016.


    Kotorynski, W. P. 1994. Dispersion in Pipes with Slowly Varying Cross-Sections. SIAM Journal on Mathematical Analysis, Vol. 25, Issue. 3, p. 915.


    Mercer, G. N. and Roberts, A. J. 1994. A complete model of shear dispersion in pipes. Japan Journal of Industrial and Applied Mathematics, Vol. 11, Issue. 3, p. 499.


    Roberts, A. J. 1993. The invariant manifold of beam deformations. Journal of Elasticity, Vol. 30, Issue. 1, p. 1.


    ×
  • The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Volume 34, Issue 1
  • July 1992, pp. 54-80

Boundary conditions for approximate differential equations

  • A. J. Roberts (a1)
  • DOI: http://dx.doi.org/10.1017/S0334270000007384
  • Published online: 01 February 2009
Abstract
Abstract

A large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] D. Armbruster , J. Guckenheimer & P. Holmes , “Kuramoto-Sivashinsky dynamics on the center-unstable manifoldSIAM J. Appl. Math. 49 (1989) 676691.

[2] T. B. Benjamin , J. L. Bona & J. J. Mahoney , “Model equations for long waves in nonlinear dispersive systemsPhil. Trans. Roy. Soc. A272 (1972) 47782.

[3] J. Carr , “Applications of centre manifold theoryApplied Math. Sci. 35 (1981).

[6] P. H. Coullet & E. A. Spiegel , “Amplitude equations for systems with competing instabilitiesSIAM J. Appl. Math. 43 (1983) 776821.

[9] K. B. Dysthe , “Note on a modification to the nonlinear Schrödinger equation for application to deep water wavesProc. Roy. Soc. Lond. A369 (1979) 105114.

[12] J. Guckenheimer & P. Holmes , Nonlinear oscillators, dynamical systems and bifurcations of vector fields, Springer-Verlag (1983).

[13] J. M. Hyman , B. Nicolaenko & S. Zaleski , “Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfacesPhysica D 23 (1983) 265292.

[15] Y. Kuramoto , “Diffusion induced chaos in reactions systemsProgr. Theoret. Phys. Suppl. 64 (1978) 346367.

[16] G. N. Mercer & A. J. Roberts , “The application of center manifold theory to the dispersion of contaminant in channels with varying flow propertiesSIAM J. Appl. Math. 50 (1990) 15471565.

[17] A. Mielke , “Saint-Venant's problem and semi-inverse solutions in nonlinear elasticityArch. Rat. Mech. & Anal. 102 (1988) 205229.

[18] A. Mielke , “On Saint-Venant's problem and Saint-Venant's principle in nonlinear elasticityTrends in Appl. Maths, to Mech. (1988) 252260.

[26] A. J. Roberts , “The utility of an invariant manifold description of the evolution of dynamical systemsSIAM J. of Math. Anal. 20 (1989) 14471458.

[30] J. Sijbrand , “Properties of center manifoldsTrans. Amer. Math. Soc. 289 (1985) 431469.

[31] G. I. Sivashinsky , “On cellular instability in the solidification of a dilute binary alloyPhysica D 8 (1983) 243248.

[33] R. Smith , “Entry and exit conditions for flow reactorsIMAJ. of Appl. Maths. 41 (1988) 140.

[34] G. I. Taylor , “Dispersion of soluble matter in a solvent flowing through a tubeProc. Roy. Soc. Lond. A219 (1953) 186203.

[35] R. Temam , “Inertial manifoldsMath. Intelligencer 12 (1990) 6874.

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