Arkiv för Matematik

  • Ark. Mat.
  • Volume 5, Number 3-4 (1964), 363-390.

Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion

Bengt Nagel

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Abstract

We study a Markovian process, the state space of which is the product of a set of n points and the real x-axids. Under certain regularity conditions this study is equivalent to investigating the solution of a set of couple diffusion equations, generalization of the Fokker-Planck (or second Kolmogorov) equation. Assuming the process homogeneous in x, but in general time-inhomogeneous, this set of equations is studied with the help of the Fourier transformation. The marginal distribution in the n discrete states corresponds to a time-inhomogeneousn-state Markov chain in continuous time. The properties of such a Markov chain are studied, especially the asymptotic behaviour in the time-periodic case. We obtain a natural generalization of the well-known asymptotic behaviour in the time-homogeneous case, finding a subdivision of the states into groups of essential states, the distribution inside easch group being asymptotically periodic and independent of the starting distribution. Next, still assuming time-periodicity, we study the asymptotic behaviour of the complete Markovian process, showing that inside each of the groups mentioned above the distribution approaches a common normal distribution in x-space, with mean value and variance proportional to t. Explicit expressions for the proportionality factors are derived.

The general theory is applied to the electrodiffusion equations, corresponding to n=2.

Article information

Source
Ark. Mat., Volume 5, Number 3-4 (1964), 363-390.

Dates
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485893457

Digital Object Identifier
doi:10.1007/BF02591137

Mathematical Reviews number (MathSciNet)
MR220351

Rights
1965 © Swets & Zeitlinger B.V.

Citation

Nagel, Bengt. Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion. Ark. Mat. 5 (1964), no. 3-4, 363--390. doi:10.1007/BF02591137. https://projecteuclid.org/euclid.afm/1485893457


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References

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