Numerical solution of the Fokker-Planck equation via chebyschev polynomial approximations with reference to first passage time probability density functions

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Abstract

Chebyschev approximations are employed to solve the one-dimensional, time-dependent Fokker-Planck (forward Kolmogrov) equation in the presence of two barriers a finite “distance” apart. Solutions are presented for the fundamental intervals (−1, +1) and (0, +1). In order to speed up the calculations, sparse matrix routines are utilized. The first passage time probability density function is also evaluated. Illustrative numerical results are presented for the Wiener process with drift, and the Ornstein-Uhlenbeck process for a variety of combinations of boundary conditions.

References (17)

  • A. Bharucha-Reid

    Elements of the Theory of Markov Processes and Their Applications

    (1960)
  • D. Cox et al.

    The Theory of Stochastic Processes

    (1965)
  • J. Crow et al.

    An Introduction to Population Genetics Theory

    (1970)
  • W. Feller

    Ann. of Math.

    (1952)
  • L. Fox et al.

    Chebyschev Polynomials in Numerical Analysis

    (1968)
  • N. Goel et al.

    Stochastic Models in Biology

    (1974)
  • J. Keilson

    J. Appl. Probability

    (1965)
  • M. Kimura

    J. Appl. Probability

    (1964)
There are more references available in the full text version of this article.

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