Similarity solutions of the one-dimensional Fokker-Planck equation.
نویسندگان
چکیده
Group theoretic methods are used to construct new exact solutions for the one-dimensional Fokker-Planck equation corresponding to a class of non-linear forcing functions f(.‘cl. An important sub-class, corresponding tof(x) = a/x + /I-Y, a < 1, /? > 0 is shown to lead to stable solutions. A discussion is given on how generalized similarity methods could be applied to higher dimensional systems. 1. INTRODUCIION WE consider the stochastic differential equation dx h; + f(u) = n(t) where n(t) is stationary Gaussian white noise <n(t)) = 0 and (&)n(t,)) = Ds(t, r1) (2) x(0) = xg The output .x(t) of (1) is a stationary Markov process and is completely specified by finding the transitional probability density P(.Y, t/x0) 3 0 which satisfies the FokkerPlanck equation !?! = D a2p ; a at a.3 T& u-NPIV (34 with the initial condition p(x, O/x,) = 6(x x0) (4) x = r is a reflecting boundary for process (l), (2) if & [ ap Dz +fWp = 0. 1 If r, and rz are reflecting boundaries, rl < r2, and x0 E R = (rl ; r2), then ,j&, t) dx = 1 We limit our discussion to those processes for whichf(x) is odd i.e. f(x) = -f( -x) 143 (5)
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ورودعنوان ژورنال:
- Physical review. A, General physics
دوره 33 6 شماره
صفحات -
تاریخ انتشار 1986