Solving Eqs. 12.13 or 12.16 gives the transitional probability functions. Since these 
equations are solved analytically only for some restricted types, several methods, such as 
a simulation method, transformation into a Schradinger equation, numerical integration, 
potential method, and so on, are used to obtain the solution. 
When, by the elapse of time, the transitional probability function p(x, tly) tends to 
p(x) as its limit and becomes stationary, independent of time and its initial value, then in 
a 
the Fokker—Planck Eqs. 12.13 and 12.16, setting > ©, = £=0 
’ 
2 ae] 2 Oras es 
Fl 
OX;0X; ja]. (OX: 
where p(x) obtained as the solution of this equation is an N—dimensional joint distribution 
related to the N—dimensional state space for this vector process X(t). 
Under some considerations and restrictions, these methods can be expanded to deal 
with nonlinear processes. 
For example, following Caughey,*? consider a nonlinear oscillation with nonlinear 
restoration under Gaussian (mean = 0) excitation as 
X(t) + BX(t) + FIX} = Ko) (12.20) 
E|fit)] = 0 (12.21) 
Wi 
Elft:) Atz)] = 5 th sf) (12.22) 
here Wo/2 shows the white spectrum of the excitation, and 6 is the Dirac’s delta 
function. From the transform 
Xi) =X) 
‘ (12.23) 
X(t) = X() 
and introducing the vector process 
X(t) = [X1(0), Xo(0))' = XO, XN, (12.24) 
from Eqs. 12.20 and 12.23 
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