Stochastic Partial Differential Equation Averaging.

[Alexey]

Whether somebody knows what equally
<int(F*Fcomp)dx>.
Where F(x,t) is complex function: F=F1+i*F2, Fcomp=F1-i*F2.
F satisfies to the next linereal stochastic partial differential equation:

i*h*Ft=-a*(Fxx-2*n*Fx/x+(n+1)*F/x/x)+U*F

int - sing of integral by dx,
Ft - first time derivative,
Ftt - second time derivative,
Fx - first dpase derivative,
Fxx - second spase derivative,
i - imaginary unity,
< > - sign of averaging on casual fluctuations U,
U - casual space - time, delta - correlated a white noise,
a, h, n – are consts. ( If it matters - there are interesting to me the cases n=1 and n=0,5)

 

[Evalesco]

I'm not sure what you're asking, but if you are asking about the integral of F*Fcomp, then it is an integral of a product of two complex functions. It is not possible to solve this integral without knowing the explicit form of the functions F and Fcomp.

 

[M98Ranger]

Thank you for sharing this information about Stochastic Partial Differential Equation Averaging. It seems like a complex topic, but I will try to provide a response based on my understanding.

From what I can gather, the equation you have provided is a linear stochastic partial differential equation that involves a complex function, F(x,t). This function is made up of two components, F1 and F2, and is described by the equation F=F1+i*F2. The equation also includes various derivatives such as Fx, Fxx, Ft, and Ftt, which represent the first and second time derivatives, as well as the first and second space derivatives, respectively.

The equation also includes some constants, a, h, and n, which I assume affect the behavior of the function and its components. It is interesting that you mention the cases where n=1 and n=0.5, as these values may have a significant impact on the behavior of the function.

In addition to these components, the equation also includes the term <int(F*Fcomp)dx>, which I believe represents an averaging of the function over casual fluctuations U. This means that the function is being averaged over a random space-time, delta-correlated white noise, represented by the term U.

In summary, the Stochastic Partial Differential Equation Averaging that you have described involves a complex function that is affected by various derivatives and constants, and is being averaged over random fluctuations. I hope this response provides some clarity on the content you have shared.