# Semilinear Wave Equation - White Noise

• Dec 29th 2010, 02:42 AM
NoClue
Semilinear Wave Equation - White Noise
Hello!

I am required to do a paper on the Semilinear Wave Equation as part of a Stochastic Partial Differential Equations seminar.
The problem is, that I've never taken a course on partial differential equations (only ordinary differential equations) or functional analysis.

Now I have the following in front of me:

Let $\displaystyle f(s,x,t)$ and $\displaystyle \sigma(s,x,t)$ be predictable random fields
depending on the parameters $\displaystyle s\in\mathbf{R}$ and $\displaystyle x\in D$. We consider
the initial-boundary value problems for a stochastic wave equation
as follows

$\displaystyle \frac{\partial^{2}u}{\partial t^{2}}=(\kappa\Delta-\alpha)u+f(u,x,t)+\dot{M}(u,x,t)$
$\displaystyle x\in D$, $\displaystyle t\in(0,T]$

$\displaystyle Bu$|$\displaystyle _{\partial D}=0$

$\displaystyle u(x,0)=g(x),$ $\displaystyle \frac{\partial u}{\partial t}(x,0)=h(x)$

where

$\displaystyle \dot{M}(s,x,t)=\sigma(s,x,t)\dot{W}(x,t)$

My problem is now, that I don't understand how this second order differential was constructed.
What are $\displaystyle (\kappa\Delta-\alpha)u$, $\displaystyle f(u,x,t)$, $\displaystyle \dot{M}(u,x,t)$ supposed to be here and what has this to do with waves?

If you can pinpoint me to some resources, which will make my life easier in understanding this. I will be very thankful!
• Dec 29th 2010, 06:10 PM
mr fantastic
Quote:

Originally Posted by NoClue
Hello!

I am required to do a paper on the Semilinear Wave Equation as part of a Stochastic Partial Differential Equations seminar.
The problem is, that I've never taken a course on partial differential equations (only ordinary differential equations) or functional analysis.

Now I have the following in front of me:

Let $\displaystyle f(s,x,t)$ and $\displaystyle \sigma(s,x,t)$ be predictable random fields
depending on the parameters $\displaystyle s\in\mathbf{R}$ and $\displaystyle x\in D$. We consider
the initial-boundary value problems for a stochastic wave equation
as follows

$\displaystyle \frac{\partial^{2}u}{\partial t^{2}}=(\kappa\Delta-\alpha)u+f(u,x,t)+\dot{M}(u,x,t)$
$\displaystyle x\in D$, $\displaystyle t\in(0,T]$

$\displaystyle Bu$|$\displaystyle _{\partial D}=0$

$\displaystyle u(x,0)=g(x),$ $\displaystyle \frac{\partial u}{\partial t}(x,0)=h(x)$

where

$\displaystyle \dot{M}(s,x,t)=\sigma(s,x,t)\dot{W}(x,t)$

My problem is now, that I don't understand how this second order differential was constructed.
What are $\displaystyle (\kappa\Delta-\alpha)u$, $\displaystyle f(u,x,t)$, $\displaystyle \dot{M}(u,x,t)$ supposed to be here and what has this to do with waves?