# Category Archives: PDE example

## Diffusion and Brownian motion

Let $$B_t$$ be a standard Brownian Motion  starting from zero and define

$p(t,x) = \frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t} }$

Given any $$x \in \mathbf R$$, define $$X_t=x + B_t$$ . Of course $$X_t$$ is just a Brownian Motion stating from $$x$$ at time 0. Fixing a smooth, bounded, compactly supported function $$f:\mathbf R \rightarrow \mathbf R$$, we define the function $$u(x,t)$$ by

$u(x,t) = \mathbf E_x f(X_t)$

where we have decorated the expectation with the subscript $$x$$ to remind us that we are starting from the point $$x$$.

1. Explain why $u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy$
2. Show by direct calculation using the formula from the previous question that for $$t>0$$, $$u(x,t)$$ satisfies the diffusion equation
$\frac{\partial u}{\partial t}= c\frac{\partial^2 u}{\partial x^2}$
for some constant $$c$$. (Find the correct $$c$$ !)
3. Again using the formula from part 1), show that
$\lim_{t \rightarrow 0} u(t,x) = f(x)$
and hence the initial condition for the diffusion equation is $$f$$.

## A PDE example

Observe that for $$k=0,1,\dots$$

$\phi_k(x) = \sin(\pi k x/2)$

form an orthonormal basis of  function for $$L^2([0,2]$$ with $$\phi(0)=\phi(2)=0$$. Here the inner-product of  two functions in  $$f,g \in L^2([0,2]$$  is

$\langle f,g\rangle =\int_0^2 f(x)g(x) dx$

Define the operator $$L$$ acting on a function $$\phi(x)$$ by

$L\phi(x)=\frac12 \frac{\partial^2 \phi}{\partial^2x}(x) – 5 \phi(x)$

To solve the equation

$\frac{\partial u}{\partial t}(x,t) = (L u)(x,t)$

with

$u(0,t)=u(2,t)=0 \qquad\text{and}\qquad u(x,0)=F(x)$

assume that $$u(x,t)$$ takes the from

$u(x,t)=\sum_{k=0}^\infty a_k(t) \phi_k(x)$

Find the equations for the $$a_k$$ and solve then find an expression for $$u(x,t)$$.