The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of convection–diffusion equation, when bulk velocity is zero.
I've tried to show b) by using the sine Fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)##
Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx##
These are equal therefore it is an orthonomal basis.
I'm not sure if this is correct so...
I'm trying to solve the diffusion equation in spherical co-ordinates with spherical symmetry. I have included the PDE in question and the scheme I'm using and although it works, it diverges which I don't understand as Crank-Nicholson should be unconditionally stable for the diffusion. The code...
Homework Statement
Hello, I am currently working on photon diffusion equation and trying to do it without using Monte Carlo technique.
Homework Equations
Starting equation integrated over t:
int(c*exp(-r^2/(4*D*c*t)-a*c*t)/(4*Pi*D*c*t)^(3/2), t = 0 .. infinity) (1)
Result...
Hello everybody!
For my water in nanoscaled-pores simulations with SPH I need a value for the characteristic velocity.
My planned approach is to estimate this value by attaining the propagation speed of a diffusion wave.
But I have problems with understanding this process since I find some...
Homework Statement
I have to calculate the stationary field inside a room.
Homework Equations
The Attempt at a Solution
I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Eeknx+Fe-knx)cos(kny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary...
The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time.
xsize = 10; % Model size, m
xnum = 10; % Number of nodes
xstp =...
Homework Statement
We let a dye diffuses into an environment of dimension L. We inject that dye into a box by one face, at t = 0 on x = 0. The linear density c follows that equation :
with the conditions :
Homework Equations / questions[/B]
a. nondimensionalize the equations and the...
I have been preforming experiments to study the diffusion of Hydrogen through Molybdenum. According to Sievert's law diatomic molecules would diffuse as atoms. But according to my experiments I notice that the flux of hydrogen is directly proportional to the pressure of hydrogen and not to the...
Hi,
I'm a second year undergrad and we've covered the heat equation,
\begin{equation}
∇^{2}\Psi = \frac{1}{c^{2}}\frac{\partial^2 \Psi}{\partial t^2}
\end{equation}
and the wave equation,
\begin{equation}
D∇^{2}u= \frac{\partial u}{\partial t}
\end{equation}
in our differential equations...
I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem.
A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...
Homework Statement
$$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$
$$U(0,t)=U_0 \quad for \quad t>0$$
$$U(y,0)=0 \quad for \quad y>0$$
$$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$
Homework Equations
This is a diffusion problem on...
Homework Statement
[/B]
We are heating a semi-infinite slab with a laser (radius of a stream is ##a##), which presents us with a steady surface heating (at ##z=0##), everywhere else on the surface the slab is isolated.
How does the temperature change with time?
Look at the limit cases: at ##t...
Consider I have a packed column of length L filled with known characteristic adsorbent. I am putting a mixture of N components in it and I am solving for concentration of each component in mobile phase at the outlet of the column. The equations which are to be generalised are as follows: An...
Homework Statement
Exercise: Hexachlorobenzene (C6Cl6) is a highly toxic waste product of pesticide manufacturing. It is resistant to biodegradation. Sediments at the bottom of a reservoir in the Upper Mississippi River catchment have been found to contain high C6Cl6 concentrations. The...