# Laplace's Equation using similarity variable and others

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• Nov 9th 2010, 01:21 PM
ricka
Laplace's Equation using similarity variable and others
I've really spent a long time struggling with these.

The first one is finding the similarity solutions of

$\displaystyle \frac{\partial u}{\partial t} = x\frac{\partial^2 u}{\partial x^2}$

using the similarity variable $\displaystyle n = xt^a$, where a is a constant I have to determine all possible values of

The other problem is solving Laplace's equation

$\displaystyle \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0$

using the similarity variable $\displaystyle n=xy^a$, where a is a constant I have to determine all possible values of
• Nov 9th 2010, 02:47 PM
Jester
Show us what you've tried so that we can help.
• Nov 9th 2010, 03:07 PM
ricka
my attempts at answering the questions
for the second one I managed to solve by the normal method of writing u as a product of a function of x only and a function of y only and working through to get:

$\displaystyle u = (c_1e^{kx} + c_2e^{-kx})(c_3cos(ky) + c_4sin(ky))$

for constants k and c subscript 1-4

but i'm not sure how to write up a solution using their similarity variable, or how to be sure how i've found all the possible constants a
• Nov 9th 2010, 03:07 PM
hmmmm
Happen to go to Edinburgh university?
• Nov 9th 2010, 03:39 PM
Jester
I think the idea here is to assume a solution of the form

$\displaystyle u = F\left(xy^a\right)$

substitute into the PDE and pick $\displaystyle a$ such that the PDE becomes an ODE in the variable $\displaystyle n = xy^a$.
• Nov 9th 2010, 04:25 PM
hmmmm
Yeah you should find the partial derivatives in terms if the similarity varibles and the other varibles then (i suggest combining to find one co-efficient in front of f' ) attempt to choose alpha such that you have an ode with constant co-efficients or in terms of the similarity vairable, I think you get 2 values forth diffusion equation and three for laplace