Help with similarity solutions to Heat Equation

[jamesufland]

I'm trying to solve the Heat equation by assuming a similarity solution of the form U=f(z) where z = x / √t also subject to U=H(x) at t=0 *H(x) is heaviside function. The question want the answers to be given in terms of the error function and also checked by using the fundamental solution of the heat equation.

I'm not that familiar with 'similarity solutions' and the 'partial derivatives' books I've taken out from the library aren't that helpful.

I'd be grateful if people could point me to useful resources to help tackle the question and even , if possible, provide some tips on how I would go about solving it

Thanks

 

[Mute]

A similarity solution in this just means that the solution to the Heat equation is a function of one variable, z, which depends on x and t, instead of depending on the two variables x and t independently. It's "similar" because if you treat the solution as depending on two variables and plot the solution U(x,t) as a function of x for a few different times, then if you rescale the x-axis by 1/t^1/2, all curves will end up lying on top of one another. (i.e., they are 'similar').

So, what your question seems to want you to do is take the heat equation and assume the solution has the form U(x,t) = f(z), where z = x/t^1/2. Plug that into the heat equation and derive an ordinary differential equation for f(z).

 

[nickthequick]

also, see the example here:

http://en.wikipedia.org/wiki/Similarity_solution


similarity solutions seem somewhat 'magical' at first. for a further look into what symmetries of differential equations tell you about the solutions, see Peter Olver's book Applications of Lie Groups to Differential Equations which has an abstract given here: http://www.springer.com/mathematics/analysis/book/978-0-387-95000-6


cheers,

Nick