Help with similarity solutions to Heat Equation

In summary, the question is asking for the solution to the Heat equation to be in terms of the error function and to be checked using the fundamental solution of the heat equation. The solution is assumed to be a similarity solution, meaning it is a function of one variable that depends on x and t. The question asks for tips on how to solve this type of equation and suggests looking into books on similarity solutions and symmetries of differential equations. Some recommended resources include the example on Wikipedia and Peter Olver's book, "Applications of Lie Groups to Differential Equations."
  • #1
jamesufland
1
0
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
 
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  • #2
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/t1/2, 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/t1/2. Plug that into the heat equation and derive an ordinary differential equation for f(z).
 
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  • #3

1. What is a similarity solution to the Heat Equation?

A similarity solution to the Heat Equation is a solution that can be expressed in terms of a single similarity variable, rather than multiple variables. This allows for a simpler and more efficient way of solving the equation.

2. How do similarity solutions help with solving the Heat Equation?

Similarity solutions help with solving the Heat Equation by reducing the number of variables needed to describe the solution. This makes the equation easier to solve and allows for a more accurate and efficient solution.

3. What are the advantages of using similarity solutions to the Heat Equation?

There are several advantages of using similarity solutions to the Heat Equation. They provide a simpler and more efficient way of solving the equation, they can be used to find solutions in cases where exact solutions are not possible, and they allow for easier comparison and analysis of different solutions.

4. Are there any limitations to using similarity solutions for the Heat Equation?

Yes, there are some limitations to using similarity solutions for the Heat Equation. They may not work for all types of boundary conditions or initial conditions, and they may not provide an exact solution in some cases. Also, the choice of similarity variable can affect the accuracy of the solution.

5. How can I determine the appropriate similarity variable for a given Heat Equation problem?

The appropriate similarity variable for a given Heat Equation problem can be determined by analyzing the problem and considering the physical properties and boundary conditions involved. It may also require some trial and error to find the most suitable similarity variable.

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