When solving problems involving the diffusion equation,
(which in one dimension is dc/dt = D(d^2c/dx^2), where c is the particle concentration, D is the diffusion constant, and x and t are space and time variables),
two solution methods are often used:
One of them involves finding a series solution in the eigenfunctions of the problem.
Another method is the Method of Images.
How does one prove that these two two methods, as applied to an arbitrary diffusion equation problem, should yield results that are identical?
Any listing of references that would be helpful in answering this question (especially for a physicist) would be greatly appreciated.
Please, remove the C++ tag from your question and use more appropriate ones. You will increase the chances of someone really knowledgeable on your subject to notice your question!
You have not imposed any initial or boundary conditions so there is no reason to expect there not to be multiple different solutions.
1. To prove the uniqueness of solutions of boundary value problems for parabolic equations, one can use the maximum principles.
2. It can also be proved using the general representation of solutions (the Green formula). For a linear problem, the difference of its two solutions is the solution of a homogeneous problem. If it is zero, then these solutions are the same.
Two different solutions of the same diffusion equation problem should Equivalent
Not identical.
There are two questions. The main question as asked in the header is different from the one asked in the text.
1) What is the method of proving that two different solutions of the same diffusion equation problem should be identical?
This is a uniqueness question and can be done using different methods. One way is to prove uniqueness of the solution via the maximum principle as mentioned by Murat, and another way is to use comparison method.
2) How does one prove that these two two methods (e.g., series solution in the eigenfunctions and method of images), as applied to an arbitrary diffusion equation problem, should yield results that are identical?
This question is different from the first one. Here, you are comparing methods as a means of obtaining solutions. Each of these methods will have a leading accuracy where both methods will be equivalent (if not identical). Beyond the leading terms, the solutions solution differs slightly but the contributions of their numerical values should be small compared to the leading terms.
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