To solve a given heat equation on the half line we can use the reflection method where the initial data is an odd extension (Dirichlet boundary conditions) /even extension (Neumann BCs). why do we choose an odd or even extension, is there any clear reasons for that?
The method is impractical and cannot be generalized but it is a good way to understand the physical meaning of a given PDE, therefore any clear explanation would be appreciated!
Hello!
It is connected with the fact that in the case of odd continuation, the corresponding solution will satisfy homogeneous Dirichler boundary conditions, and in the case of even continuation it will satisfy Neumann boundary condition. You can find proofs of these facts in 'Equations of Mathematical Physics' by A.N. Tikhonov, A.A. Samarskii, pages 254-257.
Hi Kulthoum,
You will find the answer to your question by checking for:
Best regards
The link that Dr Stéphane proposed does have an explanation and example of this method:
1. http://sites.oxy.edu/vgrigoryan/342/lecs/6.reflection_method.pdf
2. https://pdfs.semanticscholar.org/c2f6/6e2ace9152bd6e2d8b5fa879c661531b77a7.pdf
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