The Poisson equation indeed has a unique solution even when mixed boundary conditions are applied, such as Dirichlet boundary conditions on one part and Neumann boundary conditions on another. This uniqueness is guaranteed for each specific set of boundary conditions, despite the potential inconsistency between solutions derived from Dirichlet and Neumann conditions when considered separately. The discussion emphasizes the importance of understanding the specific boundary conditions applied to the Poisson equation to appreciate the uniqueness of its solutions.
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It has a unique solution for each specified set of boundary conditions.ajeet mishra said:My professor told that poission equation has a unique solution even for mixed boundary conditions( i.e. Dirichlet bc for some part and Neumann for the remaining part). But how is this possible? As different boundary conditions for the same problem will give different solutions.
But how? As there are unique solutions for drichlet and neumann conditions separately and these, in general, may not be consistent.Chestermiller said:It has a unique solution for each specified set of boundary conditions.
Can you provide a specific example (or examples) to illustrate what you are saying?ajeet mishra said:But how? As there are unique solutions for drichlet and neumann conditions separately and these, in general, may not be consistent.