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Apologies in advance if I'm being rather thick-headed...

If two simultaneous linear equations have no solution, it means that the equations are inconsistent and so graphically the two lines are parallel. But what does it mean when a boundary value problem has no solution? The cases of infinitely many solutions, and unique solution, are intuitively clear enough, but I'm having trouble coming to terms with the physical consequence of a BVP having no solution. Can someone help me put it in a physical framework? Is it possible for the given conditions of a BVP to be accurate (meaning the given system is not inconsistent), and yet for the BVP to have no solution?

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One rather philosophical way of putting it would be that the BVP is 'unphysical' - that is, that it doesn't actually model a real-world (physical) situation; the conditions are in some fashion inconsistent. –  Steven Stadnicki Feb 20 '12 at 18:15
    
This is what I had assumed initially, but various texts I've referred to seem to indicate that the boundary conditions can be consistent and accurate, and yet the BVP can still have no solution. Hope someone can help to clarify! –  Ryan Feb 21 '12 at 10:17
    
Did any of those texts give an example? It might be easier to figure out what's going on if we had something concrete to look at. –  Gerry Myerson Feb 21 '12 at 11:40
    
@Gerry Not that I remember. They didn't explicitly claim that the solution-less BVPs are physically consistent either, but certainly that was the impression I was left with. –  Ryan Mar 28 '12 at 8:12
    
I believe that the confusion here is caused by incorrect interpretation of the word "consistent". We say that simultaneous equations are inconsistent to mean that they cannot be satisfied simultaneously. This does not imply that each of these equations is somehow inconsistent. In the same way, it is perfectly possible to come up with BVPs which would not have any solutions, even though both the relevant equation and each of its associated boundary conditions (taken separately) can be perfectly meaningful, mathematically and physically. –  Aleksey Pichugin Mar 28 '12 at 12:52

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