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Separation of Variables - University of Arizona

Separation of VariablesA typical starting point to study differential equations is to guess solutions of a certain we will deal with linear PDEs, the superposition principle will allow us to form new solu-tions from linear combinations of our guesses, in many cases solving the entire problem. To beginwith, we will consider functions of two variablesu(v1,v2)(for exampleu(x,y)oru(r, )), wherethe domain is very particular: it must be of the form(v1,v2) [a,b] [c,d]. It will also be nec-essary to have homogeneous boundary conditions on opposite boundariesv1=aandv1=b(oralternativelyv2= candv2=d).

There are actually hidden boundary conditions when using polar coordinates. The first is that the solution should be finite at r= 0; we will note that some of our separated solutions do not have this property. The second is that solutions should be 2ˇ-periodic in , since = 0 and = 2ˇare the same coordinate.

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