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18 Sturm-Liouville Eigenvalue Problems - UMBC

18 Sturm-Liouville Eigenvalue ProblemsUp until now all our Eigenvalue Problems have been of the formd2 dx2+ = 0,0< x < l(1)plus a mix of boundary conditions, generally being Dirichlet or Neumanntype. This is too narrow of a viewpoint, which I wish to point out througha few 1:Simple spatial variation in diffusivity:D=D0(1 +x) the problem ut=D0[(1 +x)2ux]x0< x <1, t >0u(x,0) =f(x)0< x <1u(0,t) = 0 =u(1,t)t >0(2)From the separation of variables method,u(x,t) =T(t) (x), we obtaindTdt= D0 Tandddx[(1 +x)2d dx] + = 0,with (0) = (1) = 0. Note that by carrying through the differentiation, the equation(1 +x)2d2 dx2+ 2(1 +x)d dx+ = 0(3)is aCauchy-Eulerequation (recall the Review of ODEs appendix for a dis-cussion of the equations). If we write (x) = (1 +x)r, the characteristicequation forrbecomesr(r 1) + 2r+ = 0 r={ 1 1 4 } we have to start at = 0 atx= 0, and end at = 0 atx= 1, assumewe need oscillatory solutions like we got out of (1), since 0.

is an example of a singular Sturm-Liouville EVP, but it is close enough to the regular case that what properties are brought up below for the regular Sturm-Liouville EVP will also hold the singular Sturm-Liouville EVP too.

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