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Green's function for Cauchy-Euler equidimensional equation

  1. May 20, 2010 #1
    Hi,

    I am trying to compute the Green's function for a Cauchy-Euler equidimensional equation,
    [tex]\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x')[/tex]
    If the impulse is located at a location [tex]x'\neq x_c[/tex] then computation of Green's function is not an issue. What happens when [tex]x'= x_c[/tex] ?
    [tex]\frac{d^2G}{dx^2}+\frac{a}{(x-x_c)^2}G=A_1\delta(x-x_c)[/tex]
    The solution of the homogeneous equation is [tex](x-x_c)^{1/2\pm\nu}[/tex] where, [tex]\nu=\sqrt{1/4-a}[/tex]
    The trouble is if one tries to relate the change in slope of the Green's function with the strength of the impulse one has,
    [tex]\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{d^2G}{dx^2} dx+\int_{x_c-\epsilon}^{x_c+\epsilon}\frac{a}{(x-x_c)^2}G \!dx=A_1[/tex]
    From the Frobenius exponents we see that the only for [tex]a=0[/tex] is the second integral is a Cauchy principal value integral.
    Is it sensible to seek for a Green's function for such cases or one needs to make certain modification?
    Thank you,

    anubhab
     
  2. jcsd
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