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Numerical resolution of 2nd order non-linear differential equation

  1. Oct 28, 2004 #1
    Hi Everybody,
    Does anybody know how to solve, analytically or numerically, the following differential equation :
    [tex] \frac{d^2\Phi}{dx^2}-a.Sinh(\frac{\Phi}{U_{th}})=-b.Exp(-(\frac{x-x_{m}}{\sigma})^2})[/tex]

    The unknown function is [tex]\Phi[/tex].
    a and b are some strictly positive constants.
    q[tex]\Phi[/tex] is the energy band bending of a P-type substrate MOS capacitor versus the distance to the silicon dioxide/silicon interface.

    Uth is the thermal voltage and [tex]BExp(-(\frac{x-x_{m}}{\sigma})^2})[/tex] the (non-uniform) dopant concentration in the substrate versus the distance to the silicon dioxide/silicon interface.

    THANX.
     
  2. jcsd
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