Difference between revisions of "Solution details"
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The general form of the previous equation and its solution are defined as the following: | The general form of the previous equation and its solution are defined as the following: | ||
− | <math> \frac{d^2 y}{dx} +\frac{1-2\alpha}{x} \frac{dy}{dx}-\left [ \frac{\nu^2 \gamma^2 - \alpha^2 | + | <math> \frac{d^2 y}{dx} +\frac{1-2\alpha}{x} \frac{dy}{dx}-\left [ \frac{\nu^2 \gamma^2 - \alpha^2}{x^2} + (\beta \gamma x^{\gamma - 1})^2 \right]y = 0 </math> |
+ | and | ||
+ | |||
+ | <math> y = x^{\alpha} I_{\nu} (\beta x^{\gamma}) </math> and | ||
+ | |||
+ | <math> y = x^{\alpha} K_{\nu} (\beta x^{\gamma}) </math> | ||
+ | |||
+ | where <math> I_{\nu}</math> and | ||
+ | <math> K_{\nu}</math> are the modified Bessel of the first and the second kind. | ||
Revision as of 20:14, 26 October 2013
Asymptotic solution details for Boltzmann equation for a hole has a uniform electric field
n + - n = 0
Steps to solve Boltzmann equation <ref name="Huxley"> Huxley, L. G. H. Leonard George Holden, The diffusion and drift of electrons in gases, John Wiley and sons, 1974 , call number QC793.5.E628 H89 </ref>
for the previous equation let consider the asymptotic solution has the form:
so
where
and
In spherical coordinates:
which is symmetric in direction.
Assuming
the solution of the zenith angle direction is the Legendre polynomial, and can be written as:
and
so,
The modified Bessel functions, first and second kind, are the solutions for the previous equation but the boundary conditions determines which one to use, in this case ,
, and
as
.
so only the modified Bessel of second kind K_k are the non-zaro terms. so the the general solution for the equation can be written as :
Solution Analysis
The general form of the previous equation and its solution are defined as the following:
and
and
where
and are the modified Bessel of the first and the second kind.
The previous solution bcan be written as
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