Issue |
A&A
Volume 535, November 2011
|
|
---|---|---|
Article Number | A92 | |
Number of page(s) | 11 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361/201117885 | |
Published online | 17 November 2011 |
Online material
Appendix A: solution of the focused diffusion equation for α ≠ 2
Equation (7) can be solved by performing a Laplace transformation on the equation and transferring the corresponding homogeneous second-order differential equation into a modified Bessel differential equation. A particular solution to the inhomogeneous equation can be found with the variation-of-constants method, and the inverse Laplace transformation finally reveals the desired solution.
A.1. Laplace transformation
Performing a Laplace transformation in the time coordinate to the diffusion Eq. (7) yields, with the boundary condition limT → 0F(T) = 0 and after some simplifications, (A.1)where
(A.2)is the Laplace transform of F(T).
A.2. Homogeneous solution for α ≠ 2
With the substitution and
(A.5)Eq. (A.1) transforms into
(A.6)The corresponding homogeneous equation
(A.7)is the well known modified Bessel differential equation with the two linearly independent solutions:
where
(A.10)Accordingly,
form a fundamental system for the homogeneous differential equation corresponding to Eq. (A.1):
(A.13)
A.3. Particular solution for α ≠ 2
A particular solution to the inhomogeneous Eq. (A.1) can be found with the variation-of-constants method. According to this method two functions and
exist, that solve the linear equation system:
(A.14)The solutions are
where w is the Wronskian
(A.17)A particular solution is given by
(A.18)where
and
are integrals of
and
.
For α < 2 we choose and the particular solution reads as
(A.21)For α > 2 we choose
and the particular solution reads as
(A.24)
A.4. Inverse Laplace transformation
The general solution to the differential Eq. (A.1) is the superposition of the homogeneous solutions and
, plus the particular solution
:
(A.25)The constants c1 and c2 can be determined by the chosen spatial boundary conditions. We cannot find an inverse Laplace transformation for the exponentially growing function
and choose c1(s) = 0. Using the relation (5.16.42) from Erdelyi (1954) for a > 0,
(A.26)where
denotes the generalized Laguerre polynomials, we find
(A.27)as homogeneous solutions to the differential Eq. (7). We now perform an inverse Laplace transformation to the particular solution
. Utilizing the relation 5.16.56 from Erdelyi (1954),
where ℜ(a) > 0 and ℜ(b) > 0 has to be fulfilled, we receive
(A.28)Every superposition
(A.29)is a solution of Eq. (7). A solution fulfilling the boundary conditions (12) − (14) and (16) for
is given by
. As an aside, we note that the particular solution is an infinite sum of the homogeneous solutions (A.27):
(A.30)
Appendix B: solution of the focused diffusion equation for α = 2
For α = 2 the Laplace transformed focused diffusion Eq. (A.1) reads as (B.1)With the ansatz
(B.2)we find the corresponding homogeneous solutions:
Again, a particular solution can be found utilizing the variation-of-constants method:
(B.5)The solutions for this linear system of equations are
with the Wronskian:
(B.8)A particular solution for Eq. (B.1) is therefore given by
(B.9)where
and
are integrals of
and
.
If we choose the particular solution reads as
(B.12)Using the relation (5.6.6) from Erdelyi (1954)
(B.13)we find
(B.14)as a solution for the time-dependent focused diffusion equation for α = 2.
© ESO, 2011
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