Volume 535, November 2011
|Number of page(s)||11|
|Published online||17 November 2011|
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.
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).
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 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 .
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)
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.