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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 lim_T → 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