Abstract
In this chapter, we focus on solving the diffusion equation under no spatial constraints, that is, in free space, to find a unique solution that will satisfy both the partial differential equation and the initial condition. To such end, we will review the most widely used: the Fourier and Laplace transforms and the Green’s function formalism, presenting the complete step-by-step process for each. We also discuss the implications and consequences of the central limit theorem, which is a mainstay of statistics and probability.
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Notes
- 1.
See Sect. A.7 for further details of the Fourier transform.
- 2.
The Cauchy-Goursat theorem states that if there is a function \(f(z)=u(x,y)+iv(x,y)\), and if \(f(z)\) is analytic in a simply connected domain \(\Omega \), then for every closed curve C in \(\Omega \), the contour integral of \(f(z)\) over C is zero, namely,
$$\displaystyle \begin{aligned} \oint_{C} f(z) \; \mathrm{d}z = 0. \end{aligned} $$(3.21) - 3.
See Sect. A.8 for further details of the Laplace transform.
- 4.
The subsidiary equation is the equation in terms of s, obtained by taking the transforms of all the terms in a linear differential equation.
- 5.
Using
$$\displaystyle \begin{aligned} g(r,t) = \mathcal{L}^{-1}\big\{g(r,s)\big\}=\frac{1}{2\pi i}\; \lim_{T\to \infty} \; \int_{\gamma -iT}^{\gamma +iT} \text{e}^{st} \;g(r,s)\; \mathrm{d}s \end{aligned} $$(3.51)and
$$\displaystyle \begin{aligned} q(r,t) = \mathcal{L}^{-1}\big\{q(r,s)\big\}=\frac{1}{2\pi i}\; \lim_{T\to \infty} \; \int_{\gamma -iT}^{\gamma +iT} \text{e}^{st} \;q(r,s)\; \mathrm{d}s \end{aligned} $$(3.52) - 6.
The change of variables preserves the integral’s limits.
- 7.
The master equation is the Chapman-Kolmogorov equation expressed as a first-order differential equation in time for the probability density function of a Markovian stochastic variable.
- 8.
To fulfill this requirement, the device must be level and properly built.
- 9.
If the boundaries are not homogeneous, for example, \(y(a) = y(b) = c\), the problem would have to be manipulated into one for which the boundary conditions are homogeneous. In this explicit case, we have to write a differential equation for z with the substitution \(z = y -c\).
- 10.
The BC for Green’s function arises from the fact that Eq. (3.111) is subject to the homogeneous version of the BC associated with Eq. (3.109). For instance, if the solution of Eq. (3.109) is required to satisfy an inhomogeneous Neumann BC, then Eq. (3.111) is to be solved together with the corresponding homogeneous Neumann BC.
- 11.
Causality refers to the fact that an event cannot occur before its cause is produced. This means that the solutions to be considered must be treated carefully, since the diffusion equation itself can distinguish past from future.
- 12.
Jordan’s lemma states that if having a circular arc \(C_{R}\) with radius R at the center of the origin and a function \(f(z)\) such as \(f(z) \to 0\) uniformly as \(R\to \infty \), then
$$\displaystyle \begin{aligned} {} \lim_{R \to \infty} \int_{C_{R}} f(z)\text{e}^{i m z} \; \mathrm{d}z = 0, \qquad (m>0), \end{aligned} $$(3.120)if \(C_{R}\) lies in the first and/or second quadrants, and
$$\displaystyle \begin{aligned} {} \lim_{R \to \infty} \int_{C_{R}} f(z)\text{e}^{-i m z} \; \mathrm{d}z = 0, \qquad (m>0), \end{aligned} $$(3.121)if \(C_{R}\) lies in the third and/or fourth quadrants. Such requirements perfectly suit Green’s function of the inhomogeneous diffusion equation. For instance, in Eq. (3.120), \(m=x_{0}-x>0\) when we are to the left of \(x_{0}\), and in Eq. (3.121), \(m=x-x_{0}>0\) when we are to the right of \(x_{0}\).
- 13.
Cauchy’s residue theorem states that if C is a simple-closed contour, described in the positive sense, and if a function \(f(z)\) is analytic inside and on C except for a finite number of singular points \(z_{k}\) inside C, then
$$\displaystyle \begin{aligned} {} \oint_{C} f(z) \; \mathrm{d}x = 2\pi i \sum_{k=1}^{N}\text{Res}[f(z)]. \end{aligned} $$(3.123)
Further Reading and References
C. Constanda, Solution Techniques for Elementary Partial Differential Equations (Chapman & Hall/CRC, Boca Raton, 2010)
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Dagdug, L., Peña, J., Pompa-García, I. (2024). Solution of the Diffusion Equation in Free Space. In: Diffusion Under Confinement . Springer, Cham. https://doi.org/10.1007/978-3-031-46475-1_3
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