A fractional diffusion equation with sink term

Abstract

Approaching dynamical aspects in systems with localised sink term is fundamentally relevant from the physical-experimental point of view. However, theoretical advances have not progressed much in this direction; this is due to the great difficulty of addressing such types of problems from an analytic point of view. In this work, we investigate the systems sink points (like traps) considering a dynamics governed by the like reaction–diffusion equation in the presence of fractional derivatives. In order to do so, making use of the continuous-time random walks theory we constructed a model that contains multiples sink terms, on sequence we analysed the problem for two cases: first, considering only a single sink term and, second, considering multiples sink point associated with a fractal set. In both cases, we present the analytic solution in terms of Fox and Mittag–Leffler functions. Moreover, we perform a calculus of the mean square displacement and survival probability. The proposal and the techniques used in this work are useful to describe anomalous diffusive phenomena and the transport of particles in irregular media.

Appendix A

The Fox H function (or ${\text{ H }}$-function) may be defined in terms of the Mellin–Barnes-type integral [77]

$$\begin{aligned} \left. {{\hbox {H}}}_{p , q}^{m , n} \left[ x \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right.= & {} \left. {{\hbox {H}}}_{p,q}^{m , n} \left[ x \left| _{(b_{1},B_{1}), \ldots , (b_{q},B_{q})}^{(a_{1},A_{1}),\ldots ,(a_{p},A_{p})} \right] \right. \right. \nonumber \\= & {} \frac{1}{2\pi i}\int _{L}\chi (\xi )x^{-\xi }\mathrm{d}\xi \end{aligned}$$

(52)

with

$$\begin{aligned} \chi (\xi )= & {} \frac{\Pi _{j=1}^{m}\Gamma \left( b_{j}-B_{j}\xi \right) \Pi _{j=1}^{n}\Gamma \left( 1-a_{j}+A_{j}\xi \right) }{\Pi _{j=m+1}^{q}\Gamma \left( 1-b_{j}+B_{j}\xi \right) \Pi _{j=n+1}^{p}\Gamma \left( a_{j}-A_{j}\xi \right) } \end{aligned}$$

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where m, n, p and q are integers satisfying $0\le n\le p$ and $1\le m\le q$. It may also be defined by its Mellin transform

$$\begin{aligned} \int _{0}^{\infty }\left. {{\hbox {H}}}_{p , q}^{m , n} \left[ ax \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right. x^{\xi -1}\mathrm{d}x=a^{-\xi }\chi (\xi )\;. \end{aligned}$$

(54)

Here, the parameters have to be defined such that $A_{j}>0$ and $B_{j}>0$ and $a_j(b_{h}+\nu )\ne B_{h}(a_{j}-\lambda -1)$ where $\nu ,\lambda =0,1,2, \ldots ,$$h=1,2, \ldots ,m$ and $j=1,2, \ldots ,m$. The contour L separates the poles of $\Gamma \left( b_{j}-B_{j}\xi \right)$ for $j=1,2, \ldots ,m$ from those of $\Gamma \left( 1-a_{j}+A_{j}\xi \right)$ for $j=1,2, \ldots ,n$. The ${\text{ H }}$-function is analytic in x if either (i) $x\ne 0$ and $M>0$ or (ii) $0<|x|<1/B$ and $M=0$, where $M=\sum _{j=1}^{q}B_{j}-\sum _{j=1}^{p}A_{j}$ and $B=\prod _{j=1}^{p}A_{j}^{A_{j}}\prod _{j=1}^{q}B_{j}^{-B_{j}}$.

Some useful properties of the Fox H-function found in Refs. [77] are listed below.

(i) The H-function is symmetric in the pairs $(a_{1},A_{1}),\ldots , (a_{p},A_{p})$, likewise $(a_{n+1},A_{n+1}),\ldots ,(a_{p}, A_{p})$; in $(b_{1}, B_{1}), \ldots , (b_{q}, B_{q})$ and in $(b_{n+1}, B_{n+1}), \ldots , (b_{q}, B_{q})$.

(ii) For $k>0$

$$\begin{aligned} \left. {{\hbox {H}}}_{p, q}^{m , n} \left[ x \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right. = k\left. {{\hbox {H}}}_{p \; q}^{m \; n} \left[ x^{k} \left| _{(b_{q},kB_{q})}^{(a_{p},kA_{p})} \right] \right. \right. \end{aligned}$$

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(iii) The multiplication rule is

$$\begin{aligned} x^{k}\left. {{\hbox {H}}}_{p,; q}^{m , n} \left[ x \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right. = \left. {{\hbox {H}}}_{p ,q}^{m , n} \left[ x \left| _{(b_{q}+kB_{q},B_{q})}^{(a_{p}+kA_{p},A_{p})} \right] \right. \right. \end{aligned}$$

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(iv) For $n\ge 1$ and $q>m$,

$$\begin{aligned}&\left. {{\hbox {H}}}_{p , q}^{m , n} \left[ x \left| _{(b_{1},B_{1})\cdots (b_{q-1},B_{q-1})(a_{1},A_{1})}^{(a_{1},A_{1})(a_{2},A_{2})\cdots (a_{p},A_{p})} \right] \right. \right. \nonumber \\&\quad =\left. {{\hbox {H}}}_{p-1 ,q-1}^{m , n-1} \left[ x \left| _{(b_{1},B_{1})\cdots (b_{q-1},B_{q-1})}^{(a_{2},A_{2})\cdots (a_{p},A_{p})}\right] \right. \right. \end{aligned}$$

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(v) For $m\ge 2$ and $p>n$

$$\begin{aligned}&\left. {{\hbox {H}}}_{p , q}^{m , n} \left[ x \left| _{(b_{1},B_{1})(b_{2},B_{2})\cdots (b_{q},B_{q})}^{(a_{1},A_{1})\cdots (a_{p-1},A_{p-1})(b_{1},B_{1})} \right] \right. \right. \nonumber \\&\quad =\left. {{\hbox {H}}}_{p-1 ,q-1}^{m-1, n} \left[ x \left| _{(b_{2},B_{2})\cdots (b_{q},B_{q})}^{(a_{2},A_{2})\cdots (a_{p-1},A_{p-1})}\right] \right. \right. \end{aligned}$$

(58)

(vi) The relation between the generalised Mittag–Leffler function and the Fox H function is given by

$$\begin{aligned} E_{\alpha ,\beta }(x)=\left. {{\hbox {H}}}_{1 , 2}^{1 , 1} \left[ -x \left| _{(0,1)(1-\beta ,\alpha )}^{(0,1)} \right] \right. \right. \end{aligned}$$

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(vii) Under Fourier cosine transformation, the H function transforms as

$$\begin{aligned}&\int _{0}^{\infty }\left. {{\hbox {H}}}_{p , q}^{m , n} \left[ k \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right. \cos (kx) \mathrm{d}x\nonumber \\&\quad =\frac{\pi }{x}\left. {{\hbox {H}}}_{q+1, p+2}^{n+1, m} \left[ x \left| _{(1,1),(1-a_{p},A_{p}),\left( 1,1/2\right) }^{(1-b_{q},B_{q}),\left( 1,1/2\right) } \right] \right. \right. \end{aligned}$$

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(viii) If the poles of $\prod _{j=1}^{m}\Gamma \left( b_{j}-B_{j}\xi \right)$ are simple, the following series expansion is valid:

$$\begin{aligned}&\left. {{\hbox {H}}}_{p,q}^{m,n} \left[ x \left| _{(b_{q},B_{q})}^{(a_{p},A_{p})} \right] \right. \right. \nonumber \\&\quad = \sum _{h=1}^{m} \sum _{\nu =0}^{\infty }\frac{(-1)^{\nu }x^{(b_h+\nu )/B_{h}}}{\nu !B_h}\frac{\Pi _{j=1,j\ne h}^{m }\Gamma \left( b_{j}-\frac{B_{j}}{B_{h}}(b_{h}+\nu )\right) }{\Pi _{j=m+1}^{q}\Gamma \left( 1-b_{j}+\frac{B_{j}}{B_{h}}(b_{h}+\nu )\right) } \nonumber \\&\qquad \times \frac{\Pi _{j=1}^{n}\Gamma \left( 1-a_{j}+\frac{A_{j}}{B_{h}}(b_{h}+\nu )\right) }{\Pi _{j=n+1}^{p} \Gamma \left( a_{j}-\frac{A_{j}}{B_{h}}(b_{h}+\nu )\right) } \;. \end{aligned}$$

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