# Solutions of time fractional anomalous diffusion equations with coefficients
depending on both time and space variables

Ganbileg Bat-Ochir, Khongorzul Dorjgotov, Uuganbayar Zunderiya

###### Abstract

We derive explicit solutions for time-fractional anomalous diffusion equations
with diffusivity coefficients that depend on both space and time variables.
These solutions are expressed in Fox-H and generalized Wright functions, which
are commonly used in anomalous diffusion equations. Our study represents a
significant advancement in our understanding of anomalous diffusion with
potential applications in a wide range of fields.

##  1 Introduction

Anomalous diffusion has been the subject of extensive research in recent
years, with numerous publications that address different aspects of this
phenomenon. The growing number of applications that can be described by
anomalous diffusivity has led to increased interest in this area of study.
Until now, previous research has explored the dependence of diffusion
coefficients on space, but no published results have provided explicit
solutions when the diffusion coefficients depend on both space and time
variables. This work builds on previous research [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14], which provided exact solutions for anomalous diffusion
equations with a diffusion coefficient function that depended only on the
space variable. We extend our previous work to include the time variable in
the diffusion coefficient function. This development allows for the modeling
of processes that were previously inaccessible.

In [15], the authors proposed that the following diffusion equation

$\frac{\partial}{\partial t}P(x,t)=\frac{\partial}{\partial
x}\left(K(x,t)\frac{\partial}{\partial x}P(x,t)\right),$

where the diffusivity coefficient is taken as $K(x,t)=cx^{a}t^{b},$
$a,b,c\in\mathbb{R}$, to describe diffusion processes in turbulent media.

In [4], authors proposed time fractional version of the equation with the
coefficient $K$ depends only on $x,$ i.e. $K(x)=cx^{a},$ to describe anomalous
diffusion with an asymptotic behaviour which is regarded as a general property
of diffusion in fractal structure. Since it has been shown that the
diffusivity coefficient should not depend only on spatial variable in [15], we
consider the following generalized equation

$\frac{\partial^{\alpha}}{\partial
t^{\alpha}}u(x,t)=t^{m}\left(Ax^{d}\frac{\partial^{2}}{\partial
x^{2}}u(x,t)+Bx^{d-1}\frac{\partial}{\partial
x}u(x,t)+Cx^{d-2}u(x,t)\right),\quad x>0,~{}t>0.$ (1)

Here $m$ is a nonnegative integer, $d$ is a real number, $A$ is a positive
real number, $B$ and $C$ are real numbers.

Equation (1) can be considered as a time fractional generalization of equation
given in [15] as well as a generalization of the equation studied in [4]
considering the diffusivity coefficient depends both on time and space
variables.

The solutions that will be presented in this work will be expressed in special
functions, such as generalized Wright function and Fox-H functions. So we
refer the readers who are not familiar with these special functions should
consult with the literature [16, 17].

##  2 Main result

We recall the following special functions and their properties in preparation
for introducing exact invariant solutions of Eq. (1). The Fox H-function (e.g.
in [10, 16, 17]) is defined by means of the Mellin-Barnes type contour
integral

$H_{p,q}^{m,l}\left[z\biggr{|}\begin{array}[]{c}(a_{i},\alpha_{i})_{1,p}\\\
(b_{j},\beta_{j})_{1,q}\end{array}\right]=\frac{1}{2\pi
i}\int_{L}\frac{\prod\limits_{j=1}^{m}\Gamma(b_{j}-\beta_{j}s)\prod\limits_{i=1}^{l}\Gamma(1-a_{i}+\alpha_{i}s)}{\prod\limits_{i=l+1}^{p}\Gamma(a_{i}-\alpha_{i}s)\prod\limits_{j=m+1}^{q}\Gamma(1-b_{j}+\beta_{j}s)}z^{s}ds,$
(2)

for $z\in\mathbb{C}\setminus\\{0\\},$ where
$m,l,p,q\in\mathbb{N}_{0}=\\{0,1,2,\ldots\\}$, $0\leq m\leq q$, $0\leq l\leq
p$, $(m,l)\neq(0,0),$ $\alpha_{i},\beta_{j}\in\mathbb{R}_{+}$,
$a_{i},b_{j}\in\mathbb{R}$ ($i=1,\ldots,p;j=1,\ldots,q$). Here $L$ is a
suitable contour from $\gamma-i\infty$ to $\gamma+i\infty$, where $\gamma$ is
a real number. The integral in (2) converges if the following conditions are
met

$\omega=\sum_{i=1}^{l}\alpha_{i}-\sum_{i=l+1}^{p}\alpha_{i}+\sum_{j=1}^{m}\beta_{j}-\sum_{j=m+1}^{q}\beta_{j}>0\mbox{
and }\lvert\arg z\rvert<\frac{\pi\omega}{2}.$

The Fox H-function vanishes for large $z$ because

$H_{p,q}^{m,0}[z]\approx O\left(\exp\left(-\nu
z^{\frac{1}{\nu}}\mu^{\frac{1}{\nu}}\right)z^{\frac{2\delta+1}{2\nu}}\right),$

where
$\mu=\prod\limits_{i=1}^{p}\alpha_{i}^{\alpha_{i}}\prod\limits_{j=1}^{q}\beta_{j}^{-\beta_{j}},$
$\delta=\sum\limits_{j=1}^{q}b_{j}-\sum\limits_{i=1}^{p}a_{i}+\frac{p-q}{2}$
and $\nu=\sum\limits_{j=1}^{q}\beta_{j}-\sum\limits_{i=1}^{p}\alpha_{i}>0$.
The generalized Wright function is defined (e.g. in [10]) as

$\displaystyle{}_{p}\Psi_{q}\left[z\biggr{|}\begin{array}[]{c}(a_{i},\alpha_{i})_{1,p}\\\
(b_{j},\beta_{j})_{1,q}\end{array}\right]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\prod\limits_{i=1}^{p}\Gamma(a_{i}+\alpha_{i}k)}{\prod\limits_{j=1}^{q}\Gamma(b_{j}+\beta_{j}k)}\frac{z^{k}}{k!},$
(5)

for $z\in\mathbb{C},$ $p,q\in\mathbb{N}_{0},$ $a_{i},b_{j}\in\mathbb{C}$ and
$\alpha_{i},\beta_{j}\in\mathbb{R}\setminus\\{0\\}$
($i=1,\ldots,p;j=1,\ldots,q$). If
$\Delta=\sum\limits_{j=1}^{q}\beta_{j}-\sum\limits_{i=1}^{p}\alpha_{i}>-1$ or
$\Delta=-1$, then the series in (5) is absolutely convergent for
$z\in\mathbb{C}$ or $\lvert
z\rvert<\prod\limits_{i=1}^{p}\lvert\alpha_{i}\rvert^{-\alpha_{i}}\prod\limits_{j=1}^{q}\lvert\beta_{j}\rvert^{\beta_{j}}$,
respectively.

We know the following identities of H-function for $z>0$

$\displaystyle
H^{m,l}_{p,q}\left[z\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]$ $\displaystyle=$ $\displaystyle
H^{l,m}_{q,p}\left[\frac{1}{z}\biggr{|}\begin{array}[]{c}(1-B_{j},\beta_{j})_{1,q}\\\
(1-A_{i},\alpha_{i})_{1,p}\end{array}\right],$ (10) $\displaystyle
H^{m,l}_{p,q}\left[z\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]$ $\displaystyle=$ $\displaystyle
kH^{m,l}_{p,q}\left[z^{k}\biggr{|}\begin{array}[]{c}(A_{i},k\alpha_{i})_{1,p}\\\
(B_{j},k\beta_{j})_{1,q}\end{array}\right]\quad\mbox{ for }k>0,$ (15)
$\displaystyle
z^{\sigma}H^{m,l}_{p,q}\left[z\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]$ $\displaystyle=$ $\displaystyle
H^{m,l}_{p,q}\left[z\biggr{|}\begin{array}[]{c}(A_{i}+\sigma\alpha_{i},\alpha_{i})_{1,p}\\\
(B_{j}+\sigma\beta_{j},\beta_{j})_{1,q}\end{array}\right]\quad\mbox{ for any
}\sigma\in\mathbb{C},$ (20) $\displaystyle
H^{m+r,l}_{p+1,q+r}\left[z\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p},(1,r)\\\
\left(\frac{j}{r},1\right)_{1,r},(B_{j},\beta_{j})_{1,q}\end{array}\right]$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{\frac{r-1}{2}}}{\sqrt{r}}H^{m,l}_{p,q}\left[r^{r}z\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]\quad\mbox{ for
}r\in\mathbb{Z}_{+},l\leq p.$ (25)

Moreover the following relations hold between the special functions. The
Mittag-Leffler and Wright functions can be expressed by generalized Wright
functions

$E_{\alpha,\beta}(z)={}_{1}\Psi_{1}\left[z\left|\begin{array}[]{c}(1,1)\\\
(\beta,\alpha)\end{array}\right.\right]\quad\mbox{and}\quad\Psi\left(z;\alpha,\beta\right)=\sum\limits_{k=1}^{\infty}\frac{1}{\Gamma(\alpha
k+\beta)}\frac{z^{k}}{k!}={}_{0}\Psi_{1}\left[z\left|\begin{array}[]{c}-\\\
(\beta,\alpha)\end{array}\right.\right].$ (26)

Let us state the main result of the study as the following theorem.

######  Theorem 1.

  1. 1.

For $0<\alpha<2$ and $d\neq 2$, the equation given in (1) has the following
solution

$u(x,t)=c_{1}x^{a}H_{1,m+2}^{m+2,0}\left[\frac{x^{2-d}}{A(d-2)^{2}(\alpha+m)^{m}t^{\alpha+m}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{1}}{\alpha+m},1\right),\left(-\frac{s_{2}}{\alpha+m},1\right),\left(\frac{j}{\alpha+m},1\right)_{1,m}\end{matrix}\right.\right];$
(27)

  2. 2.

For $\alpha>2$ and $d\neq 2$, the equation given in (1) has the following
solution

$u(x,t)=x^{a}\sum_{k=1}^{[\alpha]+1}c_{k}x^{\frac{(d-2)(\alpha-k)}{\alpha+m}}t^{\alpha-k}\\\
~{}~{}~{}~{}~{}\times{}_{m+3}\Psi_{1}\left[A(d-2)^{2}(\alpha+m)^{m}x^{d-2}t^{\alpha+m}\left|\begin{matrix}\left(\frac{\alpha-
k-s_{1}}{\alpha+m},1\right),\left(\frac{\alpha-k-
s_{2}}{\alpha+m},1\right),\left(\frac{\alpha-k+i}{\alpha+m},1\right)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right],$

  3. 3.

For $d=2$, the equation given in (1) has the following solution

$u(x,t)=x^{a}\sum_{k=1}^{[\alpha]+1}c_{k}t^{\alpha-k}{}_{m+1}\Psi_{1}\left[Kt^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(\alpha-1,\alpha+m)\end{matrix}\right.\right],$

where $a$ is any real number, $K=Aa^{2}-Aa+Ba+C$ and

$s_{1,2}=\frac{\alpha+m}{2(2-d)}\left(\frac{B}{A}+2a-1\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}\right).$
(28)

By setting $\alpha=1$ in (27), we get

$u(x,t)=c_{1}x^{a}H_{1,m+2}^{m+2,0}\left[\frac{x^{2-d}}{A(d-2)^{2}(1+m)^{m}t^{1+m}}\left|\begin{matrix}(1,1+m)\\\
\left(-\frac{s_{1}}{1+m},1\right),\left(-\frac{s_{2}}{1+m},1\right),\left(\frac{j}{1+m},1\right)_{1,m}\end{matrix}\right.\right].$
(29)

If we take
$a=\frac{1}{2}\left(2d-\frac{B}{A}-3\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}\right)$,
(29) becomes

$u(x,t)=c_{1}x^{\frac{1}{2}\left(2d-\frac{B}{A}-3\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}\right)}\\\
\times
H_{1,m+2}^{m+2,0}\left[\frac{x^{2-d}}{A(d-2)^{2}(1+m)^{m}t^{1+m}}\left|\begin{matrix}(1,1+m)\\\
\left(\frac{j}{1+m},1\right)_{1,1+m},\left(\frac{d-2\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}}{d-2},1\right)\end{matrix}\right.\right].$

Then, we obtain the following corollary using (25) and (1.125) of [17] in
above the solution.

######  Corollary 1.

For $\alpha=1$ and $d\neq 2$, the equation given in (1) has the following
solution

$u(x,t)=cx^{-\frac{1}{2}\left(\frac{B}{A}-1\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}\right)}t^{-\frac{(1+m)}{d-2}\left(d-2\pm\sqrt{\left(1-\frac{B}{A}\right)^{2}-\frac{4C}{A}}\right)}EXP\left(-\frac{(1+m)x^{2-d}}{A(d-2)^{2}t^{1+m}}\right),$

where $c$ is a constant.

The corollary shows that for a special case of (1), we have solutions
expressed in terms of exponential functions.

Let us formulate the following known results as lemmas for the generalized
Wright functions [10, 18, 20] and Fox H-functions [17, 19, 20] for the
convenience to prove our main result.

######  Lemma 1.

Let $\Delta=\sum\limits_{j=1}^{q}B_{j}-\sum\limits_{i=1}^{p}A_{i}>-1.$ Then
the following equalities hold for $\alpha\in\mathbb{R}_{+}$ and
$a\in\mathbb{R}.$

(1) If $\beta_{1}>0$, $B_{1}>0$ and $A_{1}=\alpha_{1}=1,$ then we have

$\displaystyle\frac{d^{\alpha}}{dz^{\alpha}}\left(z^{B_{1}-1}{}_{p}\Psi_{q}\left[az^{\beta_{1}}\biggr{|}\begin{array}[]{c}(1,1),(A_{i},\alpha_{i})_{2,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]\right)$ $\displaystyle=$
$\displaystyle
a^{m}z^{B_{1}+m\beta_{1}-1-\alpha}{}_{p}\Psi_{q}\left[az^{\beta_{1}}\biggr{|}\begin{array}[]{c}(1,1),(A_{i}+m\alpha_{i},\alpha_{i})_{2,p}\\\
(B_{1}+m\beta_{1}-\alpha,\beta_{1}),(B_{j}+m\beta_{j},\beta_{j})_{2,q}\end{array}\right],$

where $m$ is the smallest non-negative integer such that
$B_{1}+m\beta_{1}-\alpha-1$ is not a negative integer.

(2) For $\sigma\in\mathbb{R}\setminus\\{0\\}$ and $R\in\mathbb{R},$ the
following equality holds

$\left(\frac{1}{\alpha}z\frac{d}{dz}+R\right)\left(z^{\frac{A_{1}\sigma}{\alpha_{1}}-\alpha
R}{}_{p}\Psi_{q}\left[az^{\sigma}\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]\right)\\\
=\frac{\sigma}{\alpha_{1}\alpha}z^{\frac{A_{1}\sigma}{\alpha_{1}}-\alpha
R}{}_{p}\Psi_{q}\left[az^{\sigma}\biggr{|}\begin{array}[]{c}(A_{1}+1,\alpha_{1}),(A_{i},\alpha_{i})_{2,p}\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right].$

######  Lemma 2.

Let $\nu=\sum_{j=1}^{q}\beta_{j}-\sum_{i=1}^{p}\alpha_{i}>0,$
$\mu=\sum\limits_{j=1}^{m}\beta_{j}-\sum\limits_{j=m+1}^{q}\beta_{j}-\sum\limits_{i=1}^{p}\alpha_{i}>0$,
$\alpha\in\mathbb{R}_{+}$ and $a\in\mathbb{R}\setminus\\{0\\}$. Then the
following equalities hold.

(1) If $a>0$, then

$\frac{d^{\alpha}}{dz^{\alpha}}H_{p,q}^{m,0}\left[az^{-\alpha_{p}}\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p-1},(1,\alpha_{p})\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]\\\
=z^{-\alpha}H_{p,q}^{m,0}\left[az^{-\alpha_{p}}\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p-1},(1-\alpha,\alpha_{p})\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right],\quad z>0.$

(2) If $m\geq 1,$ then

$\left(\frac{\beta_{1}}{\alpha_{p}}z\frac{d}{dz}+B_{1}\right)H_{p,q}^{m,0}\left[az^{-\alpha_{p}}\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p-1},(1,\alpha_{p})\\\
(B_{j},\beta_{j})_{1,q}\end{array}\right]\\\
=H_{p,q}^{m,0}\left[az^{-\alpha_{p}}\biggr{|}\begin{array}[]{c}(A_{i},\alpha_{i})_{1,p-1},(1,\alpha_{p})\\\
(B_{1}+1,\beta_{1}),(B_{j},\beta_{j})_{2,q}\end{array}\right].$

##  3 Proof of Main Theorem

First, we will derive solutions to the following equations

$\frac{d^{\alpha}}{dz^{\alpha}}y(z)=z^{m}\left(a_{n}z^{n}\frac{d^{n}}{dz^{n}}y+a_{n-1}z^{n-1}\frac{d^{n-1}}{dz^{n-1}}y+\cdots+a_{1}z\frac{d}{dz}y+a_{0}y\right),\quad
z>0,$ (32)

where $\alpha$ and $a_{n}$ ($i=0,\ldots,n$) are positive real numbers,
$a_{1},a_{2},\ldots,a_{n-1}$ are real numbers, $m\in\mathbb{N}$. Here the
fractional differentiation is defined in Riemann-Liouville manner. In [20], we
derived exact solutions to the class of linear fractional differential
equations of (32) when $m=0$.

In order to obtain the solutions of the equation given in (32), we consider
the following characteristic polynomial

$\widetilde{P}(s)=a_{n}\prod\limits_{j=0}^{n-1}(s-j)+a_{n-1}\prod\limits_{j=0}^{n-2}(s-j)+\cdots+a_{1}s+a_{0}$
(33)

associated with the differential operator

$P(y)=a_{n}z^{n}\frac{d^{n}}{dz^{n}}y+a_{n-1}z^{n-1}\frac{d^{n-1}}{dz^{n-1}}y+\cdots+a_{1}z\frac{d}{dz}y+a_{0}y.$

Let $s_{1},\dots,s_{n}$ be the roots of the characteristic polynomial (33).
Then we can rewrite the right hand side of (32) in factorized form as
following

$a_{n}z^{n}\frac{d^{n}}{dz^{n}}y+a_{n-1}z^{n-1}\frac{d^{n-1}}{dz^{n-1}}y+\cdots+a_{1}z\frac{d}{dz}y+a_{0}y\\\
=a_{n}\left(z\frac{d}{dz}-s_{n}\right)\cdots\left(z\frac{d}{dz}-s_{2}\right)\left(z\frac{d}{dz}-s_{1}\right)y.$

To present the solutions of the equation (32), as a preparation we show that
the following identity holds.

######  Lemma 3.

Let $a\in\mathbb{R}_{+},m\in\mathbb{N}$ and $b\in\mathbb{C}$. Then

$\frac{1}{\Gamma(1+ab+m)}\prod_{i=1}^{m}\Gamma\left(\frac{i}{a}+b+1\right)=\frac{1}{a^{m}\Gamma(1+ab)}\prod_{i=1}^{m}\Gamma\left(\frac{i}{a}+b\right).$
(34)

###### Proof.

By noticing the following identities

$\prod_{i=1}^{m}\left(\frac{i}{a}+b\right)\Gamma\left(\frac{i}{a}+b\right)=\frac{1}{a^{m}}\prod_{i=1}^{m}(i+ab)\prod_{i=1}^{m}\Gamma\left(\frac{i}{a}+b\right)=\frac{\Gamma(1+ab+m)}{a^{m}\Gamma(1+ab)}\prod_{i=1}^{m}\Gamma\left(\frac{i}{a}+b\right).$

Hence, the proof is completed. ∎

Now we are ready to give the solutions of (32).

######  Proposition 1.

The equation given in (32) has the following solutions.

  1. 1.

For $\alpha<n,$ the solution is expressed as

$y(z)=c_{1}H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{j}}{\alpha+m},1\right)_{1,n},\left(\frac{j}{\alpha+m},1\right)_{1,m}\end{matrix}\right.\right];$
(35)

  2. 2.

For $\alpha>n,$ the solution is expressed as

$y(z)=\sum_{k=1}^{[\alpha]+1}c_{k}z^{\alpha-k}{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}\left(\frac{\alpha-
k-
s_{i}}{\alpha+m},1\right)_{1,n},\left(\frac{\alpha-k+i}{\alpha+m},1\right)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right].$ (36)

Here $s_{1},\dots,s_{n}$ are the roots of the characteristic polynomial (33).

###### Proof.

Because the convergence conditions of H-functions and generalized wright
functions holds the functions in (35) and (36), it is sufficient to show that
the functions satisfy the equation in (32). Firstly, we rewrite the right hand
side of the equation (32) in the factorized form

$\displaystyle
z^{m}\left(a_{n}z^{n}\frac{d^{n}}{dz^{n}}+a_{n-1}z^{n-1}\frac{d^{n-1}}{dz^{n-1}}+\cdots+a_{1}z\frac{d}{dz}+a_{0}\right)$
$\displaystyle~{}~{}~{}~{}\left(c_{1}H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{j}}{\alpha+m},1\right)_{1,n},&\left(\frac{j}{\alpha+m},1\right)_{1,m}&\end{matrix}\right.\right]\right)$
$\displaystyle~{}=c_{1}a_{n}(\alpha+m)^{n}z^{m}\left(\frac{1}{\alpha+m}z\frac{d}{dz}-\frac{s_{n}}{\alpha+m}\right)\cdots\left(\frac{1}{\alpha+m}z\frac{d}{dz}-\frac{s_{1}}{\alpha+m}\right)$
$\displaystyle~{}~{}~{}~{}\left(H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{j}}{\alpha+m},1\right)_{1,n},&\left(\frac{j}{\alpha+m},1\right)_{1,m}&\end{matrix}\right.\right]\right).$
(37)

From Lemma 2, we get the following identity for $k=1,\ldots,n$

$\displaystyle\left(\frac{1}{\alpha+m}z\frac{d}{dz}-\frac{s_{k}}{\alpha+m}\right)\left(H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{j}}{\alpha+m},1\right)_{1,n}&\left(\frac{j}{\alpha+m},1\right)_{1,m}&\end{matrix}\right.\right]\right)$
$\displaystyle=H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
\left(-\frac{s_{j}}{\alpha+m},1\right)_{1,k-1},\left(1-\frac{s_{k}}{\alpha+m},1\right),\left(-\frac{s_{j}}{\alpha+m},1\right)_{k+1,n},&\left(\frac{j}{\alpha+m},1\right)_{1,m}&\end{matrix}\right.\right].$

Then applying the above identity in (3), the right hand side of the equation
becomes

$\displaystyle
c_{1}a_{n}z^{m}(\alpha+m)^{n}H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1,\alpha+m)\\\
(1-\frac{s_{j}}{\alpha+m},1)_{1,n}&(\frac{j}{\alpha+m},1)_{1,m}&\end{matrix}\right.\right].$

Now, let us begin the calculation of left hand side of the equation. By
applying the Lemma 2, we get

$\frac{d^{\alpha}}{dz^{\alpha}}y=c_{1}z^{-\alpha}H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1-\alpha,\alpha+m)\\\
(-\frac{s_{j}}{\alpha+m},1)_{1,n},(\frac{j}{\alpha+m},1)_{1,m}\end{matrix}\right.\right].$

Then applying (20), the left hand side is

$c_{1}a_{n}z^{m}(\alpha+n)^{m+n}H_{1,m+n}^{m+n,0}\left[\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\left|\begin{matrix}(1+m,\alpha+m)\\\
(1-\frac{s_{j}}{\alpha+m},1)_{1,n},(1+\frac{j}{\alpha+m},1)_{1,m}\end{matrix}\right.\right].$

Hence, the left hand side becomes

$\frac{d^{\alpha}}{dz^{\alpha}}y=c_{1}a_{n}z^{m}(\alpha+n)^{m+n}\frac{1}{2\pi
i}\int_{L}{\frac{\prod\limits_{j=1}^{n}\Gamma\left(1-\frac{s_{j}}{\alpha+m}-s\right)\prod\limits_{j=1}^{m}\Gamma\left(1+\frac{j}{\alpha+m}-s\right)}{\Gamma\left(1+m-(\alpha+m)s\right)}\cdot\left(\frac{z^{-(\alpha+m)}}{a_{n}(\alpha+m)^{m+n}}\right)^{s}\,d}s$

By substituting $a=\alpha+m,$ $b=-s$ in Lemma 3, we see that

$\displaystyle\frac{\prod\limits_{j=1}^{m}\Gamma\left(1+\frac{j}{\alpha+m}-s\right)}{\Gamma(1+m-(\alpha+m)s)}=\frac{1}{(\alpha+m)^{m}}\cdot\frac{\prod\limits_{j=1}^{m}\Gamma\left(\frac{j}{\alpha+m}-s\right)}{\Gamma(1-(\alpha+m)s)},$

which concludes the right and left hand sides of the equations are equal.

The second assertion of the theorem consists of $[\alpha]+1$ summands. Since
the equation is linear, it is sufficient to show that it holds for single
$k$th summand. For the fixed summand, we will take the similar steps as in the
previous case. From Lemma 2, we get the following identity for $l=1,\ldots,n$

$\displaystyle\left(\frac{1}{\alpha}z\frac{d}{dz}-\frac{s_{l}}{\alpha}\right)\left(z^{\alpha-k}{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m},1)_{1,n},(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right]\right)=\frac{\alpha+m}{\alpha}z^{\alpha-k}$
$\displaystyle~{}\times{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m},1)_{1,l-1},(\frac{\alpha-k-
s_{l}}{\alpha+m}+1,1),(\frac{\alpha-k-
s_{i}}{\alpha+m},1)_{l+1,n},(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right],$

the right hand side of the equation becomes

$\displaystyle
a_{n}\alpha^{n}z^{m}\left(\frac{1}{\alpha}z\frac{d}{dz}-\frac{s_{n}}{\alpha}\right)\cdots\left(\frac{1}{\alpha}z\frac{d}{dz}-\frac{s_{2}}{\alpha}\right)\left(\frac{1}{\alpha}z\frac{d}{dz}-\frac{s_{1}}{\alpha}\right)$
$\displaystyle~{}~{}~{}~{}\left(z^{\alpha-k}{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m},1)_{1,n},(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right]\right)$
$\displaystyle=a_{n}(\alpha+m)^{n}z^{\alpha+m-k}{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m}+1,1)_{1,n},(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right].$

The left hand side of the equation is calculated as following

$\displaystyle\frac{d^{\alpha}}{dz^{\alpha}}z^{\alpha-k}{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m},1)_{1,n},(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right]$
$\displaystyle~{}=b_{0}a_{n}(\alpha+m)^{m+n}z^{\alpha+m-k}$
$\displaystyle~{}~{}~{}~{}\times{}_{m+n+2}\Psi_{2}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(1,1),(\frac{\alpha-
k-s_{i}}{\alpha+m}+1,1)_{1,n},(\frac{\alpha-k+i}{\alpha+m}+1,1)_{1,m},(2,1)\\\
(2,1),(1+\alpha-k+m,\alpha+m)\end{matrix}\right.\right]$
$\displaystyle~{}=b_{0}a_{n}(\alpha+m)^{m+n}z^{\alpha+m-k}$
$\displaystyle~{}~{}~{}~{}\times{}_{m+n+1}\Psi_{1}\left[a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{i}}{\alpha+m}+1,1)_{1,n},(\frac{\alpha-k+i}{\alpha+m}+1,1)_{1,m},(1,1)\\\
(1+\alpha-k+m,\alpha+m)\end{matrix}\right.\right]$
$\displaystyle~{}=b_{0}a_{n}(\alpha+m)^{m+n}z^{\alpha+m-k}$
$\displaystyle~{}~{}~{}~{}\times\sum\limits_{j=0}^{\infty}\frac{\prod\limits_{i=1}^{n}\Gamma\left(\frac{\alpha-
k-
s_{i}}{\alpha+m}+1+j\right)\prod\limits_{i=1}^{m}\Gamma\left(\frac{\alpha-k+i}{\alpha+m}+1+j\right)\Gamma(1+j)}{\Gamma(1+\alpha-k+m+(\alpha+m)j)}\cdot\frac{\left(a_{n}(\alpha+m)^{m+n}z^{\alpha+m}\right)^{j}}{j!}$

which equals to the right hand side of the equation via Lemma 3 with
$a=\alpha+m,~{}b=\frac{\alpha-k}{\alpha+m}+j$. ∎

So, we have formulated exact solutions to the fractional differential
equations given in (32). In the following, we will demonstrate how to apply
the results to (1).

By applying the substitution

$u=x^{a}\varphi(x^{\frac{d-2}{\alpha+m}}t),\quad
x^{\frac{d-2}{\alpha+m}}t=z,~{}(a\in\mathbb{R})$ (38)

in (1), we have

$\frac{d^{\alpha}}{dz^{\alpha}}\varphi=z^{m}\left[\frac{A(d-2)^{2}}{(\alpha+m)^{2}}z^{2}\varphi_{zz}+\frac{d-2}{\alpha+m}\left(\frac{A(d-2)}{\alpha+m}+B\right.\right.\\\
\Big{.}\Big{.}+2Aa-A\Big{)}z\varphi_{z}+(Aa^{2}-Aa+Ba+C)\varphi\Big{]}.$ (39)

Let’s take a look at the following characteristic equation of (39)

$\frac{A(d-2)^{2}}{(\alpha+m)^{2}}s(s-1)+\frac{d-2}{\alpha+m}\left(\frac{A(d-2)}{\alpha+m}+B+A(2a-1)\right)s+Aa(a-1)+Ba+AC=0$

where $d\neq 2$. Finally, according to Proposition 1 we obtain the following
solutions to equation (39)

  1. 1.

For $0<\alpha<2,$

$\varphi(z)=c_{1}H_{1,m+2}^{m+2,0}\left[\frac{z^{-(\alpha+m)}}{A(d-2)^{2}(\alpha+m)^{m}}\left|\begin{matrix}(1,\alpha+m)\\\
(-\frac{s_{1}}{\alpha+m},1),(-\frac{s_{2}}{\alpha+m},1),(\frac{j}{\alpha+m},1)_{1,m}\end{matrix}\right.\right],$
(40)

  2. 2.

For $\alpha>2,$

$\varphi(z)=\sum_{k=1}^{[\alpha]+1}c_{k}z^{\alpha-k}{}_{m+3}\Psi_{1}\left[A(d-2)^{2}(\alpha+m)^{m}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-
k-s_{1}}{\alpha+m},1),(\frac{\alpha-k-
s_{2}}{\alpha+m},1),(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right].$ (41)

The case $d=2$, we get the following solutions to equation (39)

$\varphi(z)=\sum_{k=1}^{[\alpha]+1}c_{k}z^{\alpha-k}{}_{m+1}\Psi_{1}\left[(Aa^{2}-Aa+Ba+C)(\alpha+m)^{m}z^{\alpha+m}\left|\begin{matrix}(\frac{\alpha-k+i}{\alpha+m},1)_{1,m},(1,1)\\\
(1+\alpha-k,\alpha+m)\end{matrix}\right.\right].$ (42)

Under the substitution (38) we get our desired results in Theorem 1.

## Acknowledgments

This work was partially supported by the National University of Mongolia
(Grant No.P2018-3584) and by the Mongolian Foundation for Science and
Technology (Grant No.SHUTBIKHKHZG-2022/164).

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# Enhancing Resiliency of Integrated Space-Air-Ground-Sea Networks with
Renewable Energies: A Use Case After the 2023 Türkiye Earthquake

Bilal Karaman, Ilhan Basturk, Sezai Taskin, Ferdi Kara, Engin Zeydan, and
Halim Yanikomeroglu

###### Abstract

Natural disasters can have catastrophic consequences– a poignant example is
the series of $7.7$\- and $7.6$-magnitude earthquakes that devastated Türkiye
on February 6, 2023. To limit the damage, it is essential to maintain the
communications infrastructure to ensure individuals impacted by the disaster
can receive critical information. The disastrous earthquakes in Türkiye have
revealed the importance of considering communications and energy solutions
together to build resilient and sustainable infrastructure. Thus, this paper
proposes an integrated space-air-ground-sea network architecture that utilizes
various communications and energy-enabling technologies. This study aims to
contribute to the development of robust and sustainable disaster-response
frameworks. In light of the Türkiye earthquakes, two methods for network
management are proposed: the first aims to ensure sustainability in the pre-
disaster phase and the second aims to maintain communications during the in-
disaster phase. In these frameworks, communications technologies such as High
Altitude Platform Station(s) (HAPS) –which are among the key enablers to
unlock the potential of 6G networks–and energy technologies such as Renewable
Energy Sources (RES), Battery Energy Storage Systems (BESSs), and Electric
Vehicles (EVs) have been used as the prominent technologies. By simulating a
case study, we demonstrate the performance of a proposed framework for
providing network resiliency. The paper concludes with potential challenges
and future directions to achieve a disaster-resilient network architecture
solution.

###### Index Terms:

6G networks, disaster relief, HAPS, sustainability, earthquake, renewable
energy.

##  I Introduction

On February 6, 2023, two powerful earthquakes struck Pazarcık and Elbistan, in
Kahramanmaraş, Türkiye – with a magnitude unprecedented in recent history. The
earthquakes, with magnitudes of $7.7$ and $7.6$, respectively, caused
widespread damage in 11 provinces. The earthquakes have cost more than
$48,000$ lives and have destroyed more than $500,000$ buildings, critical
communications, and energy infrastructure, resulting in significant financial
losses. According to the Turkish Information and Communication Technologies
Authority, there are approximately 12 million mobile phone subscribers across
the 11 provinces impacted by the earthquake, corresponding to approximately
$14$% of the entire country. The total damage in the telecommunications sector
amounts to at least $$185$ million, according to the Turkish Office of
Strategy and Budget [1]. After the earthquake, satellite communications
terminals, mobile Base Stations (BSs), emergency communication vehicles, and
generators were delivered to the affected region. The power outages in this
region were the primary cause of the interruptions to mobile communication and
internet services. While mobile BSs were sent to the region, limited service
was delivered through generators that could merely provide an average of
$3$-$4$ hours of energy. As the power outages decreased, communication
services began to be delivered for longer periods of time [1].

The series of devastating earthquakes has shown that energy and communication
are among the most significant problems during natural disasters. Existing
communication infrastructures and network equipment can be damaged in the
event of a disaster or can be affected by power failures. As a result, the
cellular network or Internet connection is unavailable and the central control
authority is limited in its ability to receive timely information regarding
the disaster area, further limiting the transmission of critical information
[2]. In some instances, high-quality images need to be sent without
interruption and with minimal latency for emergency response and life-saving
treatments. Some existing energy and communications systems such as Unmanned
Aerial Vehicles (UAVs), satellite networks, Internet of Things (IoT), RES,
EVs, etc. are used as solutions to this problem separately [3, 4]. However,
instead of using these technologies individually, an integrated energy and
communication structure should be considered in the development of disaster-
resistant communication systems. Furthermore, this integrated network can be
enhanced by novel technologies, such as HAPS from a communication perspective
and BESS from an energy perspective. HAPS are promising candidates that
provide numerous advantages such as coverage adaptability, endurance, rapid
deployment, broadband capability, and low cost [5]. They also offer
sustainable solutions for infrastructure damage, energy issues (e.g. power
outages), and the restoration of damaged communication infrastructures.

In the light of above discussion, we propose the utilization of space-air-
ground-sea networks with renewable energy to provide resilient and sustainable
communications services during disasters. We consider integrated space-based
networks (including satellite), air networks (including HAPS and UAV-based
networks), ground networks (including Device-to-Device (D2D) communication),
and sea-based networks as communication solutions, and RES (including wind
turbines and solar panels), BESS, and EV as energy enablers. The disaster
management life cycle consists of pre-disaster activities (mitigation and
preparedness), response, and post-disaster recovery. Taking into account these
phases, for our integrated architecture, two methodologies are also proposed
in the pre-disaster phase for sustainability and the in-disaster phase for
communication resiliency.

##  II Current Problems of Terrestrial Networks in Post-Earthquake

###  II-A Infrastructural Damage

Ensuring the continuity of communication systems’ functionality is of critical
importance in disaster situations. BS installations are frequently implemented
on structures such as buildings and traffic lights, which are susceptible to
the impact of disasters. Depending on the intensity of an earthquake, the
backhaul links in BSs may also be damaged. For example, in the Türkiye
earthquake, $25$% of the existing BSs were disabled at time zero, and a
significant amount of the fiber infrastructure became unusable111Online:
https://m-tod.org/operatorler-deprem-bolgesinde/, Available: November 2023..
Also, while BSs on towers suffered relatively less damage, many of the BSs
located on building tops were either destroyed or damaged. The primary issue
was the BSs going out of service due to damage to the power grid. Considering
the potential damage to both communication and energy infrastructure in the
earthquake, it is crucial to develop integrated energy and communication-based
alternative solution scenarios. Note that HAPS can play an active role,
especially if the backhaul connection is damaged.

###  II-B Energy Problems (Power Outage, Gas Transportation Problems)

During a disaster, power outages can occur due to various reasons such as the
collapse of transformers, the toppling of transmission and distribution poles,
and the breakage of power lines, etc. In such circumstances, the restoration
of the power grid can be a time-consuming process. The endurance time of the
generators is $3$-$4$ hours; however, in the Türkiye earthquake, the fuel
needed for generators to operate continuously was often not delivered on time
due to challenging weather and field conditions [1]. As a consequence, the
number of disabled BSs reached $50$%, for some operators even on the second
and third days after the disaster. In this regard, RES-supported sustainable
Ground Base Station (GBS), RES-supported Eco Caravan systems, and the use of
EVs as energy sources can provide significant contributions to the operation
of terrestrial networks. In Vertical Heterogeneous Networks (VHetNets), HAPS
also leverage the advantage of having a huge payload capacity. This enables
them to utilize solar energy and batteries to sustain uninterrupted operation
for months or even years [6]. Hence, it will become possible to prevent
problems associated with power outages during disasters.

###  II-C Restoring Communication

Restoring damaged communications infrastructure after a disaster or due to
power supply problems is a vital problem that often takes a long time. The
time taken to restore operations was also too long in Türkiye earthquake. The
existing mobile BSs were not enough, and mobile BSs were even brought in from
different countries, and this process took days. As can also be seen from the
Cloudflare statistics, the full capacity could already be reached again in one
month after the earthquake 222Online:
https://blog.cloudflare.com/q1-2023-internet-disruption-summary/, Available:
November 2023.. To this end, thanks to their strong Line-of-Sight (LoS)
communications capability and a combination of radio frequency (RF) and Free-
Space Optical Communication (FSO) for backhauling, HAPS nodes are able to
provide communications services without the need for additional ground
infrastructure. However, HAPS could have challenges, e.g; in terms of
communication and sensitivity to environmental changes. Nevertheless, HAPS can
be supported by various technologies and its effectiveness in reaching remote
or densely populated disaster areas can be improved. By using technologies
such as massive multiple-input multiple-output (mMIMO) and Intelligent
Reflective Surfaces (IRS), HAPS can leverage precise beamforming and steering
techniques. This enables HAPS to provide wireless connectivity over robust LoS
connections, covering a radius of up to several hundred kilometers [7]. HAPS
can also use the S-band to react more sensitively to changes in the
environment.

##  III Proposed Space-Air-Ground-Sea Integrated Network Architecture

In this section, we propose a HAPS-supported architecture, which can be used
as both Radio Access Network (RAN) and backhaul. HAPS mainly refers to
airborne systems stationed at high altitudes, typically in the stratosphere,
above traditional aviation airspace and can be in the form of drones,
balloons, or airships. Fig. 1 shows the proposed integrated network
architecture that combines many energy technologies like RES, BESSs and EVs
and many communication technologies like Low Earth Orbit (LEO) satellite
systems, HAPS, UAVs, Integrated Sensing and Communication (ISAC) etc at
different layers, namely space, air, and terrestrial. In Fig. 1 the area in
gray represents the disaster-struck zone. The shaded areas are as follows: (i)
A represents RES supported sustainable GBS, which includes a BESS. (ii) B
represents the RES-supported Eco Caravan system as using Movable and
Deployable Resource Unit (MDRU). (iii) In C, an ISAC is indicated. (iv) D
represents the D2D communication. (v) As shown in E, EVs can play an important
role in disasters as a source of energy. In addition, EVs can be used as
mobile relays. (vi) F represents the failed BSs and (vii) G represents the
active BSs in the disaster. (viii) H represents seismic isolator-based
solutions to improve the resistance of critical power system components such
as transformers to seismic vibrations. (ix) I represents the sea-based
networks. (x) J indicates a HAPS-based communication system. Some of the
prominent communication and energy technology enablers used in the proposed
architecture for the disaster scenarios are explained in the following
subsections.

Figure 1: Proposed integrated space-air-ground-sea network topology with
energy and communication solutions.

###  III-A Communication Technology Enablers

Space-Based Networks can operate independently of terrestrial networks while
providing a reliable and resilient communications infrastructure and emergency
response support in remote or hard-to-reach areas. Satellites can play a
crucial role in disaster response communication planning, particularly in the
context of earthquakes, due to their wide coverage, resilience, and ability to
provide connectivity in remote or damaged areas. Satellite networks can be
used to establish temporary communication infrastructure in disaster-prone
areas and communication channels between emergency responders, local
communities, and other stakeholders. Rescue teams, government agencies, and
relief organizations can use satellite connectivity to share vital
information, coordinate resources, and efficiently manage rescue and relief
operations. Satellite networks can also be used to share critical information
such as maps, images, and video between emergency responders and command
centers [8] and warn and disseminate important information to local
communities. This information aids in identifying critical infrastructure
damage, assessing the extent of the disaster, and guiding resource allocation
and deployment.

Air-Based Networks contains novel technologies like HAPS and UAVs. HAPS can
play a critical role in improving the resilience of integrated space-air-
ground-sea networks during and after natural disasters such as earthquakes as
shown in Fig. 1 area J. Some of the advantages of HAPS are: (i) Because of its
surface area, it has the potential to generate almost its own energy with
solar panels. (ii) A single HAPS can be an alternative to all damaged
terrestrial BSs because it can provide a large coverage area. Here, a single
HAPS can be perceived as a multi-sector (directional) BS. (iii) For almost all
outdoor users, a direct LoS can be established that can compensate for
distance-related pathloss or scattering losses. (iv) Compared to satellites
(e.g., Starlink), no additional intermediate devices (e.g., relays) are
required. Moreover, satellites (e.g., LEO) can establish a LoS of maximum 6
minutes to any ground station. For this reason, thousands of satellites were
launched to create a constellation, but HAPS can provide stable and continuous
service because they do not make orbital movements. (v) They are very easy and
quick to run in terms of logistics (especially useful in the first days after
the earthquake, when it becomes impossible to reach the earthquake area by
land). (vi) It is an important alternative not only on the RAN side, but also
for the backhaul. A HAPS can easily forward all backhaul traffic to an
unaffected area via links such as FSO/THz communication with a remote HAPS or
ground station. During earthquakes, fiber optic lines can be damaged in the
earthquake area, so some BSs may be out of service. (vii) It is not physically
affected by weather events.

UAVs can also be a valuable tool for disaster communication planning and
warning. It can be used to build temporary communications infrastructure in
disaster-prone areas providing a versatile platform for gathering and
disseminating critical information in disaster-prone areas and supports
emergency response efforts. UAVs equipped with cameras or other sensors can
provide real-time information about the extent of damage, the location of
survivors, and other critical information that can be used to support
emergency response efforts.

Ground-Based Networks consist of various components, including cellular
networks, terrestrial Internet (that relies on physical cables, such as fiber
optics or copper wires, to transmit data packets over long distances), and
mobile ad hoc networks. Unlike air-based networks, the topology of the ground-
based networks is relatively stable and less mobile. In the event of a natural
disaster, such as a hurricane, earthquake, or flood, the ground-based
communication infrastructure can be severely damaged or destroyed as pointed
out in Section II-A. This can make it difficult for emergency responders and
survivors to communicate with each other. One way to overcome this challenge
is to use D2D communication as shown in Fig. 1 shaded area D. D2D
communication provides a decentralized and resilient communication solution in
earthquake scenarios. It is especially useful in scenarios where the use of
the power grid is inaccessible and the use of batteries for power is not
possible. ISAC can also play a critical role in improving the resilience of
integrated space-air-ground-sea networks with renewable energy, as shown in
Fig. 1 shaded area C [9]. Firstly, the extent of the damage caused by an
earthquake can be quickly estimated with ISAC. Sensors can be used to monitor
the integrity of buildings and communication systems can be used to transmit
this information in real time. Secondly, ISAC can be used to find survivors
trapped in the rubble. Sensors can be used to detect movement and signs of
life, while communication systems relay this information to rescue teams.
Third, ISAC can provide real-time information about the situation on the
ground, including the location of survivors, the extent of damage and the
status of rescue efforts. Finally, ISAC can facilitate communication between
response teams so that they can coordinate their efforts and share information
in real-time. This can ensure that resources are allocated effectively and
relief efforts are optimized.

Sea-Based Networks assumed critical significance during Türkiye earthquake,
where traditional communication infrastructure in coastal provinces collapsed.
In this network topology, various elements such as ships and unmanned surface
vehicles can facilitate communication by leveraging UAVs, HAPS, and satellite
technologies based on their respective locations as shown in Fig. 1 shaded
area I. They can complement the proposed architecture by improving coverage
and resiliency by offering diverse paths and reducing the risk of network
congestion and outages. They can also facilitate interoperability between
different types of networks (e.g., naval vessels, aircraft, and ground-based
command & control centers) via standard or customizable protocols enabling
seamless communication between different services and agencies.

###  III-B Energy Technology Enablers

The importance of energy in the design and operation of future networks cannot
be overstated, as global energy consumption is estimated to increase by $20$%
by 2030, driven by the growing number of subscribers, traffic demand,
connected devices, and services driven by information and communication
technologies [10]. Another critical factor is ensuring the resilience of power
systems to maintain uninterrupted communications in the event of disasters,
both during and after they occur. The term “power system resiliency” refers to
the ability of an electrical power system to mitigate the magnitude and
duration of disruptive events. Some of the possible solutions in this context
are: (i) The introduction of smart grids to replace centralized traditional
grid infrastructures opens up the possibility of incorporating distributed
generation units and microgrids based on renewable energies on a significant
scale, and thus has the potential to play a decisive role in disaster
scenarios. (ii) Ensuring a reliable power supply through the use of energy
storage systems, such as BESS, is of paramount importance. (iii) The use of EV
batteries in Vehicle to Load (V2L) applications is proving to be a viable
approach. (iv) Implementing solutions based on seismic isolators is a
promising approach to improve the seismic resilience of critical power system
components to strong shaking.

The United Nations aims to achieve net-zero CO2 emissions by 2050 to mitigate
the impacts of climate change [11]. In order to achieve this goal, RES are a
growing area of energy systems to enable decarbonization and sustainability
policies. With the high penetration rate of RES such as wind and solar energy
and the participation of RES-supported microgrids and RES-supported BSs, the
energy system becomes more resilient to disasters. In addition, it will be
possible to operate terrestrial communications systems for a much longer
period of time during disasters. When considering air networks, the large
surface area of HAPS, the lack of cooling systems required for communications
and energy components, and the potential to meet nearly all power needs
through the integration of Photovoltaic (PV) panels makes the use of RES in
air-based networks feasible.

###  III-C Integrated Approach

In this section, we provide some examples of the joint use of energy and
communication enablers. In Fig. 1 shaded area A, it is proposed to install GBS
towers in certain locations, especially in densely populated areas, and to
support these GBS towers with BESSs. In this context, the GBS is aimed to
produce its own energy with PV panels and wind turbines. The GBS can
autonomously charge the BESS from the RES and in case of a power outage, it
can utilize the stored energy from the BESS. Charging stations will also be
available on the GBS towers, allowing the charging of multiple UAVs. UAVs can
be directed to desired areas under the control of the Disaster Management
Center, and UAVs with low battery levels can also be diverted to charging
stations for recharging. UAV charging stations can be used not only in
emergency situations, but also to support areas with high communication needs,
agricultural or meteorological activities, observations, IoT applications, and
various purposes such as communication with different sensor nodes. Note that
MDRUs play an important role in the communication system and their energy
requirements are mostly covered by diesel generators. MDRUs can be transformed
into the Eco Caravan supported by RES, which is shown in Fig. 1 shaded area B.
The aim is to have a portable vehicle equipped with a BESS, solar and wind
energy. Thanks to EVs, the energy requirements of both GBS and MDRUs can be
met. In addition, thanks to GBS and MDRUs with BESS, wireless power
transmission technology for user devices can be evaluated in the event of a
disaster.

In disasters, terrestrial communications can suffer severe damage. In Fig. 1
area J, a hybrid architecture combining satellite and terrestrial
infrastructures with HAPS can be deployed. Integration of FSO and RF
technologies can be used to create integrated uplink networks connecting
space, air, sea, and ground systems. UAVs, especially HAPS, can serve as relay
stations in this setup to increase reliability. Alternatively, a standalone
solution relying solely on the satellite and HAPS segment can be used when
terrestrial infrastructure is not available. HAPS can be easily set up,
quickly forwarded to disaster areas, repaired, and easily replaced for
reconfiguration. Moreover, owing to their substantial payload capacity, HAPS
can be outfitted with PV modules, enabling the provision of wireless service
for extended durations, spanning several months. The wide coverage area
provided by HAPS makes this network more cost-effective compared to deploying
numerous terrestrial networks. This is how a ubiquitous and disaster-resistant
communication architecture can be implemented via HAPS.

##  IV Two Use Cases for Türkiye Earthquake

In this section, we propose two hybrid energy and communications methodologies
tailored to the earthquake in Türkiye using the architecture described above
with communication and energy components. The first methodology aims at
ensuring sustainable integrated HAPS-Terrestrial networks in the pre-disaster
situation. The second methodology aims to prioritize communication and
resiliency in the event of an earthquake. The proposed methodologies are
explained in the following subsections.

###  IV-A Methodology 1

The proposed hybrid methodology for pre-disaster depicted in Fig. 2 consists
of a zero-touch energy and network management system. This methodology is
primarily about ensuring sustainability. The hybrid methodology consists of
energy management with RES and BESS and the communication management system
with HAPS to ensure sustainability in GBS. In [10], it was shown that dynamic
traffic management and energy savings in dense urban areas can be achieved by
switching off terrestrial BSs with low network traffic and routing traffic to
HAPS. Therefore, in our hybrid methodology, while GBS has low network traffic,
the traffic is offloaded to HAPS by switching off the GBS. Besides, it is
proposed to offload the traffic for new user arrivals to HAPS when GBS is
approaching congestion status. This also eliminates the need of network
densification to meet dynamic and unprecedented data traffic in the highly-
dense areas that is conventionally handled by adding additional
permanent/temporary GBS. In this regard, the proposed methodology promises
sustainability in pre-disaster scenarios. According to detailed analysis in
[10], the proposed HAPS-integrated methodology is also economically viable in
terms of capital and operational expenditure.

Figure 2: Methodology 1 for pre-disaster.

###  IV-B Methodology 2

The proposed hybrid methodology for the disaster case shown in Fig. 3, aims to
prioritize communication in the event of an earthquake. This methodology aims
to ensure communication via the HAPS and to protect the batteries of the GBS
if the GBS is deactivated due to a power failure. In cases where communication
with HAPS is lost or the service is not available, GBS aims to primarily meet
the energy requirements with BESS considering the State of Charge (SOC). In
scenarios where the BESS is not charged, proposing a solution involves
directing volunteer EVs or mobile BESSs to the GBS for BESS charging. Another
alternative for energy needs is the use of other supporting resources such as
backup batteries on GBS, diesel generator units, etc. In the scenario where
the GBS is disabled due to an interruption in the fiber optic line (backbone),
the aim is to provide a backhaul connection with the HAPS if the GBS has
energy over the grid. In the event of a power failure in the GBS, all
communication is handled via HAPS. In cases where communication with HAPS is
lost or the service is not available, satellite communication can be
considered as an alternative.

Figure 3: Methodology 2 for in-disaster.

###  IV-C Simulation Results for the Türkiye Earthquake Case Study

In order to demonstrate the RAN capabilities of HAPS networks in Fig. 3 \-
Methodology 2, we show in Fig. 4 computer simulation results for the
Cumulative Distribution Function (CDF) of receiver power at ground users. As
stated in [12], the RAN connection between a HAPS (or HAPS constellation) and
the users on the ground can be established in both S- and Ka-bands. In S-band,
mobile handheld devices are supported, while in Ka-band Very Small Aperture
Terminals (VSAT) (i.e., $60$ cm circular polarisation) are required, which can
be permanently installed or mounted on mobile platforms (e.g., vehicles, UAVs,
etc.). During the earthquakes in Türkiye, the total affected area (11
provinces) is about $115,000$ km2 with a total of 12 million mobile
subscribers [1]. Therefore, for the S-band simulations (Fig. 4 (a)), all $12$
million mobile subscribers are randomly distributed in the area (for
simplicity) and served by a single HAPS, $2$-HAPS or $4$-HAPS. In contrast, in
the Ka-band simulations (Fig. 4 (b)), we assume that $1$ million VSAT
subscribers are randomly distributed in the same area and served by the same
amount of HAPS network. In all simulations, the parameters of HAPS (e.g.,
transmit power, antenna gain, etc.) are used based on [13]. Air-to-ground
channel characteristics (e.g., LoS probability, path loss, shadowing,
atmospheric loss, etc.) and the receiver parameters (e.g., antenna gain,
sensitivity, etc.) are also taken from [12]. From the extensive simulations,
we can conclude that even a single HAPS could restore all communications in
the region \- and thus replace all failed ground BSs -as none of the users
receive a signal lower than the receiver sensitivity. Nevertheless, by
increasing the number of HAPS, we were able to improve quality-of-service
(QoS) thanks to the $10$-$20$ dB increase in received SNR by each HAPS. As
expected, the performance deteriorates in the dense urban scenarios due to
lower LoS probability and higher losses (e.g., shadowing, clutter losses,
etc.) [12]. On the other hand, in Fig. 4 (b), we can observe an average $20$
dB higher received power than in Fig. 4 (a) in all scenarios. This can be
explained as follows. Although the path loss is greater in Ka-band, the
antenna gain of the VSAT receiver can compensate for this and provide a higher
QoS [13]. In summary, our simulation results show that even in the scenario in
which all terrestrial BSs are deactivated, the communication needs in the
earthquake-damaged region can be met with HAPS.

(a) Mobile handheld devices that operate in S-band.

(b) Very small ($\approx 60$cm) aperture terminal users operate in Ka-band.

Figure 4: CDF of received power at the ground terminals for a HAPS-RAN
communication in the whole affected region. Table I: Comparison of
Communication Enablers for different KPIs in earthquake scenarios with
proposed hybrid methodologies.

KPIs Technology | Satellite | HAPS | UAV | Terrestrial | D2D | Methodology 1 | Methodology 2
---|---|---|---|---|---|---|---
Latency | High | Medium | Low | Low | Low | Medium | Medium
Bandwidth | High | High | Medium | Medium | Low | High | High
Resiliency | High | High | Medium | Low | Medium | Medium | High
Scalability | High | High | Medium | Medium | Medium | High | High
Security | High | High | Medium | High | Medium | Medium | Medium
Interoperability | Medium | Medium | Medium | High | Low | Medium | Medium
Cost-efficiency | Medium | High | Medium | Medium | Low | High | High
Energy efficiency | Medium | High | Medium | Low | High | High | Medium

Our proposed methodologies combine terrestrial and HAPS technologies, but
their objectives are different depending on the scenario considered. To
compare our integrated hybrid methodologies with the primary communication
enablers, Table I provides comparisons in terms of various key performance
indicators (KPIs). According to the KPIs in Table 1, both Methodology 1 and
Methodology 2 perform medium in terms of interoperability, latency and
security but performs high in terms of bandwidth and scalability. Compared to
Methodology 2, the proposed Methodology 1 is better in terms of energy
efficiency. However, Methodology 2 provides better communication resilience.

##  V Challenges and Future Research Directions

The proposed architecture includes the use of different technologies,
protocols, and strategies to ensure seamless communication and energy support
in disaster scenarios. The challenges of coordinating the above elements
include ensuring seamless interoperability, overcoming communication latency
issues, and managing limited resources in a dynamic and often chaotic disaster
environment. How future networks can be managed to ensure seamless
connectivity, interoperability, and efficient resource utilization must be
considered as a future research direction. Note that there are some proposals
for solving these problems in the field of integrated networks [14, 15], which
could be used to find solutions. To achieve a disaster-resilient network
architecture, some of the future solutions are as follows: (i) Artificial
Intelligence (AI) and blockchain networks: Solutions that rely on AI-based
techniques can be used to solve various disaster communication planning
problems both in pre-disaster, in-disaster and post-disaster scenarios. In
addition, AI together with blockchain frameworks can improve remote data
protection and secure energy trading. (ii) Cloud native deployments: To
improve network scalability and flexibility of networks in disaster zones,
cloud-native deployments of network services can be used in space-air-ground-
sea networks. This can also reduce development cycles, capital, and
operational expenditures while enabling platforms to be open. Container
orchestration tools like Kubernetes can help with cloud-native deployments.
(iii) Energy harvesting: If the user’s devices are able to use radio, solar or
vibration energy from the environment, energy consumption constraints can be
significantly reduced.

##  VI Conclusions

In this paper, an integrated space-air-ground-sea network architecture is
used, which uses various communication and energy-enabling technologies, to
improve resiliency and sustainability in disaster scenarios. The integrated
communication network presented includes satellite, HAPS, UAV, and D2D
technologies. In addition, the integration of various RESs (e.g., wind
turbines and solar panels) together with innovative energy enablers (e.g.,
BESS and EVs) is proposed to ensure sustainable energy management in disaster
situations. Specifically targeting the recent earthquakes in Türkiye, two
network management methodologies are put forth. The performance of the
proposed framework is validated through the simulation of a case study
demonstrating the strength of HAPS communication in disaster areas. Future
research and practical implementation in this direction holds great potential
for the development of more resilient and sustainable disaster management
strategies.

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Bilal Karaman<EMAIL_ADDRESS>is a Research Assistant at Manisa
Celal Bayar University, Türkiye. He is also a Visiting Researcher at Carleton
University, Ottawa, Ontario, Canada.
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Ilhan Basturk [M]<EMAIL_ADDRESS>is an Associate Professor at
Manisa Celal Bayar University, Türkiye.
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Sezai Taskin<EMAIL_ADDRESS>is a Professor at Manisa Celal Bayar
University, Türkiye.
---
Ferdi Kara [SM]<EMAIL_ADDRESS>is an Assistant Professor at Zonguldak
Bulent Ecevit University, Türkiye. He is also a Postdoctoral Fellow at KTH
Royal Institute of Technology, Stockholm, Sweden.
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Engin Zeydan [SM]<EMAIL_ADDRESS>is a Senior Researcher at Centre
Tecnològic de Telecomunicacions de Catalunya (CTTC), Spain.
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Halim Yanikomeroglu [F]<EMAIL_ADDRESS>is a Chancellor’s Professor at
Carleton University, Ottawa, ON, CA.
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