# Finite-Time Capacity: Making
Exceed-Shannon Possible?

Jieao Zhu, Zijian Zhang, Zhongzhichao Wan, and Linglong Dai  All authors are
with the Beijing National Research Center for Information Science and
Technology (BNRist) as well as the Department of Electronic Engineering,
Tsinghua University, Beijing 100084, China (E-mails: zja21, zhangzj20,
<EMAIL_ADDRESS>daill@tsinghua.edu.cn). This work was supported
in part by the National Key Research and Development Program of China (Grant
No.2020YFB1807201), in part by the National Natural Science Foundation of
China (Grant No. 62031019), and in part by the European Commission through the
H2020-MSCA-ITN META WIRELESS Research Project under Grant 956256.

###### Abstract

Shannon-Hartley theorem can accurately calculate the channel capacity when the
signal observation time is infinite. However, the calculation of finite-time
capacity, which remains unknown, is essential for guiding the design of
practical communication systems. In this paper, we investigate the capacity
between two correlated Gaussian processes within a finite-time observation
window. We first derive the finite-time capacity by providing a limit
expression. Then we numerically compute the maximum transmission rate within a
single finite-time window. We reveal that the number of bits transmitted per
second within the finite-time window can exceed the classical Shannon
capacity, which is called as the Exceed-Shannon phenomenon. Furthermore, we
derive a finite-time capacity formula under a typical signal autocorrelation
case by utilizing the Mercer expansion of trace class operators, and reveal
the connection between the finite-time capacity problem and the operator
theory. Finally, we analytically prove the existence of the Exceed-Shannon
phenomenon in this typical case, and demonstrate the achievability of the
finite-time capacity and its compatibility with the classical Shannon
capacity.

###### Index Terms:

Finite-time capacity, Exceed-Shannon, Mercer expansion, signal
autocorrelation, operator theory.

##  I Introduction

The Shannon-Hartley theorem [1] has accurately revealed the fundamental
theoretical limit of information transmission rate $C$, which is also called
as the Shannon capacity, over a Gaussian waveform channel of a limited
bandwidth $W$. The expression for Shannon capacity is
$C=W\log\left(1+S/N\right)$, where $S$ and $N$ denote the signal power and the
noise power, respectively. The derivation of Shannon-Hartley Theorem heavily
depends on the Nyquist sampling principle [2]. The Nyquist sampling principle,
which is also named as the $2WT$ theorem [3], claims that one can only obtain
$2WT+o(2WT)$ independent samples within an observation time window $T$ in a
channel band-limited to $W$ [4], where $o(\cdot)$ means higher-order
infinitesimal, i.e., $o(2WT)/T\to 0,T\to\infty$.

Based on the Nyquist sampling principle, the Shannon capacity is derived by
multiplying the capacity $1/2\log(1+P/N)$ of a Gaussian symbol channel [5,
p.249] with $2WT+o(2WT)$ at first, and then dividing the result by $T$,
finally letting $T\rightarrow\infty$. In the above derivation,
$(2WT+o(2WT))/T$ is approximated by $2W$ in the final step to obtain the
Shannon capacity. Note that this approximation only holds when $T\to\infty$.
Therefore, the Shannon capacity only asymptotically holds as $T$ becomes
sufficiently large. When $T$ is of finite value, the approximation fails to
work. Thus, when the observation time $T$ is finite, i.e., the received signal
can only be observed within a finite-time window $[0,T]$, Shannon-Hartley
Theorem cannot be directly applied to calculate the capacity in a finite-time
window. To the best of our knowledge, the evaluation of the finite-time
capacity has not yet been investigated in the literature. One possible reason
is that, most of the researchers mainly focused on how to approximate the
Shannon capacity with advanced coding and modulation schemes. It is worth
noting that any real-world communication systems transmit signals in a finite-
time window, thus evaluating the finite-time capacity is of practical
significance.

In this paper, to fill in this gap, the finite-time capacity instead of the
traditional infinite-time counterpart is analyzed, where we reveal and prove
the existence of “Exceed-Shannon” phenomenon within a finite-time observation
window111Simulation codes will be provided to reproduce the results presented
in this paper:
http://oa.ee.tsinghua.edu.cn/dailinglong/publications/publications.html..
Specifically, our contributions are summarized as follows:

  * •

We derive the capacity expressions within a finite-time observation window by
using dense sampling and limiting methods. In this way, we can overcome the
continuous difficulties that appear when analyzing the information contained
in a continuous time interval. These finite-time capacity expressions make the
analysis of finite-time capacity problems possible.

  * •

We approximate the original continuous finite-time capacity expressions by
discrete matrices, and conduct numerical experiments based on the discretized
formulas. In the numerical results under a special setting, we reveal the
“Exceed-Shannon” phenomenon222In fact, the finite-time “Exceed-Shannon”
phenomenon revealed in this paper does not contradict the classical infinite-
time Shannon-Hartley theorem, since new assumptions are considered.
Specifically, in the Shannon-Hartley theorem, the sampling time is assumed to
be infinitely long, while in this paper, the sampling takes place in a finite-
time observation window. Similarly, although compressed sensing[6] can achieve
much lower sampling rate than the Nyquist sampling rate to perform accurate
sparse signal reconstruction, it does not contradict the Nyquist sampling
principle due to the new assumption of signal sparsity., i.e., the mutual
information within a finite-time observation window exceeds the Shannon
capacity.

  * •

In order to analytically prove the revealed “Exceed-Shannon” phenomenon, we
first derive an analytical finite-time capacity formula based on Mercer
expansion [7], where we can find the connection between the capacity problem
and the operator theory [8]. To make the problem tractable, we construct a
typical case in which the transmitted signal has certain statistical
properties. Utilizing this construction, we obtain a closed-form capacity
solution in this typical case, which leads to a rigorous proof of the “Exceed-
Shannon” phenomenon. Inspired by the techniques in the proof, we find that the
finite-time capacity is, in fact, a more general case of the Shannon limit,
thus the “Exceed-Shannon” phenomenon of the finite-time capacity is compatible
with the classical Shannon theory.

Organization: In the rest of this paper, the finite-time capacity is
formulated and evaluated numerically in Section II, where the “Exceed-Shannon”
phenomenon is first discovered. Then, in Section III, we derive a closed-form
finite-time capacity formula under a typical case. Based on this formula, in
Section IV, the “Exceed-Shannon” phenomenon is rigorously proved. Finally,
conclusions are drawn in Section V.

Notations: $X(t)$ denotes a Gaussian process; $R_{X}(t_{1},t_{2})$ denotes the
autocorrelation function; $S_{X}(f),S_{X}(\omega)$ are the power spectral
density (PSD) of the corresponding process $X(t)$, where $\omega=2\pi f$;
Boldface italic symbols $\bm{X}(t_{1}^{n})$ denotes the column vector
generated by taking samples of $X(t)$ on instants $t_{i},1\leq i\leq n$;
Upper-case boldface letters such as ${\bf{\Phi}}$ denote matrices;
$\mathbb{E\left[\cdot\right]}$ denotes the expectation;
$\mathbbm{1}_{A}(\cdot)$ denotes the indicator function of the set $A$;
$\mathcal{L}^{2}([0,T])$ denotes the collection of all the square-integrable
functions on window $[0,T]$; ${\rm i}$ denotes the imaginary unit.

##  II Numerical Analysis of the Finite-Time Capacity

In this section, we focus on the numerical evaluation of the finite-time
capacity. In Subsection II-A, we model the transmission problem by Gaussian
processes, and derive the capacity expressions within a finite-time
observation window by using dense sampling and limiting methods; In Subsection
II-B, we approximate the finite-time capacity by discretized matrix-based
formulas; In Subsection II-C, we reveal the “Exceed-Shannon” phenomenon by
numerically evaluating the finite-time capacity in a special setting of the
signal autocorrelations.

###  II-A The Expressions for Finite-Time Capacity

The finite-time capacity is, heuristically, defined as the maximum number of
bits that can be successfully transmitted within a finite-time window. Since
Shannon capacity is defined on pairs of random variables, it is crucial to
introduce randomness into the transmission model. Inspired by [9], we model
the transmitted signal by a zero-mean stationary Gaussian stochastic process,
denoted as $X(t)$, and the received signal by $Y(t):=X(t)+N(t)$. The process
$N(t)$, which denotes the noise, is also a stationary Gaussian process
independent of $X(t)$. The receiver is only allowed to observe the signal
within finite-time window $[0,T]$, where $T>0$ is the observation window span.
Our goal is to find the maximum number of bits that can be acquired within
this time window.

To analytically express the amount of the acquired information, we first
introduce $n$ sampling instants inside the time window, denoted by
$(t_{1},t_{2},\cdots,t_{n}):=t_{1}^{n}$, and then let $n\rightarrow\infty$ to
approximate the finite-time capacity333In this paper, we do not explicitly
distinguish between the terms “finite-time mutual information” and “finite-
time capacity”, since we consider communication schemes where the source
autocorrelation is fixed.. This approximation of capacity becomes more precise
as the sampling instants $t_{1}^{n}$ become denser. Then by defining
${\bm{X}}(t_{1}^{n})\equiv(X(t_{1}),X(t_{2}),\cdots,X(t_{n}))$ and
${\bm{Y}}(t_{1}^{n})\equiv(Y(t_{1}),Y(t_{2}),\cdots,Y(t_{n}))$, the capacity
on these $n$ samples can be expressed as

$I(t_{1}^{n})=I({\bm{X}}(t_{1}^{n});{\bm{Y}}(t_{1}^{n})),$ (1)

and the finite-time capacity is defined as

$I(T)=\lim_{n\rightarrow\infty}\sup_{\\{t_{1}^{n}\\}\subset[0,T]}{I(t_{1}^{n})}.$
(2)

Then, the transmission rate $C(T)$ can be defined by dividing the amount of
information acquired within $[0,T]$ by the time span $T$:

$\displaystyle C(t_{1}^{n})$ $\displaystyle=I(t_{1}^{n})/T,$ (3)
$\displaystyle C(T)$ $\displaystyle=I(T)/T.$

From these definitions, we can define the limit capacity as
$C(\infty)=\lim_{T\rightarrow\infty}{C(T)}$ by letting $T\rightarrow\infty$.
The quantity $C(\infty)$ characterizes the maximum average number of bits per
second one can acquire from a received noisy stochastic process.

###  II-B Discretization

Without loss of generality, we fix the sampling instants uniformly onto
fractions of $T$: $t_{i}=(i-1)T/n,1\leq i\leq n$. Since the random vectors
${\bm{X}}(t_{1}^{n})$ and ${\bm{Y}}(t_{1}^{n})$ are samples of a Gaussian
process, they are both Gaussian random vectors with mean zero and covariance
matrices ${\bm{K}}_{X}$ and ${\bm{K}}_{Y}$, where
${\bm{K}}_{X},{\bm{K}}_{Y}\in\mathbb{R}^{n\times n}$ are symmetric positive-
definite matrices. The entries of ${\bm{K}}_{X}$ and ${\bm{K}}_{Y}$ are
determined by the autocorrelation functions of Gaussian processes $X(t)$ and
$Y(t)$, denoted by $R_{X}(t_{1},t_{2})$ and $R_{Y}(t_{1},t_{2})$:

$\displaystyle({\bm{K}}_{X})_{i,j}$
$\displaystyle=R_{X}(t_{i},t_{j}):=\mathbb{E}\left[X(t_{i})X(t_{j})\right],$
(4) $\displaystyle({\bm{K}}_{Y})_{i,j}$
$\displaystyle=R_{Y}(t_{i},t_{j}):=\mathbb{E}\left[Y(t_{i})Y(t_{j})\right].$

Note that $Y(t)$ is the independent sum of $X(t)$ and $N(t)$, thus the
autocorrelation functions satisfy
$R_{Y}(t_{1},t_{2})=R_{X}(t_{1},t_{2})+R_{N}(t_{1},t_{2})$, and similarly the
covariance matrices satisfy ${\bm{K}}_{Y}={\bm{K}}_{X}+{\bm{K}}_{N}$.

The mutual information $I(t_{1}^{n})$ is defined as
$I(t_{1}^{n})=h({\bm{Y}}(t_{1}^{n}))-h({\bm{Y}}(t_{1}^{n})|{\bm{X}}(t_{1}^{n}))=h({\bm{Y}}(t_{1}^{n}))-h({\bm{N}}(t_{1}^{n}))$,
where $h(\cdot)$ denotes the differential entropy. For $n$-dimentional
Gaussian vector ${\bm{U}}$ with mean 0 and covariance matrix ${\bm{K}}$, the
differential entropy is given by

$h({\bm{U}})=\frac{1}{2}\log\left((2\pi e)^{n}\det({\bm{K}})\right).$ (5)

Plugging (5) into the definition of $I(t_{1}^{n})$, we obtain

$I(t_{1}^{n})=\frac{1}{2}\log\left(\frac{\det({\bm{K}}_{X}+{\bm{K}}_{N})}{\det{({\bm{K}}_{N})}}\right).$
(6)

In (6), by letting $n\rightarrow\infty$, we can find that $I(t_{1}^{n})$
increases monotonously when $n$ doubles, because of the data processing
inequality. Though without rigorous proof, we can assume with confidence that
$I(t_{1}^{n})$ is an increasing function of $n$. However, it remains unknown
whether $I(t_{1}^{n})$ tends to a finite limit. In fact, $I(t_{1}^{n})$ can be
arbitrarily large, since the signal outside the noise band is strictly
unpolluted by the noise, which results in infinite SNR. Thus, the capacity
will diverge to infinity. Therefore, in order to avoid capacity divergence, at
least one of the following conditions should be satisfied:

  * •

The noise process $N(t)$ is not band-limited.

  * •

The power spectral density of $X(t)$ is strictly contained inside the band of
$N(t)$.

Thus, in the following numerical analysis, we choose $N(t)$ to be band-
unlimited. This leads to the choice of reasonable autocorrelation functions of
$X(t)$ and $N(t)$ in the following subsection.

###  II-C Numerical Analysis

In order to study the properties of mutual information $I(t_{1}^{n})$ as a
function of $n$, we perform numerical analyses under different values of $n$
and $T$. The autocorrelation function and PSD of the signal process $X(t)$ and
noise process $N(t)$ are set to the special case

$\displaystyle R_{X}(t_{1},t_{2})$
$\displaystyle=\mathrm{sinc}(10(t_{1}-t_{2})),$ (7) $\displaystyle
R_{N}(t_{1},t_{2})$ $\displaystyle=\exp(-\lvert t_{1}-t_{2}\rvert),$
$\displaystyle S_{X}(f)$ $\displaystyle=0.1\times\mathbbm{1}_{\\{-5\leq f\leq
5\\}},$ $\displaystyle S_{N}(f)$ $\displaystyle=\frac{2}{1+(2\pi f)^{2}},$

where $\mathrm{sinc}(x):=\sin(\pi x)/(\pi x)$. Note that the PSD of the
transmitted process $S_{X}(f)$ is strictly band-limited, while the PSD of the
noise process is not. In fact, the noise PSD is carefully selected to ensure
the received noise has finite power on each instant $t_{i}$, allowing the
execution of numerical computations. A finitely powered process PSD must be
colored, in contrast to additive white Gaussian noise (AWGN) with white PSD
and infinite power. That is the reason why we choose $S_{N}(f)$ to be the form
of (7).

In order to compare the finite-time capacity with the classical Shannon
capacity, we have to calculate the Shannon capacity with colored noise
spectrum $S_{N}(f)$, which is a generalized version of the well-known formula
$C=W\log\left(1+S/N\right)$. The Shannon capacity $C_{\rm sh}$ of colored
noise PSD [5], measured in ${\rm nat/s}$, is expressed as

$C_{\rm
sh}:=\frac{1}{2}\int_{-\infty}^{+\infty}{\log\left(1+\frac{S_{X}(f)}{S_{N}(f)}\right)\mathrm{d}f}\quad[{\rm
nat/s}].$ (8)

Then, plugging (7) into (8) yields the numerical result for $C_{\rm sh}$.

In the numerical analysis, we calculate the finite-time transmission rate
$C(T)$ and Shannon capacity against the number of samples $n$ within the
observation window $[0,T]$. The numerical results are collected in Fig. 1. It
is shown that $I(t_{1}^{n})$ is an increasing function of $n$, and for fixed
values of $T$, the approximated finite-time capacity $I(t_{1}^{n})$ tends to a
finite limit under the correlation assumptions given by (7). The most amazing
observation is that, we can obtain more information within finite-time window
$[0,T]$ than the prediction $TC_{\rm sh}$ given by the Shannon capacity (8).
We call this phenomenon the “Exceed-Shannon” phenomenon.

Figure 1: A first glance to the “Exceed-Shannon” phenomenon, where the red
dashed horizontal line is the Shannon capacity, and the $T=1,2,8$ curves
illustrate the dependence of $C(t_{1}^{n})$ on $n$.

To analytically verify the existence of the Exceed-Shannon phenomenon, we are
going to introduce some mathematical tools in the following Section III, and
finally give an analytical proof in Section IV.

##  III A Closed-Form Finite-Time Capacity Formula

In this section, we first introduce the Mercer expansion in Subsection III-A
as a basic tool for our analysis. Then we derive the series representation of
the finite-time capacity, and the corresponding power constraint in Subsection
III-B, under the assumption of AWGN noise. The power constraint shows that the
finite-time capacity is upper-bounded, thus the series expansion of the
finite-time capacity converges absolutely.

###  III-A The Mercer Expansion

Motivated by the discovery of the Exceed-Shannon phenomenon, we go further
into the underlying mechanism behind this fact. Since the calculation of (6)
depends on the evaluation of determinants that are determined by
autocorrelation functions of Gaussian processes, it is possible to obtain
$I(t_{1}^{n})$ and $I(T)$ directly from the autocorrelation functions. In
fact, if we know the Mercer expansion [7] of the autocorrelation function
$R_{X}(t_{1},t_{2})$ on window $[0,T]$, then we can calculate
$h({\bm{X}}(t_{1}^{n}))$ more easily [10]. In the following discussion, we
assume the Mercer expansion of the source autocorrelation function
$R_{X}(t_{1},t_{2})$ to be in the following form

$\lambda_{k}\phi_{k}(t_{1})=\int_{0}^{T}{R_{X}(t_{1},t_{2})\phi_{k}(t_{2})\mathrm{d}t_{2}};k>0,k\in\mathbb{N}.$
(9)

Due to the positive-definite property of the integral kernel
$R_{X}(t_{1},t_{2})$, the eigenvalues are strictly positive: $\lambda_{k}>0$,
and the eigenfunctions form an orthonormal set:

$\int_{0}^{T}{\phi_{i}(t)\phi_{j}(t)\mathrm{d}t}=\delta_{ij}.$ (10)

The Mercer’s theorem [7] ensures the existence and uniqueness of the
eigenpairs $(\lambda_{k},\phi_{k}(t))_{k=1}^{\infty}$, and furthermore, the
kernel itself can be expanded under the eigenfunctions. The convergence is
absolute and uniform:

$R_{X}(t_{1},t_{2})=\sum_{k=1}^{+\infty}{\lambda_{k}\phi_{k}(t_{1})\phi_{k}(t_{2})}.$
(11)

The Mercer expansion enables us to analytically express an autocorrelation
function on a finite-time interval $[0,T]$, since the autocorrelation function
can be naturally treated as a positive-definite integral kernel.

###  III-B Finite-Time Capacity Formula

Based on Mercer expansion, we can obtain a closed-form formula in the
following Theorem 1.

######  Theorem 1 (Source expansion, AWGN noise)

Suppose the information source, modeled by the transmitted process $X(t)$, has
autocorrelation function $R_{X}(t_{1},t_{2})$. An AWGN noise of PSD $n_{0}/2$
is imposed onto $X(t)$, resulting in the received process $Y(t)$. The Mercer
expansion of $R_{X}(t_{1},t_{2})$ on $[0,T]$ is given by (9), satisfying (10).
Then the finite-time mutual information $I(T)$ within the observation window
$[0,T]$ between the processes $X(t)$ and $Y(t)$ can be expressed as

$I(T)=\frac{1}{2}\sum_{k=1}^{+\infty}{\log\left(1+\frac{\lambda_{k}}{n_{0}/2}\right)}.$
(12)

###### Proof:

See Appendix A. ∎

From Theorem 1, we can conclude that the finite-time capacity of AWGN channel
is uniquely determined by the Mercer spectra $\lambda_{k}$ of
$R_{X}(t_{1},t_{2})$ within $[0,T]$. However, it remains unknown whether the
series representation (12) converges. In fact, the convergence is closely
related to the signal power. In Fourier transform, the power in the time
domain is equal to the power in the frequency domain, which is known as the
Parseval’s theorem[11]. Like the Fourier transform, the transform defined by
the orthonormal basis $\\{\phi_{k}(t)\\}_{k=1}^{\infty}$ also satisfy the
Parseval’s theorem. This observation leads to a theoretic verification of
power conservation in the view of $\lambda_{k}$, which is stated in the
following Lemma 1.

######  Lemma 1 (Operator Trace Coincide with Power Constraint)

Given stationary Gaussian process $X(t)$ with mean zero and autocorrelation
$R_{X}(t_{1},t_{2})$. The Mercer expansion of $R_{X}(t_{1},t_{2})$ on $[0,T]$
is given by (9), satisfying (10). The Mercer operator
$M(\cdot):\mathcal{L}^{2}([0,T])\rightarrow\mathcal{L}^{2}([0,T])$ is defined
by the integral
$(M\phi)(s)=\int_{0}^{T}{R_{X}(s,\tau)\phi(\tau)\mathrm{d}\tau}$. Then the sum
of all the eigenvalues $\lambda_{k}$ of operator $M$ is equal to the signal
energy $PT$ within $[0,T]$:

$\mathrm{tr}(M):=\sum_{k=1}^{+\infty}{\lambda_{k}}=PT,$ (13)

where $P=R_{X}(0,0)$.

###### Proof:

See Appendix B. ∎

###### Remark 1

The convergence of the finite-time capacity series (12) is ensured by Lemma 1.
In fact, from the above Lemma 1, we can conclude that the sum of $\lambda_{k}$
is finite when $T$ is finite. It can be immediately derived that
$I(T)<\infty$, since $\log(1+x)\leq x$. Furthermore, note that the sum of
$\lambda_{k}$ is finite even for non-stationary processes (i.e., the power at
time $t$: $P(t):=\mathbb{E}\left[X^{2}(t)\right]$ is not always a constant
$P:=R_{X}(0,0)$), as long as $P(t)<\infty$ holds for any $0<t<T$. Then the
conclusion $I(T)<\infty$ holds even for non-stationary processes.

###### Remark 2

The finite-time capacity formula (12) is closely related to the operator
theory [8] in functional analysis. The sum of all the eigenvalues
$\lambda_{k}$ is called the operator trace in linear operator theory. As is
mentioned in Lemma 1, the autocorrelation function $R_{X}(t_{1},t_{2})$ can be
treated as a linear operator $M$ on $\mathcal{L}^{2}([0,T])$. Furthermore,
this operator belongs to the trace class [12] if and only if
$\int_{0}^{T}{R_{X}(t,t)\mathrm{d}t}<\infty$. Note that this condition is
automatically satisfied if $X(t)$ is a Gaussian process, since Gaussian random
variables always have finite variances.

The Mercer spectra enables us to explicitly calculate the finite-time
capacity, and furthermore, prove the “Exceed-Shannon” phenomenon. This will be
demonstrated in the next section.

##  IV Proof of the Existence of Exceed-Shannon Phenomenon

In this section, we first give two different proofs of the existence of the
Exceed-Shannon phenomenon, both in a typical case. Then we discuss the
achievability of the finite-time capacity, and the compatibility with Shannon-
Hartley Theorem.

###  IV-A Closed-Form Capacity in A Typical Case

In order to show the existence of Exceed-Shannon phenomenon, we only need to
show that the finite-time capacity is greater than Shannon capacity in a
typical case. Let us consider a finite-time communication scheme with a
finitely-powered stationary transmitted signal autocorrelation444The signal
autocorrelation $R_{X}(\tau)$ is often observed in many scenarios, such as
passing a signal with white spectrum through an RC lowpass filter., which is
specified as

$R_{X}(t_{1},t_{2})=R_{X}(\tau)=P\exp(-\alpha\lvert\tau\rvert),$ (14)

where $\tau=t_{1}-t_{2}$, in AWGN channel555In this theoretical proof of
“Exceed-Shannon” phenomenon, we assume the noise to be AWGN, to simplify the
analytical computations. Gaussian processes of white spectrum are “immoral”,
thus they can neither be power-limited, nor they can be directly sampled and
numerically represented in computers. with noise PSD being $n_{0}/2$. The
power of signal $X(t)$ is $P=R_{X}(0)$. According to Lemma 1, the trace of the
corresponding Mercer operator $M(\cdot)$ is finite. Then the finite-time
capacity given by Theorem 1 is also finite, as is shown in Remark 1. Finding
the Mercer expansion is equivalent to finding the eigenpairs
$(\lambda_{k},\phi_{k}(t))_{k=1}^{\infty}$. The eigenpairs are determined by
the following characteristic integral equation [13]:

$\lambda_{k}\phi_{k}(s)=\int_{0}^{s}{Pe^{-\alpha(s-t)}\phi_{k}(t)\mathrm{d}t}+\int_{s}^{T}{Pe^{-\alpha(t-s)}\phi_{k}(t)\mathrm{d}t}.$
(15)

Differentiating both sides of (15) twice with respect to $s$ yields the
boundary conditions and the differential equation that $\lambda_{k},\phi_{k}$
must satisfy:

$\displaystyle\lambda_{k}\phi_{k}^{\prime\prime}(s)$
$\displaystyle=(\alpha^{2}\lambda_{k}-2\alpha P)\phi_{k}(s),0<s<T,$ (16)
$\displaystyle\phi_{k}^{\prime}(0)$ $\displaystyle=\alpha\phi_{k}(0),$
$\displaystyle\phi_{k}^{\prime}(T)$ $\displaystyle=-\alpha\phi_{k}(T).$

Let $\omega_{k}>0$ denote the resonant frequency of the above harmonic
oscillator differential equation, then $\lambda_{k}=2\alpha
P/(\alpha^{2}+\omega_{k}^{2})$, and $\omega_{k}$ must satisfy the above two
boundary constraints. Let
$\phi_{k}(t)=A_{k}\cos(\omega_{k}t)+B_{k}\sin(\omega_{k}t)$ be the sinusoidal
form of the eigenfunction. Using the boundary conditions we obtain

$\displaystyle B_{k}\omega_{k}=\alpha A_{k},$ (17) $\displaystyle
B_{k}\omega_{k}\cos(\omega_{k}T)-A_{k}\omega_{k}\sin(\omega_{k}T)$
$\displaystyle=-\alpha\left(A_{k}\cos(\omega_{k}T)+B_{k}\sin(\omega_{k}T)\right).$

To ensure the existence of solution to the homogeneous linear equations (17)
with unknowns $A_{k},B_{k}$, the determinant must be zero. Exploiting this
condition, we find the equation that $\omega_{k}$ must satisfy:

$\tan(\omega_{k}T)=\frac{2\omega_{k}\alpha}{\omega_{k}^{2}-\alpha^{2}}.$ (18)

Figure 2: Images of $2\arctan(\omega/\alpha)$ and $k\pi-\omega T$ with
$T=2,\alpha=1$ and $k=1,2,3,4$. The desired resonant frequencies $\omega_{k}$
can be read from the horizontal coordinates of the intersection points of the
curve $2\arctan(\omega/\alpha)$ and the parallel lines.

By introducing an auxillary variable
$\theta_{k}=\arctan(\omega_{k}/\alpha)\in[0,\pi/2]$, equation (18) can be
simplified as $\tan(\omega_{k}T)=-\tan(2\theta_{k})$, i.e., there exists a
positive integer $m$ such that $2\arctan(\omega_{k}/\alpha)=m\pi-\omega_{k}T$.
The integer $m$ can be chosen to be equal to $k$. From the function images of
$2\arctan(\omega/\alpha)$ and $m\pi-\omega T$ (Fig. 2), we can determine
$\omega_{k}$, and then $\lambda_{k}$. To sum up, the solution to the
characteristic equation (15) are collected into (19) as follows.

$\displaystyle 2\arctan(\omega_{k}/\alpha)$ $\displaystyle=k\pi-\omega_{k}T,$
(19) $\displaystyle\lambda_{k}$ $\displaystyle=\frac{2\alpha
P}{\alpha^{2}+\omega_{k}^{2}},$ $\displaystyle\phi_{k}(t)$
$\displaystyle=\frac{1}{Z_{k}}\left(\omega_{k}\cos(\omega_{k}t)+\alpha\sin(\omega_{k}t)\right),$

where $Z_{k}$ denotes the normalization constants of $\phi_{k}(t)$ on $[0,T]$
to ensure orthonormality.

Equation (19) gives all eigenpairs $(\lambda_{k},\phi_{k})_{k=1}^{\infty}$,
from which we can calculate $I(T)$ by applying Theorem 1. As for the Shannon
capacity $C_{\rm sh}$, by applying (8) and evaluating the integral with[14],
we can obtain

$\displaystyle C_{\rm sh}$
$\displaystyle=\frac{1}{4\pi}\int_{-\infty}^{+\infty}{\log\left(1+\frac{S_{X}(\omega)}{n_{0}/2}\right)\mathrm{d}\omega}$
(20)
$\displaystyle=\frac{1}{4\pi}\int_{-\infty}^{+\infty}{\log\left(1+\frac{\frac{2P\alpha}{\alpha^{2}+\omega^{2}}}{n_{0}/2}\right)\mathrm{d}\omega}$
$\displaystyle\overset{(a)}{=}\frac{1}{2}\left(\sqrt{\alpha^{2}+\frac{4P\alpha}{n_{0}}}-\alpha\right),$

where the evaluation of the improper integral $(a)$ is given in Appendix E.

After all the preparation works above, we can rigorously prove that
$C(T)>C_{\rm sh}$ under the typical case of (14), as long as the transmission
power $P$ is smaller than a constant $\delta$. The following Theorem 2 proves
this result.

######  Theorem 2 (Existence of Exceed-Shannon phenomenon in a typical case)

Suppose $X(t)$ and $Y(t)$ are specified according to (14). The eigenpairs are
determined by (19). Then, for any fixed positive values of $T,n_{0}$ and
$\alpha$, there exists a positive number $\delta$ such that the Exceed-Shannon
inequality

$C(T):=\frac{1}{2T}\sum_{k=1}^{+\infty}\log\left(1+\frac{\lambda_{k}}{n_{0}/2}\right)>C_{\rm
sh}$ (21)

holds strictly for arbitrary $0<P<\delta$.

###### Proof:

See Appendix C. ∎

Figure 3: Theoretical verification of the Exceed-Shannon effect. The blue
lines represent the finite-time mutual information $I(T)$. The red lines are
the Shannon capacities $TC_{\rm sh}$ calculated from (8). All curves are
evaluated under hypothesis (14), where $P=1,2,4$, $n_{0}=1$ and $\alpha=1$.

To verify the above theoretical analysis, numerical experiments on $I(T)$ are
conducted based on evaluations of (19) and (20). As is shown in Fig. 3, it
seems that we can always harness more mutual information in a finite-time
observation window, compared with the Shannon capacity. Though seems
impossible, this fact is somehow unsurprising because the observations
${\bm{Y}}(t_{1}^{N})$ inside the finite-time window $[0,T]$ can always
eliminate some extra uncertainty outside the window due to the autocorrelation
of $X(t)$. Different from the finite-time capacity, the Shannon capacity
describes the circumstance of $T\rightarrow\infty$, where the fringe effect
near $t=0$ and $t=T$ becomes negligible compared with the prolonged window
period. Thus, the Shannon capacity does not take into consideration the small
extra information on the fringe, causing an underestimation of the capacity.
Fig. 3 also shows that, the extra capacity $\Delta I:=I(T)-TC_{\rm sh}$
between the finite-time result and Shannon capacity tends to a constant as
$T\rightarrow\infty$. As is discussed above, the difference may come from the
additional elimination of uncertainty at the fringe of the window. This
asymptotically constant difference results in asymptotic linearity of the
finite-time mutual information $I(T)$ as a function of $T$.

Apart from the above discussion, there is an extra interesting observation in
Fig. 3, which leads to another rigourous proof of the Exceed-Shannon
phenomenon. If we investigate the slope of curves $I(T)$ at the origin, i.e.,
the “instant transmission rate” at the origin, we find that $C(0^{+})>C_{\rm
sh}$. This observation is confirmed by the following theorem:

######  Theorem 3 (Instant Finite-Time Rate Exceeds Shannon)

Suppose $X(t)$ and $Y(t)$ are specified according to (14). The eigenpairs are
given by (19). Then the instantaneous finite-time information transmission
rate $C(0^{+})$ is given by:

$C(0^{+})=\frac{\partial I(T)}{\partial T}|_{T=0^{+}}=\frac{P}{n_{0}}$ (22)

###### Proof:

See Appendix D. ∎

From the conclusion of Theorem 3 and (20), we can reduce the Exceed-Shannon
inequality $C(0^{+})>C_{\rm sh}$ at $T=0^{+}$ to the following inequality:

$\frac{P}{n_{0}}>\frac{1}{2}\left(\sqrt{\alpha^{2}+\frac{4P\alpha}{n_{0}}}-\alpha\right),$
(23)

which can be directly verified by simple term-shifting and squaring on both
sides. This inequality implies that the average transmission rate in the
finite-time regime is strictly larger than the Shannon capacity around the
origin $T=0$.

###### Remark 3

Both Theorem 2 and Theorem 3 prove the existence of Exceed-Shannon phenomenon
rigorously. However, their differences are:

  * •

Theorem 2 proves that the Exceed-Shannon inequality holds in a small power
range $0<P<\delta$, but Theorem 3 is true for arbitrary power $P$.

  * •

Since Theorem 3 only proves $C(0^{+})>C_{\rm sh}$, it does not ensure the
Exceed-Shannon inequality when $T$ becomes larger. By contrast, Theorem 2
states that the the Exceed-Shannon phenomenon exists for arbitrary $T>0$.

###### Remark 4

In fact, Theorem 2 and Theorem 3 characterize the Exceed-Shannon phenomenon of
the finite-time capacity from two aspects. One aspect is the observation time
$T$, and the other is the transmit power $P$. Combining these two proofs of
the theorems may result in a universal proof that is independent of the choice
of parameters $T$ and $P$, which requires further study.

The conclusion of Theorem 3 is verified numerically in Fig. 5 and Fig. 5. The
blue solid lines, representing the finite-time capacity $C(T)$, are above the
red dashed lines representing the Shannon capacity, which demonstrates the
Exceed-Shannon phenomenon. The $C(T)$ curves all start at $P/n_{0}$ when
$T=0^{+}$, which coincides with the conclusion of Theorem 3. It can also be
observed from the two figures that, for fixed values of $P,n_{0}$, as $\alpha$
increases, the transmitted signal $X(t)$ tends to be less correlated, thus
being more informative. The transmission rate is then improved. This insight
also comes from the change of PSD $S_{X}(f)$. As $\alpha$ increases, the PSD
becomes flatter, i.e., a wider range of bandwidth is occupied, and thus the
rate increases accordingly.

Figure 4: Comparison between the finite-time capacity $C(T)$ and Shannon
capacity, when $\alpha=1$.

Figure 5: Comparison between the finite-time capacity $C(T)$ and Shannon
capacity, when $\alpha=2$.

###  IV-B Further Discussions on the Exceed-Shannon Phenomenon

Achievability of the finite-time capacity. It is known that for any band-
limited stationary Gaussian process $X(t)$ with PSD $S_{X}(f)$, one can
generate signals whose PSD is exactly $S_{X}(f)$ by first generating a
sequence $X(nT_{s})$ of sufficiently high sampling rate, and then passing the
generated sequence through a shaping filter. Since the transmitted signal
$X(t)$ and its generating sequence $X(nT_{s})$ determine each other uniquely
if $X(t)$ is strictly band-limited, $X(t)$ and $X(nT_{s})$ can be treated to
contain the same amount of information. Then we can conclude that, after
observing the noisy received process $Y(t),t\in[0,T]$, because of the
definition of the finite-time capacity $I(T)$, the amount of uncertainty of
the underlying transmitted sequence $X(nT_{s})$ can be reduced by exactly
$I(T)$ nats. That is to say, we link the finite-time capacity with a sequence-
to-sequence capacity. Thus, the finite-time mutual information $I(T)$ is
achievable by standard capacity-achieving techniques such as random coding
[1], as long as the sampling instants are dense enough.

Compatibility with the Shannon-Hartley theorem. Though the Exceed-Shannon
effect does imply an average data transmission rate within a finite-time
window higher than that predicted by Shannon, in fact, it is still impossible
to construct a long-time stable data transmission scheme above the Shannon
capacity by leveraging this effect. So the Exceed-Shannon phenomenon does not
contradict the Shannon-Hartley Theorem. Placing additional observation windows
cannot increase the average information rate, because the received process
$Y(t)$ observed by the subsequent additional windows has already been
implicitly altered by the previous observation. The posterior process
$Y(t)|{\bm{Y}}(t_{1}^{N})$ does not carry as much information as the original
one, thus causing a rate reduction in the later windows. It is expected that,
the average transmission rate would ultimately decrease to the Shannon
capacity as the total observation time tends to infinity (i.e.,
$C(\infty)=C_{\rm sh}$), and the analytical proof is still worth investigation
in future works.

##  V Conclusions

In this paper, we provided rigorous proofs of the existence of the “Exceed-
Shannon” phenomenon under typical autocorrelation settings of the transmitted
signal process and the noise process. Our discovery of the “Exceed-Shannon”
phenomenon revealed a possible new direction of research in information
theory, as it provided a generalization of Shannon’s renowned formula
$C=W\log(1+S/N)$ to the practical finite-time communications. It shows the
possibility that we can communicate at a higher-than-Shannon rate in a short
time. Since the finite-time capacity is a more precise estimation of the
ultimate capacity limit, the optimization target may shift from the Shannon
capacity to the finite-time capacity in the design of practical communication
systems. Thus, it has guiding significance for the performance improvement of
modern communication systems. In future works, general proofs of $C(T)>C_{\rm
sh}$, independent of the concrete autocorrelation settings, still require
further investigation. Moreover, we need to answer the question of how to
exploit this Exceed-Shannon phenomenon to improve the communication rate. In
addition, although we have discovered numerically that the finite-time
capacity agrees with the Shannon capacity when $T\rightarrow\infty$, an
analytical proof of this result is required in the future.

## Appendix A
Proof Of Theorem 1

Define $n$-by-$n$ matrix ${\bm{\Phi}}_{n}$ as

${\bm{\Phi}}_{n}=[\phi_{1}(t_{1}^{n}),\phi_{2}(t_{1}^{n}),\cdots,\phi_{n}(t_{1}^{n})],$

where $t_{i}=(i-1)T/n,1\leq i\leq n$. According to the definition of this
matrix, the following relation holds:

$\displaystyle\left(\frac{T}{n}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{\Phi}}_{n}\right)_{ij}$
$\displaystyle=\frac{T}{n}\sum_{k=1}^{n}{\phi_{i}(t_{k})\phi_{j}(t_{k})}$ (24)
$\displaystyle\rightarrow\int_{0}^{T}{\phi_{i}(t)\phi_{j}(t)\mathrm{d}t}=\delta_{ij}.$

This implies that the matrix ${\bm{\Phi}}_{n}$ satisfies the property of
asymptotic orthogonality:

$\|\frac{T}{n}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{\Phi}}_{n}-I_{n}\|_{2}\rightarrow
0,$ (25)

and the matrix ${\bm{\Phi}}_{n}$ can asymptotically diagonalize
${\bm{K}}_{X}=\left(R_{X}(t_{i},t_{j})\right)_{i,j=1}^{n}$ because of the
eigenvalue property (9):

$\displaystyle\mathbb{E}\left[\frac{T^{2}}{n^{2}}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{X}}(t_{1}^{n}){\bm{X}}^{\mathrm{T}}(t_{1}^{n}){\bm{\Phi}}_{n}\right]$
$\displaystyle=\frac{T^{2}}{n^{2}}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{K}}_{X}{\bm{\Phi}}_{n}$
(26)
$\displaystyle\rightarrow\mathrm{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{n}).$

Next, we investigate the noise realizations on sampling instants $t_{i}$: For
AWGN noise, the instantaneous power is $\infty$, i.e.,
$\mathbb{E}[N(t_{j})^{2}]=\infty$, so it is necessary to assume that the AWGN
noise is sampled after passing a rectangular-shaped impulse response filter
$\xi(t)$ with pulse width $T/n$ and gain $n/T$. This assumption is reasonable,
since the filter $\xi(t)$ tends to an ideal sampler $\delta(t)$ as
$n\to\infty$. Under this hypothesis, the noise variance for each sample can be
calculated as

$\displaystyle\mathbb{E}\left[\left(\int_{0}^{T/n}{N(t)\xi\left(\frac{T}{n}-t\right)\mathrm{d}t}\right)^{2}\right]$
(27)
$\displaystyle=\iint_{0}^{T/n}{\mathrm{d}t_{1}\mathrm{d}t_{2}}{\mathbb{E}\left[N(t_{1})N(t_{2})\right]\xi\left(t_{1}\right)\xi\left(t_{2}\right)}$
$\displaystyle\overset{(a)}{=}\iint_{0}^{T/n}{\mathrm{d}t_{1}\mathrm{d}t_{2}}{\frac{n_{0}}{2}\delta(t_{1}-t_{2})\xi\left(t_{1}\right)\xi\left(t_{2}\right)}$
$\displaystyle=\frac{n_{0}}{2}\int_{0}^{T/n}{\xi^{2}(t)\mathrm{d}t}$
$\displaystyle=\frac{n_{0}}{2}\frac{n}{T}.$

Note that the equality $(a)$ holds since the noise autocorrelation is
$\frac{n_{0}}{2}\delta(t_{1}-t_{2})$. In this way, the mutual information
within window $[0,T]$ can be calculated as

$\displaystyle I(T)=$
$\displaystyle\frac{1}{2}\lim_{n\rightarrow\infty}\log\left(\frac{\det({\bm{K}}_{X}+\frac{nn_{0}}{2T}{\bm{I}}_{n})}{\det\left(\frac{nn_{0}}{2T}{\bm{I}}_{n}\right)}\right)$
(28) $\displaystyle=$
$\displaystyle\frac{1}{2}\lim_{n\rightarrow\infty}\log\det\left({\bm{I}}_{n}+\frac{2T}{nn_{0}}{\bm{K}}_{X}\right)$
$\displaystyle\overset{(b)}{=}$
$\displaystyle\frac{1}{2}\lim_{n\rightarrow\infty}\log\det\left(\frac{T}{n}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{\Phi}}_{n}+\frac{1}{n_{0}/2}\frac{T^{2}}{n^{2}}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{K}}_{X}{\bm{\Phi}}_{n}\right)$
$\displaystyle\overset{(c)}{=}$
$\displaystyle\frac{1}{2}\lim_{n\rightarrow\infty}\log\det\left({\bm{I}}_{n}+\frac{1}{n_{0}/2}\mathrm{diag}(\lambda_{1},\cdots,\lambda_{n})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{+\infty}{\log\left(1+\frac{\lambda_{k}}{n_{0}/2}\right)},$

where $(b)$ comes from sandwiching the determinant in the bracket with both
the asymptotically orthogonal matrix $\sqrt{T/n}{\bm{\Phi}}_{n}$ on the left
and its transpose on the right, and $(c)$ comes from plugging (24) and (26)
into the previous step.

## Appendix B
Proof Of Lemma 1

Tracing both left and right hand sides of (26) and let $n\rightarrow\infty$,
we obtain

$\displaystyle\mathrm{tr}(M)$
$\displaystyle=\lim_{n\rightarrow\infty}\mathrm{tr}\left(\frac{T^{2}}{n^{2}}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{K}}_{X}{\bm{\Phi}}_{n}\right)$
(29)
$\displaystyle=\lim_{n\rightarrow\infty}\left(\frac{T}{n}\times\mathrm{tr}\left(\frac{T}{n}{\bm{\Phi}}_{n}^{\mathrm{T}}{\bm{K}}_{X}{\bm{\Phi}}_{n}\right)\right)$
$\displaystyle=\lim_{n\rightarrow\infty}\left(\frac{T}{n}\mathrm{tr}\left({\bm{K}}_{X}\right)\right)$
$\displaystyle=\lim_{n\rightarrow\infty}\left(\frac{T}{n}\times nP\right)$
$\displaystyle=PT,$

which completes the proof of Lemma 1.

## Appendix C
Proof Of Theorem 2

Plugging (20) into the right-hand side of (21), and differentiate both sides
w.r.t $P$. Notice that if $P=0$, then both sides of (21) are equal to 0. Thus,
we only need to prove that the derivative of left-hand side is strictly larger
than that of right-hand side within a small interval $P\in(0,\delta)$:

$\frac{1}{2}\sum_{k=1}^{+\infty}{\left(\frac{1}{1+\frac{2\lambda_{k}}{n_{0}}}\frac{2\lambda_{k}}{n_{0}P}\right)}>\frac{T}{n_{0}}\frac{1}{\sqrt{1+4P/(n_{0}\alpha)}}.$
(30)

Multiply both sides of (30) by $n_{0}$ and define $\mu_{k}:=\lambda_{k}/(PT)$,
and then from Lemma 1 we obtain $\sum_{k}{\mu_{k}}=1$. In this way, (30) is
equivalent to

$\sum_{k=1}^{+\infty}{\frac{\mu_{k}}{1+\frac{2\lambda_{k}}{n_{0}}}}>\frac{1}{\sqrt{1+4P/(n_{0}\alpha)}}.$
(31)

Since $\varphi(x):=1/(1+2x/n_{0})$ is convex on $(0,+\infty)$, by applying
Jensen’s inequality to the left-hand side of (31), we only need to prove that

$\frac{1}{1+\frac{2}{n_{0}}\sum_{k}{\lambda_{k}\mu_{k}}}>\frac{1}{\sqrt{1+4P/(n_{0}\alpha)}}.$
(32)

From the definition of $\mu_{k}$ we can derive that
$\lambda_{k}\mu_{k}=\lambda_{k}^{2}/(PT)$. So we go on to calculate
$\sum_{k}{\lambda_{k}^{2}}$. That is equivalent to calculate
$\mathrm{tr}(M^{2})$, where $M^{2}$ corresponds to the integral kernel:

$K_{M^{2}}(t_{1},t_{2}):=\int_{0}^{T}{P^{2}\exp(-\alpha\lvert
t_{1}-s\rvert)\exp(-\alpha\lvert t_{2}-s\rvert)\mathrm{d}s}.$ (33)

Evaluating the kernel $K_{M^{2}}$ on the diagonal $t=t_{1}=t_{2}$ yields

$\displaystyle K_{M^{2}}(t,t)$
$\displaystyle=P^{2}\int_{0}^{T}{\exp(-2\alpha\lvert t-s\rvert)\mathrm{d}s},$
(34) $\displaystyle=\frac{P^{2}}{2\alpha}(2-\exp(-2\alpha
t)-\exp(-2\alpha(T-t))).$

Integrating this kernel on the diagonal of $[0,T]^{2}$ gives
$\sum_{k}{\lambda_{k}^{2}}=\mathrm{tr}(M^{2})$:

$\displaystyle\sum_{k}{\lambda_{k}^{2}}$
$\displaystyle=\frac{P^{2}}{2\alpha}\int_{0}^{T}{\left(2-e^{-2\alpha
t}-e^{-2\alpha(T-t)}\right)\mathrm{d}t},$ (35)
$\displaystyle=\frac{P^{2}}{\alpha}\left(T-\frac{1}{2\alpha}(1-e^{-2\alpha
T})\right).$

By substituting (35) into (32), we just need to prove that

$\sqrt{1+4P/(n_{0}\alpha)}>1+\frac{2P}{n_{0}\alpha}\left(1-\frac{1-e^{-2\alpha
T}}{2\alpha T}\right).$ (36)

Define the dimensionless number $x=2P/(n_{0}\alpha)$. Since the function
$\psi(x):=(1-\exp(-x))/x$ is strictly positive and less than 1 at $x>0$, we
can conclude that, there exists a small positive $\delta>0$ such that (36)
holds for $0<P<\delta$. The number $\delta$ can be chosen as

$\delta=\frac{n_{0}\alpha\psi(2\alpha T)}{(1-\psi(2\alpha T))^{2}}>0,$ (37)

which implies that (30) holds for any $0<P<\delta$. Thus, integrating (30) on
both sides from $p=0$ to $p=P,P<\delta$ gives rise to the conclusion (21),
which completes the proof of Theorem 2.

## Appendix D
Proof Of Theorem 3

Differentiating the finite-time capacity expression for $I(T)$, i.e. (12),
with respect to $T$, then we obtain

$\frac{\partial I(T)}{\partial T}|_{T=0^{+}}=\lim_{T\rightarrow
0^{+}}\frac{1}{2}\sum_{k=1}^{+\infty}{\frac{1}{(n_{0}/2)(1+2\lambda_{k}/n_{0})}\frac{\partial\lambda_{k}}{\partial
T}},$ (38)

where $\lim_{T\rightarrow
0^{+}}\sum_{k=1}^{+\infty}\frac{\partial\lambda_{k}}{\partial T}$, by applying
Lemma 1, can be expressed as

$\displaystyle\lim_{T\rightarrow
0^{+}}\sum_{k=1}^{+\infty}\frac{\partial\lambda_{k}}{\partial T}$
$\displaystyle=\lim_{T\rightarrow 0^{+}}\frac{\partial}{\partial
T}\sum_{k=1}^{+\infty}{\lambda_{k}}$ (39)
$\displaystyle=\frac{\partial}{\partial T}(PT)|_{T=0^{+}}$ $\displaystyle=P.$

Since $\omega_{k}\uparrow+\infty$ as $T\downarrow 0^{+}$, we can safely
conclude that $\lambda_{k}\downarrow 0^{+}$. From Dirichlet’s test, the series
in (38) converges uniformly. Thus, by interchanging the infinite sum and the
limit operation, we obtain

$\displaystyle\frac{\partial I(T)}{\partial T}|_{T=0^{+}}$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{+\infty}{\lim_{T\rightarrow
0^{+}}\left(\frac{1}{(n_{0}/2)(1+2\lambda_{k}/n_{0})}\frac{\partial\lambda_{k}}{\partial
T}\right)}$ (40)
$\displaystyle=\frac{1}{n_{0}}\sum_{k=1}^{+\infty}\frac{\partial\lambda_{k}}{\partial
T}|_{T=0^{+}}$ $\displaystyle=\frac{1}{n_{0}}\lim_{T\rightarrow
0^{+}}\sum_{k=1}^{+\infty}\frac{\partial\lambda_{k}}{\partial T}$
$\displaystyle=\frac{P}{n_{0}},$

which completes the proof of Theorem 3.

## Appendix E
The Evaluation of The Improper Integral (20)

Define the improper inegral with parameters $P>0$ and $\alpha>0$:

$J(P,\alpha):=\int_{-\infty}^{+\infty}{\log\left(1+\frac{P}{\alpha^{2}+\omega^{2}}\right){\rm
d}\omega}.$ (41)

By taking the partial derivative of $J(P,\alpha)$ with respect to $P$, we
obtain

$\frac{\partial J(P,\alpha)}{\partial
P}=\int_{-\infty}^{+\infty}{\frac{1}{\omega^{2}+(\alpha^{2}+P)}{\rm
d}\omega}.$ (42)

Note that the analytic function defined as

$f(z):=\frac{1}{z^{2}+(\alpha^{2}+P)},$ (43)

has residual

${\rm Res}\left[f(z),z=z_{p}\right]=\frac{1}{2{\rm i}\sqrt{\alpha^{2}+P}},$
(44)

at pole $z_{p}={\rm i}\sqrt{\alpha^{2}+P}$ in the upper half-plane, thus the
integral in (42) can be evaluated by the residual theorem:

$\displaystyle\frac{\partial J(P,\alpha)}{\partial P}$ $\displaystyle=2\pi{\rm
i}{\rm Res}\left[f(z),z={\rm i}\sqrt{\alpha^{2}+P}\right]$ (45)
$\displaystyle=\frac{\pi}{\sqrt{\alpha^{2}+P}}.$

Since $J(0,\alpha)\equiv 0$, by integrating (42) with respect to $P$ from $0$
to $P$ yields

$\displaystyle J(P,\alpha)$
$\displaystyle=\int_{0}^{P}\frac{\pi}{\sqrt{\alpha^{2}+p}}{\rm d}p$ (46)
$\displaystyle=2\pi\sqrt{\alpha^{2}+p}|_{p=0}^{P}$
$\displaystyle=2\pi\left(\sqrt{\alpha^{2}+P}-\alpha\right).$

Then the integral in (20) can be calculated by setting $P\leftarrow
4P\alpha/n_{0}$ and $\alpha\leftarrow\alpha$ in (46):

$\displaystyle\frac{1}{4\pi}\int_{-\infty}^{+\infty}{\log\left(1+\frac{\frac{2P\alpha}{\alpha^{2}+\omega^{2}}}{n_{0}/2}\right)\mathrm{d}\omega}$
(47)
$\displaystyle=\frac{2\pi}{4\pi}\left(\sqrt{\alpha^{2}+\frac{4P\alpha}{n_{0}}}-\alpha\right)$
$\displaystyle=\frac{1}{2}\left(\sqrt{\alpha^{2}+\frac{4P\alpha}{n_{0}}}-\alpha\right),$

which completes the proof.

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# On Homogeneous System of Fractal Differential Equations

Alireza Khalili Golmankhaneh1, Donatella Bongiorno2,
1 Department of Physics, Urmia Branch,
Islamic Azad University, Urmia 63896,West Azerbaijan, Iran
<EMAIL_ADDRESS>
2 Department of Engineering, University of Palermo, Palermo 90100, Italy
<EMAIL_ADDRESS>

###### Abstract

In this paper, a homogeneous system of n $\alpha$-order linear fractal
differential equation is defined and the set of its fundamental solutions
through the corresponding Wronskian matrix is described. Finally, the
solutions of some assigned autonomous homogeneous systems are plotted to show
the details previously proved.

††footnotetext:  MSC2020:28A80, 34A38††Corresponding author Alireza Khalili
Golmankhaneh

##  1 Introduction

Fractals are geometric shapes characterized by having dimensions that are
larger and non-integer compared to their topological dimensions[1, 2].

Lightning does not follow a linear path, clouds lack perfect spherical shapes,
and mountains do not conform to cone-like structures. The complexity of
natural shapes differs fundamentally from those in traditional geometry.
Specifically, fractal geometry captures the intricate nature of these shapes,
representing a qualitative difference rather than just a matter of degree [3].

The research was conducted to examine the process of identifying and analyzing
fractal patterns, particularly those emerging from social processes [4].
Several techniques for quantifying the spatial and temporal attributes of
fractal patterns were presented [5]. It covers techniques like box-counting,
which can determine fractal dimensions in diverse contexts such as Saturn’s
rings, forest landscapes, brain imaging data, and text analysis [5]. The
analytical applications of IFS methods, demonstrating their direct uses as
well as novel analytical approaches inspired by the IFS fractal framework,
were showcased [6].

The systematic analysis, modeling, and synthesis of integrated fractal and
fractal-rate point processes are explored, presenting a rigorous yet practical
approach [7]. These processes amalgamate the scaling properties of fractals
with the discrete nature of random point processes [7]. The comprehensive
understanding of polymer synthesis and curing, by establishing structural and
physical foundations through fractal analysis, was thoroughly investigated
[8]. Theoretical concepts and practical applications of fractals and
multifractals, as discussed in [9], cater to various scientific communities,
including those involved in petroleum, chemical, civil and environmental
engineering, atmospheric research, and hydrology [10]. Fractal analysis has
been utilized in various fields, including the identification of coding
regions in DNA and the assessment of space-filling properties in tumors, blood
vessels, and neurons. Additionally, fractal concepts have been effectively
integrated into models of biological phenomena, such as epithelial cell
growth, blood vessel development, periodontal disease progression, and viral
infections [11]. Ecologists have harnessed the power of fractals to tackle
fundamental questions pertaining to scale, measurement, and hierarchy within
ecological systems [12]. Self-similar and self-affine processes are ubiquitous
in nature, manifesting across various scales and contexts, including galaxies,
landscapes, earthquakes, geological formations, aggregates, colloids, rough
surfaces, interfaces, glassy materials, polymers, proteins, and other large
molecules [13]. The explanations covered transformations in metric spaces,
dynamics within fractals, fractal dimension, interpolation techniques on
fractals, Julia sets, parameter spaces, and measures associated with fractals
[14, 15]. Fractal dimensions were initially introduced using random walks and
Sierpinski gaskets. The exploration culminated with diffusion-limited
aggregates, closely associated with critical exponents [16]. Self-similarity
and scale invariance, concepts rooted in both phase transitions and fractal
geometry, intersect in numerous domains of physics. The discussion extends to
advanced theoretical subjects such as chaos theory, percolation phenomena,
turbulence models, and renormalization group methods [17, 18]. Constitutive
equations for fractal elasticity are provided, establishing a linear elastic
relationship based on the definition of fractal elastic potential. The
physical dimensions of the second derivatives of the elastic potential are
determined by the fractal dimensions of both stress and strain [19]. Recent
physics research has introduced fractal time, characterized by self-similar
properties and fractional dimension [20, 21].
The significance of fractals has spurred researchers to analyze them using a
myriad of methods, including fractional calculus [22], harmonic analysis [23],
measure theory [24, 25, 26, 27, 28], nonstandard methodologies [29, 30],
probabilistic approaches [31] and fractional space [32]. The analysis was
conducted on an integral of the s-Riemann type, where the gauge is a positive
constant but the points involved in the s-Riemann sums were not randomly
chosen [33]. The investigation into numerical methods for solving partial
differential equations on fractals was conducted [34]. This study delves into
both strong and weak forms of the equations, utilizing standard graph
Laplacian matrices and discrete approximations of fractal sets.
While the Riemann-like method remains significant as an extension of classical
calculus, fractal calculus offers a mathematical framework tailored for
handling equations with solutions displaying fractal characteristics,
including fractal sets and curves [35, 36]. Fractal calculus stands out for
its elegant and algorithmic approaches, which are deemed superior to other
methods, rendering it particularly attractive [37].
The fractal local Mellin transform and fractal non-local transform have been
employed as tools for solving fractal differential equations [38]. The concept
of Laplace transform and local Fourier transform has been extended to fractal
curves, facilitating the solution of fractal differential equations with
constant coefficients [39, 40]. This study investigates fractal time within
the framework of economic models, employing both local and non-local fractal
Caputo derivatives [41]. The gauge integral approach has been successfully
employed in the generalization of $F^{\alpha}$-calculus (FC). This
generalization focuses on integrating functions over a specific subset [42,
43] of the real number line, encompassing singularities that arise within
fractal sets. The concept establishes a connection between the order of non-
local fractal derivatives and the fractal Hurst exponent [44]. Fractal
differential equations have been solved, and their stability conditions have
been established [45]. The Einstein field equations and their analogues were
presented, emphasizing the relevance and practicality of fractal geometry
[46]. Fractal integro-differential equations for different types of circuits
exposed to zero-mean additive white Gaussian noise on fractal sets have been
formulated, integrating a fractal time component [47, 48, 49]. The suggestion
was made that two distinct del-operators, each acting on a vector field and a
scalar field, could be defined in spaces with non-integer dimensions. These
del-operators were then utilized to derive the conventional formulation of the
Laplacian and the fundamental vector differential operators in fractional-
dimensional space [50]. Fractal functional differential equations were
formulated as a framework that offers a mathematical model for phenomena
characterized by fractal time and fractal structure [51]. The study
demonstrated that non-local fractal derivatives can effectively characterize
fractional Brownian motion on thin Cantor-like sets. Additionally, it
introduced a staircase function associated with a fractal comb, which was
utilized to define derivatives and integrals for functions defined on these
combs [52]. Fractal integral and differential forms were defined utilizing
nonstandard analysis [53]. Diffusion and fractional Brownian motion in
fractal-structured media were categorized, and fractal stochastic differential
equations were formulated [54, 55, 56]. The study [57] investigated the
analogues of the separable method and integrating factor methodology for
solving $\alpha$-order differential equations. Inspiration was drawn from
Newton, Lagrange, Hamilton, and Appell for the proposed fractal analogue of
mechanics. Furthermore, to establish the Langevin equation on fractal curves,
fractal velocity and acceleration were defined [58]. The conventional Fokker-
Planck Equation and its fractal version, incorporating fractal derivatives,
were linked to each other in the study [59]. The study explored various
mathematical models for the growth of double-sized cancer in the fractal
temporal dimension [60]. Additionally, a fractal discharging model for
batteries was developed to investigate the impact of non-locality on solution
behavior and how the system’s previous state influences its current state
[61].
In this investigation, we analyze systems of fractal differential equations
and explore their solutions.
The paper is structured as follows:
In Section 2, we offer a concise overview of fractal calculus. Section 3
delves into the study of systems of $\alpha$-order linear fractal differential
equations, accompanied by the proof of related theorems. In Section 4, we
focus on solving homogeneous fractal linear systems with constant
coefficients. Finally, Section 5 presents the conclusion.

##  2 Preliminaries

In this section, a concise overview of fractal calculus in the context of
fractal sets (see [35, 36, 37]) and the definition of fractal matrix with its
fractal derivative and its fractal integral are given.

######  Definition 1.

A flag function of a set $F\subset\mathbb{R}$ and a closed interval $I$ is
defied as follows:

$\rho(F,I)=\begin{cases}1,&\text{if }F\cap I\neq\emptyset;\\\
0,&\text{otherwise}.\end{cases}$ (1)

######  Definition 2.

Given $\delta>0$, a constant, and given $P_{[a,b]}$ a subdivision of $[a,b]$,
the coarse-grained mass of a fractal set $F\subset\mathbb{R}$ is defined as
follows:

$\gamma_{\delta}^{\alpha}(F,a,b)=\inf_{|P|\leq\delta}\sum_{i=0}^{n-1}\Gamma(\alpha+1)(t_{i+1}-t_{i})^{\alpha}\rho(F,[t_{i},t_{i+1}]),$
(2)

where $|P|=\max_{0\leq i\leq n-1}(t_{i+1}-t_{i})$, and $0<\alpha\leq 1$.

######  Definition 3.

The mass function of a fractal set $F\subset\mathbb{R}$ is the limit as
$\delta$ tends to zero of the coarse-grained mass:

$\gamma^{\alpha}(F,a,b)=\lim_{\delta\rightarrow
0}\gamma_{\delta}^{\alpha}(F,a,b).$ (3)

######  Definition 4.

The $\gamma$-dimension of the intersection between $F\subset\mathbb{R}$ and
the interval $[a,b]$ is defined as follows:

$\displaystyle\dim_{\gamma}(F\cap[a,b])$
$\displaystyle=\inf\\{\alpha:\gamma^{\alpha}(F,a,b)=0\\}$
$\displaystyle=\sup\\{\alpha:\gamma^{\alpha}(F,a,b)=\infty\\}$ (4)

######  Definition 5.

For a fractal set $F\subset\mathbb{R}$, the integral staircase function of
order $\alpha$ is defined as follows:

$S_{F}^{\alpha}(x)=\begin{cases}\gamma^{\alpha}(F,a_{0},x),&\text{if }x\geq
a_{0};\\\ -\gamma^{\alpha}(F,x,a_{0}),&\text{otherwise}.\end{cases}$ (5)

where $a_{0}$ is an arbitrary fixed real number.

######  Definition 6.

Let $F$ be a fractal subset of the real line. A function
$f:F\rightarrow\mathbb{R}$ is said to be $F$-continuous at $x\in F$ if

$f(x)=\underset{y\rightarrow x}{F_{-}\text{lim}}f(y).$ (6)

whenever the $F_{-}\text{lim}$ exists.

######  Definition 7.

Let $f$ be a function defined on a $\alpha$-perfect fractal set
$F\subset\mathbb{R}$, and let $x$ be a point of $F\subset\mathbb{R}$. The
$F^{\alpha}$-derivative of $f$ at the point $x$ is:

$D_{F}^{\alpha}f(x)=\begin{cases}\underset{y\rightarrow
x}{F_{-}\text{lim}}\leavevmode\nobreak\
\frac{f(y)-f(x)}{S_{F}^{\alpha}(y)-S_{F}^{\alpha}(x)},&\text{if }x\in F;\\\
0,&\text{otherwise}.\end{cases}$ (7)

if the fractal limit exists [35].

######  Definition 8.

Let I=[a,b]. Let $F\subset\mathbb{R}$ be an $\alpha$-perfect fractal subset of
$[a,b]$ and let $x\in F.$ The $F^{\alpha}$-integral of a bounded function $f$
defined on $F\subset\mathbb{R}$ where $S^{\alpha}_{F}(*)$ is finite on $I$, is
defined as follows:

$\displaystyle\int_{a}^{b}f(x)d_{F}^{\alpha}x$
$\displaystyle=\sup_{P_{[a,b]}}\sum_{i=0}^{n-1}\inf_{x\in F\cap
I}f(x)(S_{F}^{\alpha}(x_{i+1})-S_{F}^{\alpha}(x_{i}))$
$\displaystyle=\inf_{P_{[a,b]}}\sum_{i=0}^{n-1}\sup_{x\in F\cap
I}f(x)(S_{F}^{\alpha}(x_{i+1})-S_{F}^{\alpha}(x_{i})).$ (8)

######  Definition 9.

Let $F\subset\mathbb{R}$ and let $a_{ij}:F\rightarrow R$ where $i=1,2,...,m$
and $j=1,2,...,n$ be fractal functions. Therefore the fractal matrix $m\times
n$ is defined as follows

$\mathbf{A}(t)=\left(\begin{array}[]{ccc}a_{11}(t)&\cdots&a_{1n}(t)\\\
\vdots&&\vdots\\\ a_{m1}(t)&\cdots&a_{mn}(t)\\\ \end{array}\right).$ (9)

Moreover, whenever $n=1$ the fractal matrix will be called fractal vector and
we will write:

$\mathbf{x}(t)=\left(\begin{array}[]{c}x_{1}(t)\\\ \vdots\\\ x_{m}(t)\\\
\end{array}\right)$ (10)

If each element of $\mathbf{A}(t)$ is a $F$-continuous function, then the
fractal matrix $\mathbf{A}(t)$ is a $F$-continuous. Moreover, if each element
of $\mathbf{A}(t)$ is a $F^{\alpha}$-differentiable function, then
$\mathbf{A}(t)$ is $F^{\alpha}$-differentiable and the fractal derivative of
$\mathbf{A}(t)$ is the following:

$D_{F}^{\alpha}\mathbf{A}(t)=\left(D_{F}^{\alpha}a_{ij}(t)\right).$ (11)

Finally, whenever $F\subset[a,b]$, the fractal integral of a fractal matrix
$\mathbf{A}(t)$, is given as follows:

$\int_{a}^{b}\mathbf{A}(t)d_{F}^{\alpha}t=\left(\int_{a}^{b}a_{ij}(t)d_{F}^{\alpha}t\right).$
(12)

##  3 Homogeneous System of $\alpha$-Order Linear Fractal Differential
Equation

In this section, we introduce a homogeneous system of $\alpha$-order linear
fractal differential equations and through the corresponding Wronskian matrix,
we discuss some characteristic properties of the set of its solutions. The
general form of a homogeneous system of $n$ $\alpha$-order fractal
differential linear equations is as follows:

$\begin{split}D_{F}^{\alpha}x_{1}(t)&=p_{11}(t)x_{1}(t)+...+p_{1n}(t)x_{n}(t)\\\
\vdots&\\\
D_{F}^{\alpha}x_{n}(t)&=p_{n1}(t)x_{1}(t)+...+p_{nn}(t)x_{n}(t)\end{split}$
(13)

In short form, Eq. (13) can be expressed as:

$D_{F}^{\alpha}\mathbf{x}(t)=\mathbf{P}(t)\mathbf{x}(t),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\in F,$ (14)

where

$\mathbf{x}(t)=\left(\begin{array}[]{c}x_{1}(t)\\\ \vdots\\\ x_{n}(t)\\\
\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{P}(t)=\left(\begin{array}[]{ccc}p_{11}(t)&\cdots&p_{1n}(t)\\\
\vdots&&\vdots\\\ p_{n1}(t)&\cdots&p_{nn}(t)\\\ \end{array}\right).$ (15)

We assume that each element of the fractal matrix $\mathbf{P}(t)$ is a real
value $F$-continuous function on $F$. Moreover, we will use the following
notation:

$\mathbf{x}^{1}(t)=\left(\begin{array}[]{c}x_{11}(t)\\\ x_{21}(t)\\\ \vdots\\\
x_{n1}(t)\\\
\end{array}\right),\cdots,\mathbf{x}^{k}(t)=\left(\begin{array}[]{c}x_{1k}(t)\\\
x_{2k}(t)\\\ \vdots\\\ x_{nk}(t)\\\ \end{array}\right),\cdots$ (16)

where $x_{ij}(t)=x_{i}^{(j)}(t)$ denotes the $i^{th}$ component of the
$j^{th}$ solution $\mathbf{x}^{(j)}(t)$ of Eq.(14). We say that the solution
of a given system of $\alpha$-order linear fractal differential equation is
the collection of all $F^{\alpha}$-differentiable complex vector functions
defined on $F$: $\mathbf{x}^{(j)}(t)$ such that, by substituting each
$\mathbf{x}^{(j)}(t)$ for the individual $x_{ij}(t)$, the system (14) is
satisfied. Now let us observe that by the linearity of the system, if the
fractal vector functions $\mathbf{x}^{(1)}(t),\leavevmode\nobreak\
\mathbf{x}^{(2)},...,\mathbf{x}^{(n)}(t)$ are solutions of the system (14),
then the linear combination
$c_{1}\mathbf{x}^{(1)}(t)+c_{2}\mathbf{x}^{(2)}(t)+...+c_{n}\mathbf{x}^{(n)}(t)$
is also a solution for any constants $c_{1},c_{2},...,c_{n}$, so the set of
all the solutions of Eq.(14) is a vector space.
Now let us consider the following fractal matrix:

$\mathbf{X}(t)=\left(\begin{array}[]{ccc}x_{11}(t)&\cdots&x_{1n}(t)\\\
\vdots&&\vdots\\\ x_{n1}(t)&\cdots&x_{nn}(t)\\\ \end{array}\right)$ (17)

whose columns are the fractal vectors
$\mathbf{x}^{(1)}(t),...,\mathbf{x}^{(n)}(t)$. The Wronskian of the solutions
is defined by:

$W[\mathbf{x}^{(1)}(t),...,\mathbf{x}^{(n)}(t)]=\det\mathbf{X}(t),\leavevmode\nobreak\
\leavevmode\nobreak\ \forall t\in F.$ (18)

######  Theorem 1.

If the Wronskian of the solutions is not zero in each point of $F$, then there
exist constants $c_{1},c_{2},...,c_{n}$ such that each solution
$\mathbf{x}=\mathbf{x}(t)$ of the system Eq.(14) can be expressed as:

$\mathbf{x}(t)=c_{1}\mathbf{x}^{(1)}(t)+...+c_{n}\mathbf{x}^{(n)}(t)$ (19)

This is called the general solution of the system Eq.(14), and the set
${\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)}$ is denoted as the
fundamental set of solutions on $F$.

###### Proof.

To prove this theorem, let us consider a point $t_{0}\in F$, and let
$\mathbf{y}(t)=\mathbf{x}(t)$. We have to show that there exist values
$c_{1},\ldots,c_{n}$ such that:

$c_{1}\mathbf{x}^{(1)}(t_{0})+\ldots+c_{n}\mathbf{x}^{(n)}(t_{0})=\mathbf{y}(t_{0})$
(20)

This equation can be expressed component-wise as:

$\begin{split}c_{1}x_{11}(t_{0})+...+c_{n}x_{1n}(t_{0})&=y_{1}(t_{0})\\\
&\vdots\\\ c_{1}x_{n1}(t_{0})+...+c_{n}x_{nn}(t_{0})&=y_{n}(t_{0})\end{split}$
(21)

By the conjugacy of $F^{\alpha}$-calculus and the ordinary calculus [35, 36,
37]), the necessary and sufficient condition for Eq.(21) to have a unique
solution is the non-vanishing of the determinant of the coefficients, which is
the Wronskian evaluated at $t_{0}$. Therefore, there exists a unique solution
of Equation (14) in the form
$\mathbf{x}(t)=c_{1}\mathbf{x}^{(1)}(t)+\ldots+c_{n}\mathbf{x}^{(n)}(t)$. This
completes the proof. ∎

######  Theorem 2.

If $\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ are solutions of Eq.(14)
on the fractal set $F$, then at each point of $F\subset\mathbb{R}$,
$W[\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)]$ is either identically zero
or never vanishes.

###### Proof.

To prove this theorem, we establish that the Wronskian of
$\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ satisfies the differential
equation:

$D_{F}^{\alpha}W=(p_{11}(t)+p_{22}(t)+\ldots+p_{nn}(t))W$ (22)

Thus, we have:

$W(t)=c\exp\left(\int(p_{11}(t)+\ldots+p_{nn}(t))d_{F}^{\alpha}t\right)$ (23)

where $c$ is an arbitrary constant. The conclusion of the theorem follow
immediately. ∎

######  Theorem 3.

Let

$\mathbf{e}^{(1)}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ \vdots\\\ 0\\\
\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{e}^{(2)}=\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\ \vdots\\\ 0\\\
\end{array}\right),\cdots,\mathbf{e}^{(n)}=\left(\begin{array}[]{c}0\\\ 0\\\
0\\\ \vdots\\\ 1\\\ \end{array}\right).$ (24)

Further, let $\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ be the solutions
of the following system of fractal differential equation

$D_{F}^{\alpha}\mathbf{x}(t)=\mathbf{P}(t)\mathbf{x}(t),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall t\in F,$ (25)

that satisfy the initial conditions:

$\mathbf{x}^{(1)}(t_{0})=\mathbf{e}^{(1)},\ldots,\mathbf{x}^{(n)}(t_{0})=\mathbf{e}^{(n)},$
(26)

where $t_{0}\in F$. Then $\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ form
a fundamental set of solutions of the system Eq.(25).

###### Proof.

Linear Independence:
By contradiction let us suppose that there exist constants
$c_{1},c_{2},\ldots,c_{n}$ not all zero such that

$c_{1}\mathbf{x}^{(1)}(t)+c_{2}\mathbf{x}^{(2)}(t)+\cdots+c_{n}\mathbf{x}^{(n)}(t)=\mathbf{0},$
(27)

for all $t\in F$. Now by evaluating this expression at $t=t_{0}$, we have that

$c_{1}\mathbf{x}^{(1)}(t_{0})+c_{2}\mathbf{x}^{(2)}(t_{0})+\cdots+c_{n}\mathbf{x}^{(n)}(t_{0})=\mathbf{0},$
(28)

and by the initial conditions (26), we get

$c_{1}\mathbf{e}^{(1)}+c_{2}\mathbf{e}^{(2)}+\cdots+c_{n}\mathbf{e}^{(n)}=\mathbf{0},$
(29)

which implies that $c_{1}=c_{2}=\cdots=c_{n}=0$.
Spanning the Solution Space:
To show that $\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ span the
solution space, let us denoted by $\mathbf{x}(t)$ any solution of the system
Eq.(25) with initial conditions (26), and let us define
$\mathbf{y}^{(i)}(t)=\mathbf{x}(t)-\mathbf{x}^{(i)}(t)$ for $i=1,2,\ldots,n$.
Then $\mathbf{y}^{(i)}(t)$ satisfies the same differential equation as
$\mathbf{x}(t)$ and has the initial condition
$\mathbf{y}^{(i)}(t_{0})=\mathbf{0}$. By the uniqueness of solutions to
initial value problems (see ), we have $\mathbf{y}^{(i)}(t)=\mathbf{0}$ for
all $i=1,2,\ldots,n$. This implies that
$\mathbf{x}(t)=\mathbf{x}^{(1)}(t)+\mathbf{x}^{(2)}(t)+\cdots+\mathbf{x}^{(n)}(t)$.
Hence, any solution $\mathbf{x}(t)$ of Eq.(25) with initial conditions (26)
can be expressed as a linear combination of
$\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$. Thus,
$\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ span the solution space.
Since $\mathbf{x}^{(1)}(t),\ldots,\mathbf{x}^{(n)}(t)$ are both linearly
independent and span the solution space, they form a fundamental set of
solutions of the system Eq.(25). Thus, the theorem is proved. ∎

######  Theorem 4.

Consider the system of fractal differential equation as:

$D_{F}^{\alpha}\mathbf{x}(t)=\mathbf{P}(t)\mathbf{x}(t),$ (30)

where each element of $\mathbf{P}$ is a real-valued $F$-continuous function on
$F$. If $\mathbf{x}(t)=\mathbf{u}(t)+i\mathbf{v}(t)$ is a complex-valued
solution of Eq.(30), then its real part $\mathbf{u}(t)$ and its imaginary part
$\mathbf{v}(t)$ are also solutions of this equation.

###### Proof.

To prove this theorem, we substitute $\mathbf{u}(t)+i\mathbf{v}(t)$ for
$\mathbf{x}(t)$ in Eq.(30), thereby obtaining

$D_{F}^{\alpha}\mathbf{x}(t)-\mathbf{P}(t)\mathbf{x}(t)=D_{F}^{\alpha}\mathbf{u}(t)-\mathbf{P}(t)\mathbf{u}(t)+i(D_{F}^{\alpha}\mathbf{v}(t)-\mathbf{P}(t)\mathbf{v}(t))=0$
(31)

Thus we have

$D_{F}^{\alpha}\mathbf{u}(t)-\mathbf{P}(t)\mathbf{u}(t)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
D_{F}^{\alpha}\mathbf{v}(t)-\mathbf{P}(t)\mathbf{v}(t)=0,$ (32)

Therefore, $\mathbf{u}(t)$ and $\mathbf{v}(t)$ are solutions of Eq.(30). ∎

##  4 Homogeneous System of $\alpha$-Order Linear Fractal Differential
Equation with Constant Coefficients

A homogeneous fractal linear system with constant coefficients is a system of
linear differential equations in which each element of the fractal matrix is
constant. Mathematically, such a system can be represented as:

$D_{F}^{\alpha}\mathbf{x}(t)=\mathbf{A}\mathbf{x}(t),$ (33)

where $\mathbf{A}$ is a $n\times n$ constant matrix. By [57] it follows that
to solve Eq.(33), we consider a solution of the form

$\mathbf{x}(t)=\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t)),$ (34)

where $r$ and $\boldsymbol{\xi}$ are to be determined. Substituting Eq.(34)
into Eq.(33), we obtain

$r\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t))=\mathbf{A}\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t)),$
(35)

or equivalently,

$(\mathbf{A}-r\mathbf{I})\boldsymbol{\xi}=0,$ (36)

where $\mathbf{I}$ is the $n\times n$ identity matrix therefore, to find
solutions of Eq.(33), we solve the system of algebraic equations given by
Eq.(36). The eigenvalues $r_{1},...,r_{n}$ are the roots of the $n^{th}$
degree polynomial equation

$\det(\mathbf{A}-r\mathbf{I})=0.$ (37)

The corresponding eigenvectors to eigenvalues $r_{1},...,r_{n}$ provide the
general solution of the fractal system given by Eq.(33).

######  Example 1.

Consider the fractal system of differential equations given by:

$D_{F}^{\alpha}\mathbf{x}(t)=\left(\begin{array}[]{cc}1&1\\\ 4&1\\\
\end{array}\right)\mathbf{x}(t).$ (38)

To find the solution explicitly, we assume that
$\mathbf{x}(t)=\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t))$ and substitute it
into Eq.(38), we have:

$\left(\begin{array}[]{cc}1-r&1\\\ 4&1-r\\\
\end{array}\right)\left(\begin{array}[]{c}\xi_{1}\\\ \xi_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ \end{array}\right).$ (39)

This equation leads to the characteristic equation:

$\begin{vmatrix}1-r&1\\\ 4&1-r\end{vmatrix}=(r-3)(r+1)=0,$ (40)

which gives the eigenvalues $r_{1}=3$ and $r_{2}=-1$. These eigenvalues are
also the roots of the characteristic polynomial. Substituting $r=3$ and $r=-1$
into Eq.(39), we obtain the eigenvectors:

$\boldsymbol{\xi}^{(1)}(t)=\left(\begin{array}[]{c}1\\\ 2\\\
\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\boldsymbol{\xi}^{(2)}=\left(\begin{array}[]{c}1\\\ -2\\\ \end{array}\right).$
(41)

Thus, the corresponding solutions of Eq.(38) are:

$\mathbf{x}^{(1)}(t)=\left(\begin{array}[]{c}1\\\ 2\\\
\end{array}\right)\exp(3S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{x}^{(1)}=\left(\begin{array}[]{c}1\\\ -2\\\
\end{array}\right)\exp(-S_{F}^{\alpha}(t)).$ (42)

The Wronskian of these solutions is:

$W[\mathbf{x}^{(1)}(t),\mathbf{x}^{(2)(t)}]=\begin{vmatrix}\exp(3S_{F}^{\alpha}(t))&\exp(-S_{F}^{\alpha}(t))\\\
2\exp(3S_{F}^{\alpha}(t))&-2\exp(-S_{F}^{\alpha}(t))\end{vmatrix}=-4\exp(2S_{F}^{\alpha}(t)).$
(43)

This Wronskian is not zero, indicating that the solutions
$\mathbf{x}^{(1)}(t)$ and $\mathbf{x}^{(2)}(t)$ form a fundamental set.

Therefore, the general solution of the system given by Eq.(38) is:

$\begin{split}\mathbf{x}(t)&=c_{1}\mathbf{x}^{1}(t)+c_{2}\mathbf{x}^{2}(t)\\\
&=c_{1}\left(\begin{array}[]{c}1\\\ 2\\\
\end{array}\right)\exp(3S_{F}^{\alpha}(t))+c_{2}\left(\begin{array}[]{c}1\\\
-2\\\ \end{array}\right)\exp(-S_{F}^{\alpha}(t))\end{split}$ (44)

where $c_{1}$ and $c_{2}$ are arbitrary constants. Setting $c_{2}=0$ results
in:

$\mathbf{x}=c_{1}\mathbf{x}^{1}(t),$ (45)

or in scalar form:

$x_{1}(t)=c_{1}\exp(3S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=2c_{1}\exp(3S_{F}^{\alpha}(t)).$ (46)

Choosing $c_{1}=0$, yields:

$\mathbf{x}(t)=c_{2}\mathbf{x}^{2}(t),$ (47)

or:

$x_{1}(t)=c_{2}\exp(-S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=-2c_{2}\exp(-S_{F}^{\alpha}(t)).$ (48)

Figure 1: We have plotted $x_{1}$ for $c_{1}=1$ (red) and $c_{2}=2$ (blue).
Figure 2: We have plotted $x_{2}(t)$ for $c_{1}=1/2$ (red) and $c_{2}=2$
(blue).

In Figures 1 and 2, we have visualized Eqs.(46) and (48) for various value of
the constants $c_{1}$ and $c_{2}$.

######  Example 2.

Consider the system of fractal differential equations given by:

$D_{F}^{\alpha}\mathbf{x}(t)=\left(\begin{array}[]{cc}-3&\sqrt{2}\\\
\sqrt{2}&-2\\\ \end{array}\right)\mathbf{x}(t).$ (49)

To find the solution, we assume
$\mathbf{x}(t)=\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t))$. Substituting this
into Eq.(49), we obtain the algebraic system:

$\left(\begin{array}[]{cc}-3-r&\sqrt{2}\\\ \sqrt{2}&-2-r\\\
\end{array}\right)\left(\begin{array}[]{c}\xi_{1}\\\ \xi_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ \end{array}\right).$ (50)

So, the eigenvalues and eigenvectors are:

$r_{1}=-1,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \boldsymbol{\xi}^{1}=\left(\begin{array}[]{c}1\\\
\sqrt{2}\\\ \end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ r_{2}=-4,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\boldsymbol{\xi}^{2}=\left(\begin{array}[]{c}-\sqrt{2}\\\ 1\\\
\end{array}\right).$ (51)

Thus, the general solution is:

$\begin{split}\mathbf{x}(t)&=c_{1}\mathbf{x}^{1}(t)+c_{2}\mathbf{x}^{2}(t)\\\
&=c_{1}\left(\begin{array}[]{c}1\\\ \sqrt{2}\\\
\end{array}\right)\exp(-S_{F}^{\alpha}(t))+c_{2}\left(\begin{array}[]{c}-\sqrt{2}\\\
1\\\ \end{array}\right)\exp(-4S_{F}^{\alpha}(t)).\end{split}$ (52)

or if $c_{2}=0$

$x_{1}(t)=c_{1}\exp(-S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=\sqrt{2}\leavevmode\nobreak\ c_{1}\exp(-S_{F}^{\alpha}(t))$ (53)

and if $c_{1}=0$

$x_{1}(t)=-\sqrt{2}\leavevmode\nobreak\
c_{2}\exp(-4S_{F}^{\alpha}(t)),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=c_{2}\exp(-4S_{F}^{\alpha}(t))$ (54)

Figure 3: Graph of $x_{1}(t)$ for $c_{1}=1$ and $c_{2}=1$ Figure 4: Graph of
$x_{2}(t)$ for $c_{1}=1$ and $c_{2}=1$

In Figures 3 and 4, we have plotted Eq.(53) and Eq.(54) for different values
of $c_{1}$ and $c_{2}$.

######  Example 3.

Consider the system of fractal differential equations given by:

$D_{F}^{\alpha}\mathbf{x}(t)=\left(\begin{array}[]{cc}-\frac{1}{2}&1\\\
-1&-\frac{1}{2}\\\ \end{array}\right)\mathbf{x}.$ (55)

To find the solution, we assume
$\mathbf{x}(t)=\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t))$. Substituting this
into Eq.(55) we have:

$\left(\begin{array}[]{cc}-\frac{1}{2}-r&1\\\ -1&-\frac{1}{2}-r\\\
\end{array}\right)\left(\begin{array}[]{c}\xi_{1}\\\ \xi_{2}\\\
\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\ \end{array}\right).$ (56)

Then, the eigenvalues and eigenvectors are:

$r_{1}=-\frac{1}{2}+i,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \boldsymbol{\xi}^{1}=\left(\begin{array}[]{c}1\\\ i\\\
\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ r_{2}=-\frac{1}{2}-i,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\boldsymbol{\xi}^{2}=\left(\begin{array}[]{c}1\\\ -i\\\ \end{array}\right),$
(57)

and the general solution is:

$\begin{split}\mathbf{x}(t)&=c_{1}\mathbf{x}^{1}(t)+c_{2}\mathbf{x}^{2}(t)\\\
&=c_{1}\left(\begin{array}[]{c}1\\\ i\\\
\end{array}\right)\exp((-\frac{1}{2}+i)S_{F}^{\alpha}(t))+c_{2}\left(\begin{array}[]{c}1\\\
-i\\\ \end{array}\right)\exp((-\frac{1}{2}-i)S_{F}^{\alpha}(t)).\end{split}$
(58)

We can express the real-valued solution as:

$\begin{split}\mathbf{u}(t)&=\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\left(\begin{array}[]{c}\cos(S_{F}^{\alpha}(t))\\\
-\sin(S_{F}^{\alpha}(t))\\\ \end{array}\right),\\\
\mathbf{v}(t)&=\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\left(\begin{array}[]{c}\sin(S_{F}^{\alpha}(t))\\\
\cos(S_{F}^{\alpha}(t))\\\ \end{array}\right).\end{split}$ (59)

Then the general solution is

$\mathbf{x}(t)=c_{1}\mathbf{u}(t)+c_{2}\mathbf{v}(t)$ (60)

or if $c_{1}=0$

$x_{1}(t)=c_{2}\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\sin(S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=c_{2}\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\cos(S_{F}^{\alpha}(t))$
(61)

or if $c_{2}=0$

$x_{1}(t)=c_{1}\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\cos(S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=-c_{1}\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\sin(S_{F}^{\alpha}(t))$
(62)

Their Wronskian is:

$\begin{split}W[\mathbf{u},\mathbf{v}](t)&=\begin{vmatrix}\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\cos(S_{F}^{\alpha}(t))&\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\sin(S_{F}^{\alpha}(t))\\\
-\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\sin(S_{F}^{\alpha}(t))&\exp\left(-\frac{S_{F}^{\alpha}(t)}{2}\right)\cos(S_{F}^{\alpha}(t))\\\
\end{vmatrix}\\\ &=\exp\left(-S_{F}^{\alpha}(t)\right)\end{split}$ (63)

which is never zero. Thus, $\mathbf{u}(t)$ and $\mathbf{v}(t)$ are fundamental
solutions of Eq.(55).

Figure 5: Graph of $x_{1}$ for different values of $c_{1}$ and $c_{2}$. Figure
6: Graph of $x_{2}$ for different values of $c_{1}$ and $c_{2}$.

Graphs depicting the behavior of $x_{1}$ and $x_{2}$ in Eqs.(61) and (62) with
respect to $t$ are illustrated in Figures 5 and 6.

######  Example 4.

Consider the system of fractal differential equations given by:

$D_{F}^{\alpha}\mathbf{x}(t)=\left(\begin{array}[]{cc}1&-1\\\ 1&3\\\
\end{array}\right)\mathbf{x}(t).$ (64)

To solve Eq.(64), let
$\mathbf{x}(t)=\boldsymbol{\xi}\exp(rS_{F}^{\alpha}(t))$. Substituting this
into Eq.(64), we find the double eigenvalue:

$r_{1}=r_{2}=2,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \boldsymbol{\xi}=\left(\begin{array}[]{c}1\\\ -1\\\
\end{array}\right)$ (65)

One independent solution is then:

$\mathbf{x}^{(1)}(t)=\left(\begin{array}[]{c}1\\\ -1\\\
\end{array}\right)\exp(2S_{F}^{\alpha}(t))$ (66)

To find the second independent solution, we assume:

$\mathbf{x}=\boldsymbol{\xi}S_{F}^{\alpha}(t)\exp(2S_{F}^{\alpha}(t))+\boldsymbol{\eta}\exp(2S_{F}^{\alpha}(t))$
(67)

where $\boldsymbol{\xi}$ and $\boldsymbol{\eta}$ are constant vectors to be
determined. Substituting Eq.(67) into Eq.(64), we get:

$(\mathbf{A}-2\mathbf{I})\boldsymbol{\xi}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\mathbf{A}-2\mathbf{I})\boldsymbol{\eta}=\boldsymbol{\xi}$ (68)

By solving Eq.(68), we obtain:

$r_{1}=r_{2}=2,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \boldsymbol{\xi}=\left(\begin{array}[]{c}1\\\ -1\\\
\end{array}\right)$ (69)

and

$\boldsymbol{\eta}=\left(\begin{array}[]{c}0\\\ -1\\\
\end{array}\right)+k\left(\begin{array}[]{c}1\\\ -1\\\ \end{array}\right)$
(70)

where $k$ is an arbitrary constant. Substituting $\boldsymbol{\xi}$ and
$\boldsymbol{\eta}$ into Eq.(67), we find the second independent solution:

$\mathbf{x}^{(2)}(t)=\left(\begin{array}[]{c}1\\\ -1\\\
\end{array}\right)S_{F}^{\alpha}(t)\exp(2S_{F}^{\alpha}(t))+\left(\begin{array}[]{c}0\\\
-1\\\ \end{array}\right)\exp(2S_{F}^{\alpha}(t))$ (71)

The Wronskian of these solutions is:

$W[\mathbf{x}^{(1)}(t),\mathbf{x}^{(2)}(t)]=-\exp(4S_{F}^{\alpha}(t))$ (72)

Thus, $\mathbf{x}^{(1)}(t)$ and $\mathbf{x}^{(2)}(t)$ form a fundamental set
of solutions. The general solution of Eq.(64) is then given by:

$\begin{split}\mathbf{x}(t)&=c_{1}\mathbf{x}^{(1)}(t)+c_{2}\mathbf{x}^{(2)}(t)\\\
&=c_{1}\left(\begin{array}[]{c}1\\\ -1\\\
\end{array}\right)\exp(2S_{F}^{\alpha}(t))+c_{2}\bigg{[}\left(\begin{array}[]{c}1\\\
-1\\\ \end{array}\right)S_{F}^{\alpha}(t)\exp(2S_{F}^{\alpha}(t))\\\
&+\left(\begin{array}[]{c}0\\\ -1\\\
\end{array}\right)\exp(2S_{F}^{\alpha}(t))\bigg{]}\end{split}$ (73)

or if $c_{1}=0$ we have

$x_{1}(t)=c_{2}S_{F}^{\alpha}(t)\exp(2S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=-c_{2}(S_{F}^{\alpha}(t)+1)\exp(2S_{F}^{\alpha}(t))$ (74)

if $c_{2}=0$ we have

$x_{1}(t)=c_{1}\exp(2S_{F}^{\alpha}(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
x_{2}(t)=-c_{1}\exp(2S_{F}^{\alpha}(t))$ (75)

Figure 7: Graph of $x_{1}$ for different values of $c_{1}=1$ and $c_{2}=1$.
Figure 8: Graph of $x_{2}$ for different values of $c_{1}=-1$ and $c_{2}=-1$.

Graphs of $x_{1}$ and $x_{2}$ in Eqs.(74) and (75) are shown in Figures 7 and
8.

######  Example 5.

Consider the system of fractal differential equations represented as:

$D_{F}^{\alpha}\mathbf{x}(t)=\left(\begin{array}[]{cc}2&0\\\ 0&-3\\\
\end{array}\right)\mathbf{x}(t).$ (76)

Expressing Eq.(76) in the scalar form, we obtain that:

$D_{F}^{\alpha}x_{1}(t)=2x_{1}(t),\quad D_{F}^{\alpha}x_{2}(t)=2x_{2}(t).$
(77)

The solution to Eq.(77) is:

$\mathbf{x}(t)=\left(\begin{array}[]{c}c_{1}\exp(2S_{F}^{\alpha}(t))\\\
c_{2}\exp(-3S_{F}^{\alpha}(t))\\\
\end{array}\right)=c_{1}\left(\begin{array}[]{c}\exp(2S_{F}^{\alpha}(t))\\\
0\\\
\end{array}\right)+c_{2}\left(\begin{array}[]{c}\exp(-3S_{F}^{\alpha}(t))\\\
0\\\ \end{array}\right).$ (78)

Alternatively, we can express it as:

$\mathbf{x}^{1}(t)=\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)\exp(2S_{F}^{\alpha}(t)),\quad\mathbf{x}^{2}(t)=\left(\begin{array}[]{c}0\\\
1\\\ \end{array}\right)\exp(-3S_{F}^{\alpha}(t)).$ (79)

The Wronskian of these two solutions is:

$W[\mathbf{x}^{1}(t),\mathbf{x}^{2}(t)]=\begin{vmatrix}\exp(2S_{F}^{\alpha}(t))&0\\\
0&\exp(-3S_{F}^{\alpha}(t))\\\ \end{vmatrix}=\exp(-S_{F}^{\alpha}(t)),$ (80)

which is never zero. Hence, $\mathbf{x}^{1}(t)$ and $\mathbf{x}^{2}(t)$
constitute a fundamental set of solutions.

##  5 Conclusion

In this paper we have introduced the notion of a fractal matrix and its
corresponding homogeneous system of fractal differential equations, through
the use of $F^{\alpha}$-calculus, introduced by Parvate and Gangal [35] and
subsequently studied by Khalili Golmankhaneh [37]. As it is well known the
systems of linear differential equations are of great importance in
literature, both because many physical phenomena can be modeled by equations
of this type and because even non-linear phenomena can, in a first
approximation, be described by linear equations. Therefore, motivated by this
fact, we dealt with the detailed study of the set of solutions of homogeneous
system of $\alpha$-order linear differential equations:
$D^{\alpha}_{F}\mathbb{x}(t)=\mathbb{P}(t)\mathbb{x}(t),$ having non-integer
orders of differentiation in the system. Moreover, guided by the fact that,
the solutions to such systems often exhibit complex and self-similar patterns
over different scales, akin to fractal geometries, we have explicitly
represented the solutions of homogeneous system of $\alpha$-order linear
differential equations with constant coefficients:
$D^{\alpha}_{F}\mathbb{x}(t)=\mathbb{A}\mathbb{x}(t),$ illustrating everything
with significant examples.
Declaration of Competing Interest:
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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