# Generalized Bernoulli Differential Equation

Hector Carmenate  Mathematics and Natural Sciences Department, Padron Campus,
Miami Dade College, 627 SW 27th Ave, Miami, 33135, Fl, United States; E-mail:
<EMAIL_ADDRESS>Paul Bosch  Facultad de Ingeniería, Universidad del
Desarrollo, Ave. La Plaza 680, San Carlos de Apoquindo, Las Condes, 7550000,
Santiago, Chile; E-mail<EMAIL_ADDRESS>Juan E. Nápoles  Facultad de Ciencias
Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste,
Corrientes 3400, Argentina; E-mail<EMAIL_ADDRESS>José M.
Sigarreta  Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos
E. Adame No.54 Col. Garita, Acapulco Gro. 39650, Mexico; E-mail:
<EMAIL_ADDRESS>
(January 2023)

###### Abstract

In this paper we propose and solve a generalization of the Bernoulli
Differential Equation, by means of a generalized fractional derivative. First
we prove a generalization of Gronwall’s inequality, which is useful for
studying the stability of systems of fractional differential equations and we
state results about the qualitative behavior of the trivial solution of the
proposed equation. After that, we prove and state the main results about the
solution of the generalized Bernoulli Differential Equation and also we give
some examples that show the advantage of considering this fractional
derivative approach. We also present a finite difference method as an
alternative to the solution of the generalized Bernoulli equation and prove
its validity by means of examples.

keywords: generalized local derivative, boundedness of solutions, Gronwall
inequality

##  1 Introduction

Fractional calculus is a branch of mathematics that studies the
differentiation and integration operators of generalized orders, fractional
calculus had its birth almost at the same time as traditional calculus. But
its application has already been widely demonstrated, for some applications
refer to the following works [16, 27, 17]. Fractional Calculus has had
different difficulties in the study of systems of fractional differential
equations, although some approaches can be found in [8, 11, 29].

The global fractional derivatives collect information on an interval and keep
track of the history of the process, can be said to possess a certain memory;
which makes it possible to model non-local and distributed responses that
commonly appear in natural and physical phenomena, although it is known that
they have certain limitations. [22] defines a conformable fractional
derivative that eliminates these disadvantages; more recently, a non-
conformable local fractional derivative is introduced in [13]. We can consider
fractional local derivatives to be a new tool, which has been proven its
usefulness in multiple applications for several autors, see for example [30,
9, 13, 16].

This paper relies on use of new differential operators, depending on a general
kernel function $T(t,\alpha)$, which include at once several local derivatives
earlier introduced and studied in many different sources. This new tool is
very powerful because we can model a phenomenon from two points of view:
Considering different kernels and varying the order involved in said kernel.

It’s know that one of the most paradigmatic nonlinear equations is the
Bernoulli differential equation (which is know was introduced by Jacob in his
work [6]), this equation can be considered as a statement of the principle of
conservation of energy in fluids. In this paper we’ll study Bernoulli’s
equation under the generalized fractional local derivative operator, but first
we prove a generalized Gronwell’s inequality and we obtain stability
conditions for systems of differential fractional equations under the approach
of the generalized local fractional derivative, after that we find solvability
and stability conditions for the generalized Bernoulli’s equation proposed and
finally, we give some examples showing particular cases of this Bernoulli’s
equation seen from different approaches.

There are cases in which the fractional differential equation becomes too
complicated to solve by classical methods and other alternatives must be
analyzed, numerical methods are a very useful tool when we must solve this
type of equations. The finite difference methods replace the derivative
operator by an appropriate quotient in differences that allows to approximate
the solution quite efficiently. Here we use a finite difference method to
solve the generalized Bernoulli equation and show the results obtained by
means of examples.

##  2 Generalized Derivation and Integration Operators

In this section we present a definition of a generalized local fractional
derivative introduced in [10] as well as a fractional integration operator
introduced in [9] and some of its most important properties that will be
useful in the next section.

We will now present some advantages of studying the Bernoulli equation using
the generalized differentiation operator.

The derivative we are considering generalizes many of the properties of the
local derivatives existing so far, it also allows the computation of higher
order derivatives and is not limited only to functions defined on the positive
half-line. It is important to emphasize that the choice of the kernel
$T(t,\alpha)$ leads to different practical applications. Thanks to the
generality of the theoretical results obtained, we can state that they do not
depend on the choice of the kernel. In the same direction, different
applications of the generalized fractional derivatives are shown, as well as
their relations with other types of local derivatives(conformable or not), for
example [31]:

  1. 1.

If $\alpha\in(0,1]$ and $T(t,\alpha)=t^{1-\alpha}$, then the conformable
fractional derivative defined in [22] is obtained.

  2. 2.

If $\alpha\in(0,1]$ and $T(t,\alpha)=k(t)^{1-\alpha}$, then the general
conformable fractional derivative defined in [2] is derived.

  3. 3.

If $\alpha\in(0,1]$ and $T(t,\alpha)=e^{t^{-\alpha}}$, then the non-
conformable fractional derivative defined in [13] is obtained.

  4. 4.

If $\alpha\in(0,1]$ and
$T(t,\alpha)={\left(t+\frac{1}{\Gamma(\alpha)}\right)}^{1-\alpha}$, then we
obtain the beta-derivative defined in [3].

In [10] the definition of generalized fractional derivative is given as
follows.

Given $s\in\mathbb{R}$, we denote by $\lceil s\rceil$ the _upper integer part_
of $s$, i.e., the smallest integer greater than or equal to $s$.

###### Definition 1

Given an interval $I\subseteq\mathbb{R}$, $f:I\rightarrow{\mathbb{R}}$,
$\alpha\in\mathbb{R}^{+}$ and a positive continuous function $T(t,\alpha)$ on
$I$, the _derivative_ $G_{T}^{\alpha}f$ of $f$ of order $\alpha$ at the point
$t\in I$ is defined by

$\displaystyle G_{T}^{\alpha}f(t)$ $\displaystyle=\lim_{h\to
0}\frac{1}{h^{\lceil\alpha\rceil}}$
$\displaystyle\times\sum_{k=0}^{\lceil\alpha\rceil}(-1)^{k}\binom{\lceil\alpha\rceil}{k}f\big{(}t-khT(t,\alpha)\big{)}.$
(1)

If $a=\min\\{t\in I\\}$ (respectively, $b=\max\\{t\in I\\}$), then
$G_{T}^{\alpha}f(a)$ (respectively, $G_{T}^{\alpha}f(b)$) is defined with
$h\to 0^{-}$ (respectively, $h\to 0^{+}$) instead of $h\to 0$ in the limit.

If we choose the function $T(t,\alpha)=t^{\lceil\alpha\rceil-\alpha}$, then we
obtain the following particular case of the function $G_{T}^{\alpha}$ which is
a conformable derivative.

###### Definition 2

Let $I$ be an interval $I\subseteq(0,\infty)$, $f:I\rightarrow{\mathbb{R}}$
and $\alpha\in\mathbb{R}^{+}$. The _conformable derivative_ $G^{\alpha}f$ of
$f$ of order $\alpha$ at the point $t\in I$ is defined by

$\displaystyle G^{\alpha}f(t)$ $\displaystyle=\lim_{h\rightarrow
0}\frac{1}{h^{\lceil\alpha\rceil}}$
$\displaystyle\times\sum_{k=0}^{\lceil\alpha\rceil}(-1)^{k}\binom{\lceil\alpha\rceil}{k}f\big{(}t-kht^{\lceil\alpha\rceil-\alpha}\big{)}.$
(2)

Note that, if $\alpha=n\in\mathbb{N}$ and $f$ is smooth enough, then
Definition 2 coincides with the classical definition of the $n$-th derivative.

In [22] is defined a conformable derivative $T_{\alpha}$ which is a particular
case of $G^{\alpha}$ when $\alpha\in(0,1]$ and $T(t,\alpha)=t^{1-\alpha}$. See
[1], [19] and [20] for more information on $T_{\alpha}$.

The following results in [10] contain some basic properties of the derivative
$G_{T}^{\alpha}$.

###### Theorem 3

Let $I$ be an interval $I\subseteq\mathbb{R}$, $f:I\rightarrow{\mathbb{R}}$
and $\alpha\in\mathbb{R}^{+}$.

$(1)$ If there exists $D^{\lceil\alpha\rceil}f$ at the point $t\in I$, then
$f$ is $G_{T}^{\alpha}$-differentiable at $t$ and
$G_{T}^{\alpha}f(t)=T(t,\alpha)^{\lceil\alpha\rceil}D^{\lceil\alpha\rceil}f(t)$.

$(2)$ If $\alpha\in(0,1]$, then $f$ is $G_{T}^{\alpha}$-differentiable at
$t\in I$ if and only if $f$ is differentiable at $t$; in this case, we have
$G_{T}^{\alpha}f(t)=T(t,\alpha)f^{\prime}(t)$.

###### Theorem 4

Let $I$ be an interval $I\subseteq\mathbb{R}$, $f,g:I\rightarrow{\mathbb{R}}$
and $\alpha\in\mathbb{R}^{+}$. Assume that $f,g$ are
$G_{T}^{\alpha}$-differentiable functions at $t\in I$. Then the following
statements hold:

$(1)$ $af+b\,g$ is $G_{T}^{\alpha}$-differentiable at $t$ for every
$a,b\in\mathbb{R}$, and
$G_{T}^{\alpha}(af+b\,g)(t)=a\,G_{T}^{\alpha}f(t)+b\,G_{T}^{\alpha}g(t)$.

$(2)$ If $\alpha\in(0,1]$, then $fg$ is $G_{T}^{\alpha}$-differentiable at $t$
and $G_{T}^{\alpha}(fg)(t)=f(t)G_{T}^{\alpha}g(t)+g(t)G_{T}^{\alpha}f(t)$.

$(3)$ If $\alpha\in(0,1]$ and $g(t)\neq 0$, then $f/g$ is
$G_{T}^{\alpha}$-differentiable at $t$ and
$G_{T}^{\alpha}(\frac{f}{g})(t)=\frac{g(t)G_{T}^{\alpha}f(t)-f(t)G_{T}^{\alpha}g(t)}{g(t)^{2}}$.

$(4)$ $G_{T}^{\alpha}(\lambda)=0$, for every $\lambda\in\mathbb{R}.$

$(5)$
$G_{T}^{\alpha}(t^{p})=\frac{\Gamma(p+1)}{\Gamma(p-\lceil\alpha\rceil+1)}t^{p-\lceil\alpha\rceil}T(t,\alpha)^{{\lceil\alpha\rceil}}$,
for every $p\in\mathbb{R}\setminus\mathbb{Z}^{-}\\!.$

$(6)$
$G_{T}^{\alpha}(t^{-n})=(-1)^{\lceil\alpha\rceil}\frac{\Gamma(n+\lceil\alpha\rceil)}{\Gamma(n)}\,t^{-n-\lceil\alpha\rceil}T(t,\alpha)^{{\lceil\alpha\rceil}}$,
for every $n\in\mathbb{Z}^{+}.$

###### Theorem 5

Let $\alpha\in(0,1]$, $g$ a $G_{T}^{\alpha}$-differentiable function at $t$
and $f$ a differentiable function at $g(t)$. Then $f\circ g$ is
$G_{T}^{\alpha}$-differentiable at $t$, and $G_{T}^{\alpha}(f\circ
g)(t)=f^{\prime}(g(t))\,G_{T}^{\alpha}g(t)$.

In [9] is defined an integral operator in the following way.

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a,t\in I$ and
$\alpha\in\mathbb{R}$. The integral operator $J_{T,a}^{\alpha}$ is defined for
every locally integrable function $f$ on $I$ as

$J_{T,a}^{\alpha}(f)(t)=\int_{a}^{t}\frac{f(\omega)}{T(\omega,\alpha)}\;d\omega.$

The following results appear in [9].

###### Proposition 6

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a\in I$, $0<\alpha\leq 1$ and
$f$ a differentiable function on $I$ such that $f^{\prime}$ is a locally
integrable function on $I$. Then, we have for all $t\in I$

$J_{T,a}^{\alpha}\big{(}{G}_{T}^{\alpha}(f)\big{)}(t)=f(t)-f(a).$

###### Proposition 7

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a\in I$ and $\alpha\in(0,1]$.

$G_{T}^{\alpha}\big{(}J_{T,a}^{\alpha}(f)\big{)}(t)=f(t),$

for every continuous function $f$ on $I$ and $a,t\in I$.

In [22] it is defined the integral operator $J_{T,a}^{\alpha}$ with $T$ given
by $T(t,\alpha)=t^{1-\alpha}$, and [22, Theorem 3.1] shows

$G^{\alpha}J_{t^{1-\alpha}\\!,\,a}^{\alpha}(f)(t)=f(t),$

for every continuous function $f$ on $I$, $a,t\in I$ and $\alpha\in(0,1]$.
Hence, Proposition 7 extends to any $T$ this important equality.

The following result summarizes some elementary properties of the integral
operator $J_{T,a}^{\alpha}$.

###### Theorem 8

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a,b\in I$ and
$\alpha\in\mathbb{R}$. Suppose that $f,g$ are locally integrable functions on
$I$, and $k_{1},k_{2}\in\mathbb{R}$. Then we have

$(1)$
$J_{T,a}^{\alpha}\big{(}k_{1}f+k_{2}g\big{)}(t)=k_{1}J_{T,a}^{\alpha}f(t)+k_{2}J_{T,a}^{\alpha}g(t),$

$(2)$ if $f\geq g$, then $J_{T,a}^{\alpha}f(t)\geq J_{T,a}^{\alpha}g(t)$ for
every $t\in I$ with $t\geq a$,

$(3)$ $\left|J_{T,a}^{\alpha}f(t)\right|\leq
J_{T,a}^{\alpha}\left|f\right|(t)$ for every $t\in I$ with $t\geq a$,

$(4)$
$\int_{a}^{b}\frac{f(\omega)}{T(\omega,\alpha)}d\omega=J_{T,a}^{\alpha}f(t)-J_{T,b}^{\alpha}f(t)=J_{T,a}^{\alpha}f(t)(b)$
for every $t\in I$.

###### Remark 9

The above results generalize Proposition 1, Proposition 2 and Theorem 1 of
[14], respectively, obtained with $0<\alpha\leq 1$.

Given $a,b\in I$ $(b>a)$, let us denote by $F_{a,b}$ the usual inner product
in $L^{2}[a,b]$

$F_{a,b}(f,g)=\int_{a}^{b}f(t)g(t)\,dt.$

###### Proposition 10

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a,b\in I$ with $a<b$ and
$\alpha\in\mathbb{R}$. The adjoint of $J_{T,a}^{\alpha}$ in $L^{2}[a,b]$ with
respect to the inner product $F_{a,b}$ is the operator

$A^{\alpha}_{T,a,b}(f)(t)=\frac{1}{T(t,\alpha)}\int_{t}^{b}f(s)\,ds.$

###### Proof 1

We obtain, by using integration by parts,

$\displaystyle
F_{a,b}\big{(}J_{T,a}^{\alpha}(f),g\big{)}=\int_{a}^{b}g(t)J_{T,a}^{\alpha}(f)(t)\,dt$
$\displaystyle=\Big{[}-J_{T,a}^{\alpha}(f)(t)\int_{t}^{b}g(s)\,ds\Big{]}_{t=a}^{t=b}$
$\displaystyle+\int_{a}^{b}\frac{f(t)}{T(t,\alpha)}\int_{t}^{b}g(s)\,ds\,dt$
$\displaystyle=\int_{a}^{b}\frac{f(t)}{T(t,\alpha)}\int_{t}^{b}g(s)\,ds\,dt=\int_{a}^{b}f(t)\,A^{\alpha}_{T,a,b}(g)(t)\,dt$
$\displaystyle=F_{a,b}\big{(}f,A^{\alpha}_{T,a,b}(g)\big{)}.$

###### Proposition 11

Let $I$ be an interval $I\subseteq\mathbb{R}$, $a,b\in I$ with $a<b$ and
$\alpha\in\mathbb{R}$. Then $J_{T,a}^{\alpha}$ is a Hilbert-Schmidt integral
operator on $L^{2}[a,b]$, and so, a continuous and compact operator.

###### Proof 2

Let us denote by $\chi_{{}_{A}}$ the _characteristic function_ of the set $A$
(i.e., $\chi_{{}_{A}}(t)=1$ if $t\in A$ and $\chi_{{}_{A}}(t)=0$ otherwise).
Then

$J_{T,a}^{\alpha}(f)(t)=\int_{a}^{t}\frac{f(\omega)}{T(\omega,\alpha)}\;d\omega=\int_{a}^{b}k(t,\omega)f(\omega)\,d\omega,$

with

$k(t,\omega)=\chi_{{}_{[a,t]}}(\omega)\,\frac{1}{T(\omega,\alpha)}\,.$

We have

$\displaystyle\int_{a}^{b}\int_{a}^{b}|k(t,\omega)|^{2}\,d\omega\,dt$
$\displaystyle\leq\int_{a}^{b}\int_{a}^{b}\frac{1}{T(\omega,\alpha)^{2}}\,d\omega\,dt$
$\displaystyle=\int_{a}^{b}\frac{b-a}{T(\omega,\alpha)^{2}}\,d\omega<\infty,$

since $T(\omega,\alpha)$ is a positive continuous function on $I$. Thus,
$J_{T,a}^{\alpha}$ is a Hilbert-Schmidt integral operator on $L^{2}[a,b]$, and
so, a continuous and compact operator.

The integral operator $J_{T,a}^{\alpha}$ can be applied in order to solve
fractional differential equations. Let $p$ and $q$ be continuous functions on
an interval $I\subseteq\mathbb{R}$, $a\in I$ and $\alpha\in(0,1]$. Then the
solution of the linear fractional equation

$G_{T}^{\alpha}y+p(t)y=q(t),$ (3)

on $I$ is given by

$y(t)={e}^{-J_{T,a}^{\alpha}(p)(t)}\left(J_{T,a}^{\alpha}\big{(}q{e}^{J_{T,a}^{\alpha}(p)}\big{)}(t)+C\right).$
(4)

###  2.1 Generalized Bernoulli Differential Equation

Next, we prove a fractional version of Gronwall’s inequality which will be
useful in the study of the stability of systems of fractional differential
equations. A version of this inequality was proved in [1].

###### Theorem 12

Let $r$, $c$, $d$ and $k$ be continuous functions on the interval $[a,b]$,
$d,k\geq 0$, and $\alpha\in(0,1]$, such that

$r(t)\leq c(t)+d(t)J_{T,a}^{\alpha}(kr)(t),$

for all $t\in[a,b]$. Then we have

$r(t)\leq
c(t)+d(t)J_{T,a}^{\alpha}(cke^{-J_{T,a}^{\alpha}(dk)})(t)e^{J_{T,a}^{\alpha}(dk)(t)},$

for every $t\in[a,b]$.

###### Proof 3

Let us define $R(t)=J_{T,a}^{\alpha}(kr)(t)$. Thus, $R(a)=0$,

$r(t)\leq c(t)+d(t)R(t)$

for all $t\in[a,b]$, and Proposition 7 gives

$G_{T}^{\alpha}R(t)=k(t)r(t)\leq c(t)k(t)+d(t)k(t)R(t),$ (5)

for every $t\in[a,b]$, since $k\geq 0$. Let us define
$E(t)=e^{-J_{T,a}^{\alpha}(dk)(t)}$. Theorem 5 and Proposition 7 give

$\displaystyle G_{T}^{\alpha}E(t)$
$\displaystyle=G_{T}^{\alpha}\big{(}e^{-J_{T,a}^{\alpha}(dk)}\big{)}(t)$
$\displaystyle=G_{T}^{\alpha}\big{(}-J_{T,a}^{\alpha}(dk)\big{)}(t)e^{-J_{T,a}^{\alpha}(dk)(t)}$
$\displaystyle=-d(t)k(t)E(t).$

Since $R$ and $E$ are $G_{T}^{\alpha}$-differentiable on $[a,b]$, Theorem 4
and (5) give

$\displaystyle G_{T}^{\alpha}(ER)(t)$
$\displaystyle=E(t)G_{T}^{\alpha}R(t)+R(t)G_{T}^{\alpha}E(t)$
$\displaystyle=E(t)G_{T}^{\alpha}R(t)-d(t)k(t)R(t)E(t)$ $\displaystyle\leq
c(t)k(t)E(t)+d(t)k(t)R(t)E(t)$ $\displaystyle-d(t)k(t)R(t)E(t)=c(t)k(t)E(t),$

for every $t\in[a,b]$, since $E\geq 0$. Since $E(t)R(t)$ is differentiable on
$[a,b]$ by Theorem 3, we have that Theorem 8 and Proposition 6 give

$\displaystyle J_{T,a}^{\alpha}\big{(}ckE\big{)}(t)$ $\displaystyle\geq
J_{T,a}^{\alpha}\big{(}G_{T}^{\alpha}(ER)\big{)}(t)$
$\displaystyle=E(t)R(t)-E(a)R(a)=E(t)R(t),$

for any $t\in[a,b]$. Thus, since $d,E\geq 0$,

$\displaystyle r(t)$ $\displaystyle\leq c(t)+d(t)R(t)\leq
c(t)+d(t)\frac{J_{T,a}^{\alpha}\big{(}ckE\big{)}(t)}{E(t)}$
$\displaystyle=c(t)+d(t)J_{T,a}^{\alpha}\big{(}cke^{-J_{T,a}^{\alpha}(dk)}\big{)}(t)\,e^{J_{T,a}^{\alpha}(dk)(t)},$

for every $t\in[a,b]$.

The argument in the proof of Theorem 12 also gives the following converse
inequality.

###### Theorem 13

Let $r$, $c$, $d$ and $k$ be continuous functions on the interval $[a,b]$,
$d,k\geq 0$, and $\alpha\in(0,1]$, such that

$r(t)\geq c(t)+d(t)J_{T,a}^{\alpha}(kr)(t),$

for all $t\in[a,b]$. Then we have

$r(t)\geq
c(t)+d(t)J_{T,a}^{\alpha}(cke^{-J_{T,a}^{\alpha}(dk)})(t)e^{J_{T,a}^{\alpha}(dk)(t)},$

for every $t\in[a,b]$.

If $c\geq 0$, then the conclusion of Theorem 12 can be simplified.

###### Theorem 14

Let $r$, $c$, $d$ and $k$ be continuous functions on the interval $[a,b]$,
$c,d,k\geq 0$, and $\alpha\in(0,1]$, such that

$r(t)\leq c(t)+d(t)J_{T,a}^{\alpha}(kr)(t),$

for all $t\in[a,b]$. Then we have

$r(t)\leq c(t)+d(t)J_{T,a}^{\alpha}(ck)(t)\,e^{J_{T,a}^{\alpha}(dk)(t)},$

for every $t\in[a,b]$.

###### Proof 4

Theorem 12 gives

$r(t)\leq
c(t)+d(t)J_{T,a}^{\alpha}\big{(}cke^{-J_{T,a}^{\alpha}(dk)}\big{)}(t)\,e^{J_{T,a}^{\alpha}(dk)(t)},$

for every $t\in[a,b]$. Since $d\geq 0$, it suffices to prove that

$J_{T,a}^{\alpha}\big{(}cke^{-J_{T,a}^{\alpha}(dk)}\big{)}(t)\leq
J_{T,a}^{\alpha}(ck)(t)$ (6)

for any $t\in[a,b]$.

Since $dk\geq 0$, Theorem 8 gives $J_{T,a}^{\alpha}(dk)\geq 0$ and so,
$e^{-J_{T,a}^{\alpha}(dk)}\leq 1$. Since $ck\geq 0$, we have
$cke^{-J_{T,a}^{\alpha}(dk)}\leq ck$, and Theorem 8 gives (6).

The following result appears in [9].

###### Lemma 15

Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}\in(0,1]$, $x=(x_{1},\dots,x_{n})$,
$I\subseteq\mathbb{R}$ an interval, $\Omega\subseteq\mathbb{R}^{n}$ an open
set, $t_{0}\in I$ and $x_{0}\in\Omega$. Let
$F=(F_{1},\dots,F_{n}):I\times\Omega\rightarrow\mathbb{R}^{n}$ be in $(C,Lip)$
on some open neighborhood of the point $(t_{0},x_{0})$, and consider the
initial value problem

$G_{T}^{\alpha_{j}}x_{j}=F_{j}(t,x),\quad 1\leq j\leq n,\qquad
x(t_{0})=x_{0}.$ (7)

Then there exists $h>0$ such that (7) has a unique solution on the interval
$[t_{0}-h,t_{0}+h]\cap I$. Furthermore, if $\Omega=\mathbb{R}^{n}$ and $F$ is
in $(C,Lip)$ on $J\times\mathbb{R}^{n}$ for each compact interval $J\subseteq
I$, then (7) has a unique solution on $I$.

Let $\alpha\in(0,1]$, $a\in\mathbb{R}$, $t_{0}\geq a$,
$x_{0}\in\mathbb{R}^{n}$, and
$F:[a,\infty)\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ in $(C,Lip)$ on
$[a,\infty)\times\mathbb{R}^{n}$. Let us consider the initial value problem

${G}_{T}^{\alpha}x(t)=F(t,x),\quad\quad x(t_{0})={x}_{0},$ (8)

Lemma 15 guarantees that (8) has a unique solution on $[a,\infty)$. The study
of boundedness of solutions of a differential equation, either fractional or
not, plays an important role in qualitative theory. In addition, the
qualitative behavior of solutions plays an important role in many real-world
phenomena related to applied research. Based on the previous results, we can
obtain stability results for the solutions of fractional differential
equations.

###### Definition 16

If $F(t,0)=0$ for every $t\geq a$, then the trivial solution $x\equiv 0$ of
(8) is said to be _stable_ if for any $\varepsilon>0$, there exists
$\delta=\delta(t_{0},\varepsilon)>0$ such that if $\left|x_{0}\right|<\delta$,
then $\left|x(t)\right|<\varepsilon$ for every $t\geq t_{0}$; it is _uniformly
stable_ if there exists $\delta=\delta(\varepsilon)>0$ such that if
$\left|x_{0}\right|<\delta$, then $\left|x(t)\right|<\varepsilon$ for every
$t\geq t_{0}\geq a$.

Propositions 6 and 7 give the following result.

###### Proposition 17

Let $\alpha\in(0,1]$, $a\in\mathbb{R}$, $t_{0}\geq a$,
$x_{0}\in\mathbb{R}^{n}$, and
$F:[a,\infty)\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ in $(C,Lip)$ on
$[a,\infty)\times\mathbb{R}^{n}$. Then the problem (8) is equivalent to

$x(t)={x}_{0}+J_{T,t_{0}}^{\alpha}F(s,x(s))(t).$ (9)

###### Theorem 18

Let $\alpha\in(0,1]$, $a\in\mathbb{R}$ and
$F:[a,\infty)\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ in $(C,Lip)$ on
$[a,\infty)\times\mathbb{R}^{n}$. Assume that $F(t,0)=0$ for every $t\geq a$,
$\left|F(t,x)\right|\leq k(t)|x|$ for every $t\geq a$, $x\in\mathbb{R}^{n}$
and some continuous function $k$ such that

$\int_{a}^{\infty}\frac{k(t)}{T(t,\alpha)}\,dt<\infty.$

Then the trivial solution $x\equiv 0$ of (8) is uniformly stable.

###### Proof 5

Let us define

$N=\int_{a}^{\infty}\frac{k(t)}{T(t,\alpha)}\,dt<\infty.$

Theorem 5, Proposition 6 and $k\geq 0$ give for every $t\geq t_{0}\geq a$

$\displaystyle
J_{T,t_{0}}^{\alpha}\big{(}ke^{-J_{T,t_{0}}^{\alpha}(k)}\big{)}(t)\,e^{J_{T,t_{0}}^{\alpha}(k)(t)}$
(10)
$\displaystyle=J_{T,t_{0}}^{\alpha}\big{(}G_{T}^{\alpha}(-e^{-J_{T,t_{0}}^{\alpha}(k)})\big{)}(t)\,e^{J_{T,t_{0}}^{\alpha}(k)(t)}$
$\displaystyle=\big{(}e^{-J_{T,t_{0}}^{\alpha}(k)(t_{0})}-e^{-J_{T,t_{0}}^{\alpha}(k)(t)}\big{)}\,e^{J_{T,t_{0}}^{\alpha}(k)(t)}$
$\displaystyle=\big{(}1-e^{-J_{T,t_{0}}^{\alpha}(k)(t)}\big{)}\,e^{J_{T,t_{0}}^{\alpha}(k)(t)}$
$\displaystyle=e^{J_{T,t_{0}}^{\alpha}(k)(t)}-1$ $\displaystyle\leq
e^{J_{T,a}^{\alpha}(k)(t)}-1$ $\displaystyle\leq e^{N}-1.$

Fix $\varepsilon>0$ and let $x=x(t)$ be the solution of the Cauchy problem
(8), where ${x}_{0}$ satisfies

$\left|{x}_{0}\right|<\delta=\delta(\varepsilon)=e^{-N}\varepsilon.$ (11)

Proposition 17 gives

$x(t)={x}_{0}+J_{T,t_{0}}^{\alpha}F(s,x(s))(t),$

for every $t\geq t_{0}$. Thus, Theorem 8 gives

$\displaystyle\left|x(t)\right|$
$\displaystyle\leq\left|{x}_{0}\right|+\left|J_{T,t_{0}}^{\alpha}F(s,x(s))(t)\right|$
$\displaystyle\leq\left|{x}_{0}\right|+J_{T,t_{0}}^{\alpha}\left|F(s,x(s))\right|(t)$
$\displaystyle\leq\left|{x}_{0}\right|+J_{T,t_{0}}^{\alpha}(k\left|x\right|)(t),$

for every $t\geq t_{0}$. Hence, Theorem 12, (10) and (11) give

$\displaystyle\left|x(t)\right|$
$\displaystyle\leq\left|x_{0}\right|+\left|x_{0}\right|J_{T,t_{0}}^{\alpha}\big{(}ke^{-J_{T,t_{0}}^{\alpha}(k)}\big{)}(t)\,e^{J_{T,t_{0}}^{\alpha}(k)(t)}$
$\displaystyle\leq\left|x_{0}\right|+\left|x_{0}\right|(e^{N}-1)=\left|x_{0}\right|e^{N}<\varepsilon,$

for all $t\geq t_{0}$. Since $\delta$ does not depend on $t_{0}$, the trivial
solution of (8) is uniformly stable.

We will present the basic results of the Stability Theory for the Generalized
Bernoulli Differential Equation:

${G}_{T}^{\alpha}y+p(x)y=q(x){y}^{n},\qquad n\neq 0,1.$ (12)

First, we will pose the problem for a much more general system that 12:

$G_{T}^{\alpha}y(x)=f(x,y(x)),$ (13)

$y(x_{0})=y_{0},$ (14)

where $f\in C(\mathbb{R}_{+}\times\mathbb{R},\mathbb{R})$, $x_{0}>0$. It is
further assumed that for $(x_{0},y_{0})\in$
int$(\mathbb{R}_{+}\times\mathbb{R})$ the initial value problem (13)-(14) has
a solution $y(x)\in C^{\alpha}(I)$ for all $x>x_{0}>0$. In addition, it is
assumed that $f(x,0)=0$ for all $x>x_{0}>0$.

Several types of stability can be discussed for the solutions of differential
equations (of integer or fractional order) that describe dynamic systems. The
most important type the one related to the stability of the solutions near a
point of equilibrium. This can be discussed by Lyapunov’s theory. In simple
terms, if solutions that start near an equilibrium point ${y}_{e}$ stay close
to ${y}_{e}$ for all $x$ then ${y}_{e}$ it’s stable Lyapunov. More strongly,
if ${y}_{e}$ is Lyapunov stable and all solutions starting near ${y}_{e}$
converge to ${y}_{e}$, then ${y}_{e}$ it is asymptotically stable.

The notion of exponential stability (for linear equations or systems)
guarantees a minimum rate of decay, that is, an estimate of the speed with
which the solutions converge. The stability of a solution of fractional
diferential equations can be defined in exactly the same way. The following
results form a certain theory of stability for equations of the type of (13).
Firstly, we will state the following lemma, which will be a basis for
obtaining the desired stability results.

###### Lemma 19

Let $I$ be an interval with $[t_{0},\infty)\subseteq I$ and $\alpha\in(0,1]$.
Let $p$ be a continuous function on $[t_{0},\infty)$, and consider the linear
equation

${G}_{T}^{\alpha}y+p(t)y=0.$ (15)

$(1)$ If $\liminf_{t\to\infty}J_{T,t_{0}}^{\alpha}(p)(t)>-\infty$, then the
trivial solution $y(t)\equiv 0$ is stable.

$(2)$ If $\lim_{t\to\infty}J_{T,t_{0}}^{\alpha}(p)(t)=\infty$, then
$y(t)\equiv 0$ is asymptotically stable.

Now consider the following initial value problem

${G}_{T}^{\alpha}y+p(x)y=0,y(x_{0})=y_{0},$ (16)

and show that the trivial solution $y(t)\equiv 0$ of (16) is stable if and
only if

$J_{T,0}^{\alpha}(p)(+\infty)=+\infty,$ (17)

then, by means of the properties of the integral operator, we can see that the
general solution of (16) is

$y(x)={y}_{0}{e}^{-{J}_{{T,x}_{0}}^{\alpha}p(x)},$ (18)

from which the statement 17 is easily obtained.

We are now in a position to establish the main results on the stability of
equations of type (13).

###### Theorem 20

Consider the differential equation

${G}_{T}^{\alpha}y+p(x)y=g(x,y),\quad x>0,$ (19)

where $p(x)$ is a positive and continuous function such that $0<p\leq p(x)$,
and $g$ is a continuous function with $g(x,0)\equiv 0$ for all $x$. If

$\lim_{z\to 0}\sup_{x>0}\frac{\left|g(x,z)\right|}{z}=0,$ (20)

then the trivial solution $y(x)\equiv 0$ is an asymptotically stable solution.

###### Proof 6

Choose any $\varepsilon\in(0,p)$. By (20), there exists $\delta>0$ such that

$\sup_{x>0}\left|g(x,z)\right|\leq\varepsilon\left|z\right|,\quad for\quad
all\quad\left|z\right|\leq\delta.$

Let ${y}_{0}\in(0,\delta)$. Since no two solutions can intersect, we have
$y(x)>0$ (or $y(x)<0$) for all $x>0$. Consider first that $y(x)>0$, taking
$y({x}_{0})={y}_{0}$ we obtain from (19) that $-p(x)y+g(x,y)\leq-
py+\varepsilon y<0$, i.e., the solution $y(x)$ is decreasing. Therefore, we
have ${G}_{T}^{\alpha}y(t)\leq-(p-\varepsilon)y$ but this is a particular case
of (16), so we have
$0<y(x)\leq{y}_{0}{e}^{-{J}_{{T,x}_{0}}^{\alpha}(p-\varepsilon)(x)}\rightarrow
0,\quad as\quad x\rightarrow+\infty$. Similarly, we can prove the
$y(x)\rightarrow 0$ as $x\rightarrow+\infty$ if ${y}_{0}\in(-\delta,0)$. Hence
the trivial solution $y(x)\equiv 0$ is asymptotically stable.

The previous theorem, allows us to establish conditions to ensure the
asymptotic stability of the Bernoulli Equation (12). In this way we have the
following

###### Corollary 21

Under assumption of Theorem 20 on function $p(x)$, be $q(x)$ a bounded
function, then the trivial solution $y(x)\equiv 0$ of (12) is asymptotically
stable.

The following result shows how the integral operator $J_{T,a}^{\alpha}$ can be
applied in order to solve fractional differential equations.

###### Theorem 22

Let $p$ and $q$ be continuous functions on an interval $I\subseteq\mathbb{R}$,
$a\in I$, $\alpha\in(0,1]$ and $n\in\mathbb{R}\setminus\\{1\\}$, and consider
the Bernoulli fractional equation

$G_{T}^{\alpha}y+p(t)y=q(t)y^{n}.$ (21)

Then the following statements hold.

$(1)$ For each $C\in\mathbb{R}$, the function

$\displaystyle y(t)=$
$\displaystyle[{e}^{-(1-n)J_{T,a}^{\alpha}(p)(t)}((1-n)J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}(t)$
(22) $\displaystyle+C)]^{1/(1-n)}$

is a solution of the Bernoulli fractional equation (21).

$(2)$ For each $t_{0}\in I$ and $y_{0}\in\mathbb{R}$, let $y(t;t_{0},y_{0})$
be the function in (22) with

$C=y_{0}^{1-n}{e}^{(1-n)J_{T,a}^{\alpha}(p)(t_{0})}-(1-n)J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}(t_{0}).$

If $(y_{0}^{1-n})^{1/(1-n)}=y_{0}$ and $y(t;t_{0},y_{0})$ is defined in some
right or left neighborhood $U\subseteq I$ of $t_{0}$, then $y(t;t_{0},y_{0})$
is a solution of (21) on $U$ satisfying the initial condition
$y(t_{0};t_{0},y_{0})=y_{0}$.

$(3)$ For each $t_{0}\in I$ and $y_{0}>0$ there exists a unique solution of
(21) satisfying the initial value $y(t_{0})=y_{0}$, given by
$y(t;t_{0},y_{0})$.

$(4)$ If $n\in(1,\infty)\cap\mathbb{Q}$ and $1/(1-n)=r/s$ with $s$ an odd
integer, then for each $t_{0}\in I$ and $y_{0}\in\mathbb{R}$ there exists a
unique solution of (21) satisfying the initial value $y(t_{0})=y_{0}$, given
by $y(t;t_{0},y_{0})$.

###### Proof 7

For each $C\in\mathbb{R}$, let us define the function

$\displaystyle z(t)=$
$\displaystyle{e}^{-(1-n)J_{T,a}^{\alpha}(p)(t)}((1-n)J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}(t)$
(23) $\displaystyle+C).$

Therefore, $y=z^{1/(1-n)}$. Theorems 4 and 5 and Proposition 7 give

$\displaystyle G_{T}^{\alpha}z$
$\displaystyle=(1-n)G_{T}^{\alpha}\big{(}-J_{T,a}^{\alpha}(p)\big{)}(t)\,e^{-(1-n)J_{T,a}^{\alpha}(p)(t)}$
$\displaystyle\times\left((1-n)J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}(t)+C\right)$
$\displaystyle\quad+(1-n)e^{-(1-n)J_{T,a}^{\alpha}(p)(t)}G_{T}^{\alpha}\left(J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}\right)(t)$
$\displaystyle=-(1-n)p(t)\,e^{-(1-n)J_{T,a}^{\alpha}(p)(t)}$
$\displaystyle\times\left((1-n)J_{T,a}^{\alpha}\big{(}q{e}^{(1-n)J_{T,a}^{\alpha}(p)}\big{)}(t)+C\right)$
$\displaystyle\quad+(1-n)e^{-(1-n)J_{T,a}^{\alpha}(p)(t)}q(t){e}^{(1-n)J_{T,a}^{\alpha}(p)(t)}$
$\displaystyle=-(1-n)p(t)z+(1-n)q(t).$

Hence, Theorem 5 gives

$\displaystyle G_{T}^{\alpha}y$
$\displaystyle=\frac{1}{1-n}\,z^{n/(1-n)}G_{T}^{\alpha}z$
$\displaystyle=\frac{1}{1-n}\,z^{n/(1-n)}\Big{(}-(1-n)p(t)z+(1-n)q(t)\Big{)}$
$\displaystyle=y^{n}\big{(}-p(t)y^{1-n}+q(t)\big{)}$
$\displaystyle=-p(t)y+q(t)y^{n},$

and $y$ is a solution of the Bernoulli fractional equation (21).

If $(y_{0}^{1-n})^{1/(1-n)}=y_{0}$ and $y(t;t_{0},y_{0})$ is defined in some
right or left neighborhood $U\subseteq I$ of $t_{0}$, then it is clear that
$y(t;t_{0},y_{0})$ is a solution of (21) on $U$ satisfying the initial
condition $y(t_{0};t_{0},y_{0})=y_{0}$.

By Theorem 3, the fractional differential equation (21) is equivalent to

$y^{\prime}=\frac{-1}{T(t,\alpha)}\,p(t)y+\frac{1}{T(t,\alpha)}\,q(t)y^{n}=:f(t,y).$

Since $y_{0}>0$, the function $f(t,y)$ is continuous in a neighborhood $V$ of
$(t_{0},y_{0})$ and, also, it is Lipschitz in the second variable in $V$.
Therefore, Picard’s Theorem gives that there exists a unique solution of (21)
satisfying the initial value $y(t_{0})=y_{0}$. Since $y_{0}>0$, we have
$(y_{0}^{1-n})^{1/(1-n)}=y_{0}$, and the second item of this theorem gives
that this solution is $y(t;t_{0},y_{0})$.

Assume now that $n\in(1,\infty)\cap\mathbb{Q}$ and $1/(1-n)=r/s$ with $s$ an
odd integer. For each $y_{0}\in\mathbb{R}$, the function $f(t,y)$ is
continuous in a neighborhood $V$ of $(t_{0},y_{0})$; since $n>1$, $f$ is
Lipschitz in the second variable in $V$. Hence, Picard’s Theorem gives that
there exists a unique solution of (21) with $y(t_{0})=y_{0}$. Since
$1/(1-n)=r/s$ with $s$ an odd integer, we have $(y_{0}^{1-n})^{1/(1-n)}=y_{0}$
for every $y_{0}\in\mathbb{R}$, and the second item of this theorem gives that
this solution is $y(t;t_{0},y_{0})$.

###### Remark 23

The results obtained, relative to the Bernoulli Equation, generalize and
complete those obtained in [25], obtained using the conformable derivative of
[22].

The generalized derivative can be considered as a good tool to solve certain
types of problems, therefore, by having the Bernoulli differential equation
under this approach, we have a greater possibility to study and solve certain
type of problems, as we will see in the next examples.

Consider the functions $p(t)=q(t)=1$, the interval $I=[0.5,2]$ and $n=2$,
according with Equation (21) we obtain,

$G_{T}^{\alpha}y+y=y^{2},$ (24)

and consider the function $T(t,\alpha)=t^{1-\alpha}$ and $C=-1$, then by
Theorem (22) we obtain the following solution,

$y(t)=\left(1-2e^{\frac{t^{\alpha}-(0.5)^{\alpha}}{\alpha}}\right)^{-1},$ (25)

the solutions associated for differents values of $\alpha$ are shown in Figure
(1).

Figure 1: Solution of Equation 24 with $T(t,\alpha)=t^{1-\alpha}$ for
$\alpha=0.1,0.3,0.5,0.7,0.9,1$ .

As we can see, the solutions approach the curve associated with the value of
$\alpha=1$, which corresponds to the case of the Bernoulli equation with
derivative of order one, this is because the kernel $T(t,\alpha)=t^{1-\alpha}$
is associated with a conformable fractional derivative, however this does not
happen in all cases, as we will show in the next example.

Now if we choose the functions $p(t)=q(t)=e^{t}$, the interval $I=[0.5,6]$ and
$n=2$, replacing this values in Equation (21) we obtain

$G_{T}^{\alpha}y+e^{t}y=e^{t}y^{2},$ (26)

then, if we choose the kernel $T(t,\alpha)=e^{t-\alpha}$ and $C=1$, by Theorem
(22) we obtain,

$y(t)=\frac{1}{1-3e^{(t-0.5)e^{\alpha}}},$ (27)

in Figure (2) the solutions associated are shown for different values of
$\alpha$.

Figure 2: Solution of Equation (26) with $T(t,\alpha)=e^{t-\alpha}$ for
$\alpha=0.1,0.3,0.5,0.7,0.9,1$ .

In this second example we can see that the solutions approximate in ascending
order the solution associated with the case $\alpha=1$, in this case the
kernel $T(t,\alpha)=e^{t-\alpha}$ corresponds to a non-conformable fractional
derivative, however, we could obtain similar results to the first example, if
we choose a suitable function $T(t,\alpha)$ in such a way that this function
corresponds to a conformable derivative, as we can see, one of the advantages
of considering the generalized derivative is precisely the freedom to choose
the kernel in such a way that it is appropriate for the study and solution of
the problem under consideration, in addition to the fact that it generalizes
the definitions of fractional derivative mentioned at the beginning of this
work.

###  2.2 A Finite Difference Method for the Generalized Bernoulli Differential
Equation

The equation,

$\left\\{\begin{array}[]{l}G_{T}^{\alpha}y+p(t)y=q(t)y^{n},\,\,a\leq t\leq
b,\\\ y(a)=\beta,\end{array}\right.$ (28)

according to Theorem 3 is equivalent to

$y^{\prime}+\frac{1}{T(t,\alpha)}\,p(t)y=\frac{1}{T(t,\alpha)}\,q(t)y^{n}.$

Since we do not have any information at the right end of the interval, we
cannot proceed as in the usual finite differences, where usually a tridiagonal
matrix is constructed.

So, the strategy will be to transform the process to an iterative method based
on the idea of the fixed point theory, then use the two-point finite
difference formula to find the first value on the right using the initial
condition as the initial value, then the three-point difference formula to
generate the next three values and achieve a better order of convergence, and
then finally move to the five-point difference formula with an order of
convergence $\mathcal{O}(h^{4})$ to finish the iteration.

It is worth mentioning that the method can be easily extended to other types
of non-linearities, can also be divided into three independent methods that
work with different degrees of convergence. By switching from forward to
backward difference, you can decide whether you want to solve a nonlinear
equation or not, although you have to choose between speed and convergence.

First, we select an integer $N>0$ and divide the interval $[a,b]$ into $(N+1)$
equal subintervals whose endpoints are the mesh points $x_{i}=a+ih$, for
$i=0,1,\ldots,N+1$, where $h=(b-a)/(N+1)$.

Since the only information we do have is the $y_{0}=y(a)=\beta$ value given by
the initial condition, we will start by using the two point forward difference
formula to determine $y_{1}$, with truncation error $\mathcal{O}(h)$

$y^{\prime}\left(t_{i}\right)=\frac{y\left(t_{i}+h\right)-y\left(t_{i}\right)}{h}-\frac{h}{2}y^{\prime\prime}(\xi_{0}),\,\,(t_{i}<\xi_{0}<t_{i}+h).$

By making $y\left(t_{i+1}\right)=y\left(t_{i}+h\right)$ the equation (21) can
be written as

$\frac{1}{h}\left[y\left(t_{i+1}\right)-y\left(t_{i}\right)\right]+\frac{p(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})=\frac{q(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n},$

or,

$y\left(t_{i+1}\right)=\frac{-hp(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})+\frac{hq(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n}+y\left(t_{i}\right).$
(29)

Starting at $i=0$, the equation (29) defines an iterative process that
converges to the solution, this is the well known Euler method.

We can also use the finite difference backward formula (replacing $h$ by
$-h$), then the iterative process is defined as follows

$y\left(t_{i}\right)-y\left(t_{i-1}\right)+\frac{hp(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})-\frac{hq(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n}=0.$
(30)

Where we must solve a nonlinear equation at each step of the iterative process
to determine $y\left(t_{i}\right)$, the $fsolve$ command in Matlab R2022B can
be used for this purpose, taking as an initial approximation the initial
condition to start the process, that can then be updated with the previous
point.

The equation (30) improves the convergence with respect to the equation (29)
in exchange for consuming more time. So we must decide between a better
convergence or a shorter execution time.

We have already found a solution to our problem, but recall that our goal was
to use one of the above equations to find $y_{1}$. Using the three-point
midpoint difference formula, with truncation error $\mathcal{O}(h^{2})$, we
can find $y_{2}$.

$y^{\prime}\left(t_{i}\right)=\frac{1}{2h}\left[y\left(t_{i}+h\right)-y\left(t_{i}-h\right)\right]-\frac{h^{2}}{6}y^{(3)}\left(\xi_{1}\right),\,\,(t_{i}-h<\xi_{1}<t_{i}+h),$

and since we know $y_{0}$ and $y_{1}$ and taking $i=1$ to start, the equation
$\eqref{eq:FB}$ becomes,

$\frac{1}{2h}\left[y\left(t_{i+1}\right)-y\left(t_{i-1}\right)\right]+\frac{p(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})=\frac{q(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n},$

or,

$y\left(t_{i+1}\right)=\frac{-2hp(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})+\frac{2hq(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n}+y\left(t_{i-1}\right).$
(31)

With (31) we again obtain an iterative process that converges to the solution,
with better results than previously found. Now let’s try with the three-point
backward difference formula

$y^{\prime}\left(t_{i}\right)=\frac{1}{2h}\left[3y\left(t_{i}\right)-4y\left(t_{i-1}\right)+y\left({t_{i-2}}\right)\right]-\frac{h^{2}}{3}y^{(3)}\left(\xi_{1}\right),\,\,(t_{i}<\xi_{1}<t_{i}+2h),$

the process is determined as follows

$3y\left(t_{i}\right)-4y\left(t_{i-1}\right)+y\left(t_{i-2}\right)+\frac{hp(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})-\frac{hq(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n}=0.$
(32)

Starting at $i=2$ we must again solve for $y\left(t_{i}\right)$ at each
iteration and the same effects as above will be obtained.

Finally, let us use the initial condition $y_{0}$, take $y_{1}$ from the first
iteration of (29) or (30) and $y_{3}$ and $y_{4}$ from (31) or (32) to develop
a last method from the backward five-point finite difference formula with
truncation error $\mathcal{O}(h^{4})$. It is not recommended to use a higher
order approximation scheme due to Runge’s phenomenon, instead other techniques
including for example irregular grids, Chebyshev polynomials or spectral
methods can be used.

The backward five-point difference formula is determined by the formula

$\displaystyle y^{\prime}\left(t_{i}\right)$
$\displaystyle=\frac{1}{12h}\left[25y\left(t_{i}\right)-48y\left(t_{i}-h\right)+36y\left(t_{i}-2h\right)-16y\left(t_{i}-3h\right)+3y\left(t_{i}-4h\right)\right]$
$\displaystyle-\frac{h^{4}}{5}y^{(3)}\left(\xi_{2}\right)\,\,\,(t_{i}<\xi_{2}<t_{i}+4h),$

we obtain then,

$\displaystyle 25y\left(t_{i}\right)$
$\displaystyle-48y\left(t_{i-1}\right)+36y\left(t_{i-2}\right)-16y\left(t_{i-3}\right)+3y\left(t_{i-4}\right)+\frac{12hp(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})$
(33) $\displaystyle-\frac{12hq(t_{i})}{T(t_{i},\alpha)}\,y(t_{i})^{n}=0.$

Which again must be solved using some method for nonlinear equations.

Let’s solve again equation (24) given in example 1 using the finite difference
method (FDM), Table 1 and Figure 3 show the solution and error between the
finite difference method and the exact solution of equation (24) for
$\alpha=0.5$, we have zoomed in on the Figure 3 to show the difference between
the curves.

Figure 3: Comparison between the FDM and the exact solution for (24). Pts | Exact Sol | FDM | Error
---|---|---|---
0.5 | -1 | -1 | 0
0.500299940011998 | -0.999152308016293 | -0.999152149990093 | 1.58026200125505e-07
0.500599880023995 | -0.998305946878573 | -0.998305948338594 | 1.46002121503841e-09
…… | …… | …… | ……
1.99970005998800 | -0.138412976596618 | -0.138413036296161 | 5.96995426982439e-08
2.00000000000000 | -0.138379560668148 | -0.138379620351763 | 5.96836149668878e-08
Table 1: Comparison between the FDM and the exact solution for (24) with
$N=5000$ interval divisions.

The above comparison shows that the proposed finite difference method is
indeed feasible and that it delivers a solution quite close to the exact
solution with considerably small error and low run time. In the following
example we will show that the method can also be applied to more complicated
problems, where it is sometimes difficult to obtain an exact solution.

There are cases in which the functions $p(t)$ and $q(t)$ make it difficult the
use of the Theorem (22) because complicated integrals must be solved, herein
lies the usefulness of numerical methods. If we choose the functions
$p(t)=e^{-t^{2}}$, $q(t)=t$, $I=[-3,3]$ and $n=2$, we get the equation

$G_{T}^{\alpha}y+e^{-t^{2}}y=ty^{2}.$ (34)

Taking $T(t,\alpha)=e^{(1-\alpha)t}$ and $C=1$, we can find the solution to
the previous equation through the FDM, and since we don’t have an exact
solution for comparison we may approximate the integrals in Theorem (22) by a
numerical integration method. This will allow us to validate again the
proposed numerical method, as we do not have an explicit solution of the
fractional equation. Table 2 shows both solutions for $\alpha=0.5$,
graphically, the difference between the two solution curves is hardly
noticeable. Figure 4 displays the solutions associated for differents values
of $\alpha$ for the FDM.

Pts | Integ | FDM | abs(FDM-Integ)
---|---|---|---
-3 | 1 | 1 | 0
-2.99880023995201 | 0.984132376988178 | 0.983900701320463 | 0.000231675667714537
-2.99760047990402 | 0.968775560311577 | 0.968798608504813 | 2.30481932365079e-05
…… | …… | …… | ……
2.99880023995201 | 0.0115194694289867 | 0.0115187854523517 | 6.83976635032849e-07
3.00000000000000 | 0.0115195756295232 | 0.0115188916402543 | 6.83989268873003e-07
Table 2: Comparison between FDM and aproximation of the integrals in the exact
solution. Figure 4: Solution of Equation (34) with
$T(t,\alpha)=e^{(1-\alpha)t}$ for $\alpha=0.1,0.3,0.5,0.7,0.9,1$.

In this example we can see that the solutions approximate in ascending order
the case $\alpha=1$ with the kernel $T(t,\alpha)=e^{(1-\alpha)t}$, that
corresponds to a conformable derivative.

We have been able to compare two different approaches, one in which the
derivative is approximated by an appropriate difference quotient and another
in which the integral has been approximated within the explicit solution. The
approximation of the integral has been performed by numerical methods using
Matlab R2022B, although it has proved to be an alternative to FDM, it has
required much more time to deliver the solution, showing an increase of time
as the complexity of the integrals and the number of nodes increases.

The examples shown above prove that the numerical proposed method (FDM) is a
valid alternative when complicated integrals must be solved in the exact
solution, which broadens the number of problems we can solve.

##  3 Conclusions

In this article we proposed and solved a generalization of the Bernoulli
differential equation under the generalized derivative approach, we found
solubility conditions and results about the qualitative behavior of the
trivial solution. For this, a generalization of Gronwell’s inequality was
proved as well as its reciprocal and a particular case of this inequality.
After that, we shown by means of examples how this fractional derivative
approach has some advantages over some other definitions, for example, this
derivative generalizes certain definitions of fractional derivative known in
the literature and further, it allows us to choose the kernel of the
derivative depending on the problem under consideration, so, we can solve
different problems under different derivatives approaches by choosing a
suitable kernel function $T(t,\alpha)$. We also proposed and tested the
reliability of a finite difference method by means of examples, for the case
in which the explicit solution involves complex integrals, comparing the
solution obtained by FDM with the approximation of the integrals in the
explicit solution of the fractional equation, thereby extending the number of
problems we can solve.

## Availability of data and material

None

## Competing interests

The authors declare no competing interests.

## Funding

None

## Authors’ contributions

All the authors contributed equally to the work.

## Acknowledgements

None

## References

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