A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION

. In this paper we study a generalized form of the Bernoulli Differential Equation, employing a generalized conformable derivative. Initially, we establish a generalized variant of Gronwall’s inequality, essential for assessing the stability of generalized differential equation systems, and offer insights into the qualitative behavior of the trivial solution of the proposed equation. Sub-sequently, we present and prove the main results concerning the solution of the generalized Bernoulli Differential Equation, complemented by illustrative examples that underscore the advantages of this generalized derivative approach. Furthermore, we introduce a finite difference method as an alternative technique to approximate the solution of the generalized Bernoulli equation and demonstrate its validity through practical examples.


Introduction
Fractional calculus, a branch of mathematics exploring differentiation and integration operators of generalized orders, emerged nearly simultaneously with traditional calculus.While its inception paralleled that of classical calculus, fractional calculus has proven its versatility across a myriad of applications.Notable works such as [21,4,12,5,33,22] serve as testament to its widespread utility.However, tackling systems of fractional differential equations has presented unique hurdles, prompting the exploration of diverse methodologies, as evidenced by studies such as [13,16,35].
The global fractional derivatives, which 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.In [28], a conformable fractional derivative is defined, offering advantages in its own right.More recently, a non-conformable local fractional derivative was introduced in [18].The conformable fractional derivative, serving as a local operator, sets itself apart from non-local counterparts like Caputo or Riemann-Liouville.Viewing fractional local derivatives as a new perspective, they have demonstrated utility in various applications by several authors, as evidenced in [10,14,18,21,17,24,29].
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.This paper relies on the use of new differential operators, which depend on a general kernel function T (t, α).These operators encompass several local derivatives that have been introduced and studied in various sources.This new tool is powerful because it allows us to model a phenomenon from two perspectives: by considering different kernels and by varying the order associated with each kernel as shown in several studies [20,30,32,34].

Submitted
It is known that one of the most paradigmatic nonlinear equations is the Bernoulli differential equation, which was introduced by Jacob in his work [6].This equation can be viewed as a manifestation of the principle of conservation of energy in fluids.In this paper, we will investigate Bernoulli's equation using the generalized conformable derivative operator.To begin, we establish a generalized form of Gronwall's inequality and derive stability conditions for systems of generalized differential equations utilizing the generalized conformable derivative approach.Subsequently, we determine solubility and stability conditions for the proposed generalized Bernoulli's equation.Finally, we provide examples illustrating particular cases of this Bernoulli's equation viewed from different perspectives.
There are cases in which the generalized 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.Finite difference methods replace the derivative operator with an appropriate quotient in differences, which allows us to approximate the solution quite efficiently.Here, we use a finite difference method to estimated the solution of the generalized Bernoulli's equation and showcase the results obtained through examples.

Preliminaries
In this section we present a definition of a generalized conformable derivative introduced in [15] as well as a fractional integration operator shown in [14] and some of its most important properties that will be useful in the next section.
In [15] the definition is given as follows.
Given s ∈ R, we denote by ⌈s⌉ the upper integer part of s, i.e., the smallest integer greater than or equal to s.
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.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, α) 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 derivatives are shown, as well as their relations with other types of local derivatives(conformable or not), as shown in [36]:
(4) If α ∈ (0, 1] and , then we obtain the beta-derivative defined in [3]. If we choose the function T (t, α) = t ⌈α⌉−α , then we obtain the following case of the function G α T which is a conformable derivative.Definition 2.2.Let I be an interval I ⊆ (0, ∞), f : I → R and α ∈ R + .The conformable derivative G α f of f of order α at the point t ∈ I is defined by Note that, if α = n ∈ N and f is smooth enough, then Definition 2.2 coincides with the classical definition of the n-th derivative.
The following results in [15] contain some basic properties of the derivative G α T .Theorem 2.3.Let I be an interval I ⊆ R, f : I → R and α ∈ R + .
(1) If there exists T -differentiable functions at t ∈ I. Then the following statements hold: T -differentiable at t, and . In [14] is defined an integral operator in the following way.Let I be an interval I ⊆ R, a, t ∈ I and α ∈ R. The integral operator J α T,a is defined for every locally integrable function f on I as The following results appear in [14].
Proposition 2.6.Let I be an interval I ⊆ R, a ∈ I, 0 < α ≤ 1 and f a differentiable function on I such that f ′ is a locally integrable function on I.Then, we have for all t ∈ I J α T,a G α T (f ) (t) = f (t) − f (a).Proposition 2.7.Let I be an interval I ⊆ R, a ∈ I and α ∈ (0, 1].
, for every continuous function f on I and a, t ∈ I.
In [28] it is defined the integral operator J α T,a with T given by T (t, α) = t 1−α , and [28,Theorem 3.1] shows , for every continuous function f on I, a, t ∈ I and α ∈ (0, 1].Hence, Proposition 2.7 extends to any T this important equality.
The following result summarizes some elementary properties of the integral operator J α T,a .Theorem 2.8.Let I be an interval I ⊆ R, a, b ∈ I and α ∈ R. Suppose that f, g are locally integrable functions on I, and k 1 , k 2 ∈ R. Then we have Remark 2.9.The above results generalize Proposition 1, Proposition 2 and Theorem 1 of [19], respectively, obtained with 0 < α ≤ 1.
The following propositions are presented in order to serve as a basis for future research. Given For theoretical completeness we show the following result.Proof.Let us denote by χ A the characteristic function of the set A (i.e., χ and so, a continuous and compact operator.□

Generalized Gronwall's inequality
Next, we prove a generalized version of Gronwall's inequality which will be useful in the study of the stability of systems of generalized differential equations.A version of this inequality was proved in [1].
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for all t ∈ [a, b], and Proposition 2.7 gives . Theorem 2.5 and Proposition 2.7 give 3, we have that Theorem 2.8 and Proposition 2.6 give The argument in the proof of Theorem 3.1 also gives the following converse inequality.
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on some open neighborhood of the point (t 0 , x 0 ), and consider the initial value problem Then there exists h > 0 such that (3.3) has a unique solution on the interval 3) has a unique solution on I.
Lemma 3.4 guarantees that (3.4) has a unique solution on [a, ∞).The study of boundedness of solutions of a differential equation, either generalized 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 generalized differential equations.
Definition 3.5.If F (t, 0) = 0 for every t ≥ a, then the trivial solution x ≡ 0 of (3.4) is said to be stable if for any ε > 0, there exists , and there exists a continuous function k such that Then the trivial solution x ≡ 0 of (3.4) is uniformly stable.

The generalized Bernoulli differential equation
The Bernoulli Differential Equation stands out as one of the most crucial nonlinear equations, playing a pivotal role in solving intricate mathematical problems.Its practical significance is evident in its ability to describe and analyze phenomena across various fields such as Physics, Biology, and Engineering.Specifically, it enables the examination of flow behaviors in systems where pressure, velocity, and height play crucial roles.Moreover, it serves as a cornerstone for studying system dynamics, enzyme kinetics, and population dynamics [8,9,37].
The Bernoulli Differential Equation has garnered recent attention within the realm of Global fractional operators, as evidenced by studies such as [11].This section builds upon prior research by examining the Bernoulli Differential Equation through the lens of local operators, including both Conformable and Non-Conformable approaches.Furthermore, this research raises the prospect of comparing the effectiveness of both global and local methodologies in practical applications, as illustrated, for instance, in the study [23].
In this section we introduce the generalized Bernoulli differential equation, defined as: We will start by presenting the basic results of the Stability Theory for this equation.
First, we will pose the problem for a much more general system than 4.1: where f ∈ C(R + × R, R), t 0 > 0. It is further assumed that for (t 0 , y 0 ) ∈ int(R + × R) the initial value problem (4.2)-( 4.3) has a solution y(t) ∈ C α (I) for all t > t 0 > 0. In addition, it is assumed that f (t, 0) = 0 for all t > t 0 > 0.
Several types of stability can be discussed for solutions of differential equations (of integer or fractional order) describing dynamical systems.The most important type concerns the stability of solutions near an equilibrium point.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 t then y e it's Lyapunov stable.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 concept of exponential stability, applicable to linear equations or systems, ensures a minimum rate of decay.This implies an assessment of how rapidly the solutions converge.
The stability of a solution of generalized 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 (4.2).Firstly, we will state the following lemma, which will be a basis for obtaining the desired stability results.
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Submitted
and show that the trivial solution y(t) ≡ 0 of (4.5) is stable if and only if then, by means of the properties of the integral operator, we can see that the general solution of (4.5) is y(t) = y 0 e −J α T ,t 0 p(t) , (4.7) from which the statement (4.6) is easily obtained.
We are now in a position to establish the main results on the stability of equations of type (4.2). then the trivial solution y(t) ≡ 0 is an asymptotically stable solution.
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.Submitted: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION 11 The following result shows how the integral operator J α T,a can be applied in order to solve generalized differential equations.Theorem 4.4.Let p and q be continuous functions on an interval I ⊆ R, a ∈ I, α ∈ (0, 1] and n ∈ R \ {1}, and consider the generalized Bernoulli differential equation Then the following statements hold. (1) For each C ∈ R, the function is a solution of the Bernoulli generalized equation (4.10).
(2) For each t 0 ∈ I and y 0 ∈ R, let y(t; t 0 , y 0 ) be the function in (4.11) with = y 0 and y(t; t 0 , y 0 ) is defined in some right or left neighborhood U ⊆ I of t 0 , then y(t; t 0 , y 0 ) is a solution of (4.10) on U satisfying the initial condition y(t 0 ; t 0 , y 0 ) = y 0 .
(3) For each t 0 ∈ I and y 0 > 0 there exists a unique solution of (4.10) satisfying the initial value y(t 0 ) = y 0 , given by y(t; t 0 , y 0 ).
Proof.For each C ∈ R, let us define the function (4.12) Therefore, y = z 1/(1−n) .Theorems 2.4 and 2.5 and Proposition 2.7 give and y is a solution of the Bernoulli generalized equation (4.10).
If (y 1−n 0 ) 1/(1−n) = y 0 and y(t; t 0 , y 0 ) is defined in some right or left neighborhood U ⊆ I of t 0 , then it is clear that y(t; t 0 , y 0 ) is a solution of (4.10) on U satisfying the initial condition y(t 0 ; t 0 , y 0 ) = y 0 .
By Theorem 2.3, the generalized differential equation (4.10) is equivalent to 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 (4.10) satisfying the initial value y(t 0 ) = y 0 .Since y 0 > 0, we have (y 1−n 0 ) 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 ∈ (1, ∞) ∩ Q and 1/(1 − n) = r/s with s an odd integer.For each y 0 ∈ 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 (4.10) with y(t 0 ) = y 0 .Since 1/(1 − n) = r/s with s an odd integer, we have (y 1−n 0 ) 1/(1−n) = y 0 for every y 0 ∈ R, and the second item of this theorem gives that this solution is y(t; t 0 , y 0 ).□ Remark 4.5.The results obtained, relative to the Bernoulli Equation, generalize and complete those obtained in [31], obtained using the conformable derivative of [28].

4.1.
Examples.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 (4.10) we obtain,  As we can see, the solutions approach the curve associated with the value of α = 1, which corresponds to the case of the Bernoulli equation with derivative of order one.This is due to the fact the kernel T (t, α) = t 1−α is associated with a generalized conformable 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) = t, the interval I = [−1, 1] and n = 2, replacing this values in Equation (4.10) we obtain then, if we choose the kernel T (t, α) = t −α and C = −1, by Theorem (4.4) we obtain, The function, is the solution associated to the Bernoulli differential equation with the classical derivative.We wanted to show both solutions to demonstrate how, in this example, the solutions of the generalized equation are approaching the solution associated with α = 0.But they differ from the solution of the equation using the classical derivative, because a non-conformable kernel has been used.In Figure (2) the Equation (4.17This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560. Submitted: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 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, it generalizes the definitions of the operators mentioned at the beginning of this paper.

A finite difference method for the generalized Bernoulli differential equation
Usually numerical methods are used to solve differential equations for which there is no exact solution.In other cases, even with an exact solution, difficult situations arise, such as the solution of an integral that cannot be solved in terms of elementary functions.That is why, the objective of the numerical method presented below is to find an approximate solution to those cases in which, even with an exact solution, it is not possible to visualize the solution.The second example, shown by the equation (5.7) is one of those cases.
The finite difference method is often used to solve differential equations of various orders with boundary conditions.In general, first the differential operators are replaced by appropriate quotients in differences and a matrix is constructed, which is usually tridiagonal.Then the boundary conditions that form the first and last row of the matrix are added.Finally the system of equations is solved by an appropriate method that minimizes the approximation error.We have intended to follow this line of thought to develop our method.In this case, since we do not have any information at the right end of the interval, we cannot proceed as in the usual finite differences.
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 This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION 15 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 O(h 4 ) to finish the iteration.

Submitted
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.
The equation, according to Theorem 2.3 is equivalent to First, we select an integer N > 0 and divide the interval [a, b] into (N + 1) equal subintervals whose endpoints are the mesh points t i = a+ih, for i = 0, 1, . . ., N +1, where h = (b − a)/(N + 1) and y i = y(t i ).
Since the only information we do have is the y 0 = y(a) = β value given by the initial condition, we will start by using the two point forward difference formula to determine y 1 , with truncation error O(h) By making y (t i+1 ) = y (t i + h) the equation (4.10) can be written as or, Starting at i = 0, the equation (5.2) 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 Where we must solve a nonlinear equation at each step of the iterative process to determine y (t i ), the f solve 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 (5.3) improves the convergence with respect to the equation (5.2) in exchange for consuming more time.So we must decide between a better convergence or a shorter execution time.
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.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 O(h 2 ), we can find y 2 .

Submitted
and since we know y 0 and y 1 and taking i = 1 to start, the equation (5.1) becomes, or, With (5.4) 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 the process is determined as follows 3y (t i ) − 4y (t i−1 ) + y Starting at i = 2 we must again solve for y (t i ) 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 (5.2) or (5.3) and y 3 and y 4 from (5.4) or (5.5) to develop a last method from the backward five-point finite difference formula with truncation error 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 (5.6) Which again must be solved using some method for nonlinear equations.5.1.Some numerical examples.Let's solve again equation (4.13) given in example 1 using this variation of 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 (4.13) for α = 0.5, we have zoomed in on the Figure 3 to show the difference between the curves.1.Comparison between the FDM and the exact solution of (4.13) 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 (4.4) because complicated integrals must be solved, herein lies the This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.(5.7) Taking T (t, α) = e (1−α)t and C = 1, we can find the solution to the previous equation through the FDM.Since we don't have an exact solution for comparison, we may approximate the integrals in Theorem (4.4) 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 generalized equation.Table 2 shows both solutions for α = 0.5, graphically, the difference between the two solution curves is hardly noticeable.This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.Submitted: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION 19 In this example we can see that the solutions approximate in ascending order the case α = 1 with the kernel T (t, α) = e (1−α)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.

Conclusions
In this article we proposed and solved a generalization of the Bernoulli differential equation under the generalized conformable derivative approach.We also found solubility conditions and results about the qualitative behavior of the trivial solution.To this end, a generalization of Gronwall's inequality was proved as well as its reciprocal and a particular case of this inequality.Thereafter, we shown by means of examples how this generalized 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.Allowing us to choose the kernel of the derivative depending on the problem under consideration, with the goal of solve different problems under different derivatives approaches.We also proposed and tested the reliability of a finite difference method by means of examples.We exposed the case in which the explicit solution involves complex integrals.Afterwards we compare the solution obtained by (FDM) with the approximation of the integrals in the explicit solution of the generalized equation.
: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION 3 a, b ∈ I (b > a), let us denote by F a,b the usual inner product in L 2 [a, b] F a,b (f, g) = b a f (t)g(t) dt.Proposition 2.10.[7]Let I be an interval I ⊆ R, a, b ∈ I with a < b and α ∈ R. The adjoint of J α T,a in L 2 [a, b] with respect to the inner product F a,b is the operator

Proposition 2 . 11 .
Let I be an interval I ⊆ R, a, b ∈ I with a < b and α ∈ R. Then J α T,a is a Hilbert-Schmidt integral operator on L 2 [a, b],and so, a continuous and compact operator.

Theorem 4 . 2 .
Consider the differential equationG αT y + p(t)y = g(t, y), t > 0, (4.8)where p(t) is a positive and continuous function such that 0 < p ≤ p(t), and g is a continuous function with g(t, 0) ≡ 0 for all t.If lim

. 13 ) 1 . ( 4 . 14 )
And consider the function T (t, α) = t 1−α and C = −1, then by Theorem (4.4) we obtain the following solution,y(t) = 1 − 2e t α −(0.5) α α −This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.Submitted: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 A GENERALIZED BERNOULLI DIFFERENTIAL EQUATION 13 The solutions associated for different values of α are shown in Figure (1).
) and solutions associated to(4.15)  for different values of α are shown.

Figure 3 .
Figure 3.Comparison between the FDM and the exact solution of (4.13).

Figure 4 Table 2 .
displays the solutions associated for differents values of α for the FDM.Comparison between FDM and aproximation of the integrals in the exact solution.
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This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.
This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.
Submitted: December 2, 2023 Accepted: April 3, 2024 Published (early view): August 27, 2024 This peer-reviewed unedited article has been accepted for publication.The final copyedited version may differ in some details.Volume, issue, and page numbers will be assigned at a later stage.Cite using this DOI, which will not change in the final version: https://doi.org/10.33044/revuma.4560.