Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition

1.   Introduction

Similar to how fractional exponents evolve from integer exponents, classical calculus concepts like integral and derivative operators serve as the foundation for fractional calculus ^[1,2]. Many people are aware that depending on the geometrical and physical factors, integer-order derivatives and integrals have different meanings. This assumption, however, is disproved when dealing with fractional-order integration and differentiation, which covers a constantly expanding domain in both theory and practical applications to real-life challenges ^[3]. The study of fluid flow, rheology, diffusive transport, electrical networks, electromagnetic theory, probability, and research on viscoelastic materials are just a few of the scientific and engineering domains where it has lately been employed^[4,5,6]. Fractional differential equations (FDEs) have drawn attention from several researches as a result of its frequent occurrence in disciplines such as physics, chemistry, and engineering. The commonly used Laplace transform approach, the iterative method, the Fourier transform technique, and the operational method are just a few of the strategies developed to deal with FDEs ^[7,8,9]. The majority of these techniques, however, are only relevant to a few types of FDEs, particularly those that are linear and have constant coefficients. The existence of solutions for fractional semilinear differential or integrodifferential equations is one of the theoretical fields being investigated by many authors. There has been a significant development in nonlocal problems for FDEs or inclusions ^[10,11]. Reimann-Liouvile fractional derivative-based linear FDEs with variable coefficients have been solved using the decomposition approach. FDEs have recently received a lot of attention from academics. This is because FDEs are frequently used in engineering and science, including in the study of diffusion in porous media, nonlinear earth oscillation, fractional biological neurons, traffic flow, polymer rheology, modeling of neural networks, and viscoelastic panels in supersonic gas flow^{[12,13,14,15,16]}.

In the real world, there may be situations that cannot be fully captured by either wholly continuous or entirely discrete phenomena. In these cases, we require a shared domain to adequately support both conditions. In order to unify continuous and discrete calculus, Stefan Hilger created a usual state known as time scale T^[17,18,19]. This domain is based on the unification of these requirements^{[20,21,22,23]}. To solve this type of model, which combines differential and difference equations, dynamic equations on a time scale were developed ^[24,25]. Many scholars worked on dynamic equations that involve local beginning and boundary conditions and might be either linear or nonlinear. Because fractional calculus is accurate and has an advantage in the physical interpretation, several authors have studied the dynamic equation using this method^[26,27,28].

We have seen a number of equations in the real world where the systems are permitted to experience a brief disturbance, the length of which may be insignificant compared to the overall process duration. In this situation, jump discontinuities may appear in the solution of these equations at time $\varsigma_{1} < \varsigma_{2} < \varsigma_{3} < \cdots$, given in the form

Impulsive dynamic equations are dynamic equations with jump discontinuities as solutions ^[29,30,31]. Dynamic impulsive equations on time scales have caught the attention of many academics recently. However, there are very few publications that investigate impulsive dynamic equations using fractional calculus on time scales with nonlocal initial conditions ^[32,33].

Neutral differential equations ^[34,35] appear when max$\{n_1, n_2, \cdots, n_k\} = n$. The past and present values of the function are determined by Neutral differential equations, which differs from retarded differential equations in that they depend on derivatives with delays. Elastic networks are simulated by neutral type differential equations in high speed computers for the express purpose of joining switching circuits^[36]. Due to their extensive use in practical mathematics, neutral differential equations have recently attracted a lot of attention^[37,38]. Many scientists have sought to create neutral differential systems, taking note of varied fixed point strategies, mild solutions, and nonlocal situations. Also, in recent years, neural networks have been extensively studied and have been applied in many fields, such as combinatorial optimization, multiagent systems, fault diagnosis, and industrial automation. However, the practical applications of neural networks have been limited due to some inherent dynamic properties, such as the information latching phenomenon ^[39,40,41].

While the authors of ^[24,25] used the tools of the delta (Hilger) derivative to investigate the fractional dynamic equation with local initial condition and instantaneous and non-instantaneous impulses, authors of ^[42] investigated the nonlocal initial condition's impulsive dynamic equation. In ^[43], the authors have discussed the impulsive fractional dynamic equation on time scales with a nonlocal initial condition. The exploration and elucidation of results pertaining to a generalized neutral fractional impulsive dynamic equation over time scales, specifically incorporating nonlocal initial conditions, is the main contribution of the this study.

As a result of the work described above, we assert that it's important to investigate the impulsive neutral fractional dynamic equation with nonlocal initial condition of the type:

where ${\it{k}} \in \mathbb{N} \cup \{{\it{0}}\}$ and

for $\mathscr{T} \in {\mathfrak{T}}$. Let the left dense(ld) continuous function be

and ${}^{C}{\it{D}}^{{\mathfrak{w}}}$ is Caputo nabla derivative (C$\nabla$D). We assume that

which indicates the impulse at specific time, and the terms

and

represent the function's right and left limits ${\it{u}}$ at $\varsigma = \varsigma_{{\it{k}}}$ in relation to time scales. $\mathcal{I}_{{\it{k}}}$ are continuous real valued functions on $\mathscr{R}$ $\forall$ ${\it{k}} = {\it{1}}, {\it{2}}, \cdots, m$, and $\mathcal{I}_{{\it{k}}}(\varsigma_{{\it{k}}}, {\it{u}}(\varsigma_{{\it{k}}}^{-}))$ are the action of impulses on the time scale interval $\mathscr{I}$.

2.   Preliminaries

Definition 2.1. ^[44] A function $\rho$: ${\mathfrak{T}}\rightarrow \mathscr{R}$ defined by

is said to be a backward jump operator. Any $\varsigma \in {\mathfrak{T}}$ is said to be ld if $\rho(\varsigma) = \varsigma$ and if $\rho(\varsigma) = \varsigma -1$, then $\varsigma$ is said to be a left scattered point on ${\mathfrak{T}}$.

Remark 2.2. If ${\mathfrak{T}}$ is a minimum right scattered point ${\it{y}}$, then set ${\mathfrak{T}}_{\upsilon} = {\mathfrak{T}} \backslash\{{{\it{y}}}\}$, otherwise ${\mathfrak{T}}_{\upsilon} = {\mathfrak{T}}$.

Definition 2.3. ^[42] A function

is said to be an ld continuous function, if ${\it{x}}(\cdot, {\it{p}}, {\it{q}})$ is ld continuous on ${\mathfrak{T}}$ for each ordered pair $(\varsigma, \theta) \in \mathscr{R} \times \mathscr{R}$ and ${\it{x}}(\varsigma, \cdot, \cdot)$ is continuous on $\mathscr{R} \times \mathscr{R}$ for fixed point $\varsigma \in {\mathfrak{T}}$.

Proposition 2.4. ^[17] Assume ${\it{g}}$ is an increasing continuous function on $[0, \mathscr{T}] \cap {\mathfrak{T}}$. If $\mathbb{G}$ is an addition to ${\it{g}}$ in $[0, \mathscr{T}]$, $\mathscr{T} \in \mathscr{R}$, one can obtain

then,

for ${\it{s}}, {\it{t}} \in [0, \mathscr{T}] \cap {\mathfrak{T}}$, such that ${\it{s}} < {\it{t}}$.

Definition 2.5. (^[44], Higher order nabla derivative) Consider an ld continuous function ${\mathfrak{H}}$: ${\mathfrak{T}}_{\upsilon} \rightarrow \mathscr{R}$ over ${\mathfrak{T}}$. Here, ${\mathfrak{H}}_{\nabla}$ is differentiable over ${\mathfrak{T}}_{\upsilon}^{(2)} = {\mathfrak{T}}_{\upsilon \upsilon}$ along

where ${\mathfrak{H}}_{\nabla \nabla} = {\mathfrak{H}}_{\nabla}^{(2)}$ is the second order nabla derivative. Also, proceeding upto ${\it{n}}^{th}$ order, one can get ${\mathfrak{H}}_{\nabla}^{({\it{n}})}$: ${\mathfrak{T}}_{\upsilon}^{({\it{n}})}\rightarrow \mathscr{R}$.

Definition 2.6. ^[44] Consider ld continuous function ${\mathfrak{H}}$: ${\mathfrak{T}}_{\upsilon}^{({\it{n}})}\rightarrow \mathscr{R}$, such that ${\mathfrak{H}}_{\nabla}^{({\it{n}})}(\varsigma)$ (${\it{n}}^{th}$ order of nabla derivative) appears, then the C$\nabla$D is

If ${\mathfrak{w}} \in (0, 1)$, we obtain

Definition 2.7. ^[44] On the set ${\mathfrak{T}}_{\upsilon}$, consider ${\mathfrak{H}}$ to be any ld continuous function, then the Riemann-Liouville nabla derivative (RL$\nabla$D) is

Definition 2.8. ^[17] Assume ${\mathfrak{H}}$: $\mathscr{I}_{\mathcal{J}}\rightarrow \mathscr{R}$, then the RL$\nabla$D fractional integral of ${\mathfrak{H}}$ is

The RL$\nabla$D integral always satisfies the condition

Lemma 2.9. ^[17] Assume the ld continuous function is ${\it{u}}(\varsigma)$, then

Theorem 2.10. ^[25] Assume ${\it{D}} \subset {\mathfrak{C}}({\mathfrak{T}}, \mathscr{R})$. Let ${\it{D}}$ be bounded and equicontinuous simultaneously, then it is relatively compact.

Theorem 2.11. ^[42] A function ${\mathfrak{H}}(\mathcal{B})$ is relatively compact in $\mathcal{A}$ for ${\mathfrak{H}}$: $\mathcal{A}\rightarrow \mathcal{B}$, which is completely continuous.

Theorem 2.12. (^[24], Nonlinear alternatives Leray-Schauder's type) Let $\mathcal{C} \subset \mathcal{X}$ be closed and convex and $\mathcal{X}$ be as Banach space. Let ${\it{G}}$: $\mathcal{U}\rightarrow \mathcal{C}$ be a compact map and $\mathcal{U}$ be a relatively open subset of $\mathcal{C}$ with $0 \in \mathcal{U}$, then

(i) ${\it{G}}$ has a fixed point in $\mathcal{U}$; or

(ii) there is a point ${\mathfrak{u}} \in \delta {\it{U}}$ and $\gamma \in (0, 1)$ with ${\it{u}} = \gamma {\it{G}}({\mathfrak{u}})$.

Theorem 2.13. ^[42] For ${\mathfrak{w}} \in (0, 1)$, ${\it{u}}$ is a solution for ${\mathfrak{L}}$: $\mathscr{I}_{\mathcal{J}} \times \mathscr{R}\times \mathscr{R}\rightarrow \mathscr{R}$, then

if ${\it{u}}$ is the solution of equation

Definition 2.14. ^[45] Let $\mathscr{X}$ be a Banach space and

where $\mathscr{L}(\mathscr{X}),$ a family of linear, bounded operators $\mathscr{L}(\mathscr{X})$: $\mathscr{X} \rightarrow \mathscr{X}$ for all ${\it{t}} \geq 0$. A is called a semigroup if, and only if,

3.   Analysis between RL$\nabla$D and C$\nabla$D

Proposition 3.1. Let ${\it{m}} - {\it{1}} < {\mathfrak{w}} < {\it{m}}, {\it{m}} \in \mathbb{N}$ for any ${\mathfrak{w}} \in \mathscr{R}$, such that ${}^{C} {\it{D}}_{\varsigma_{o}}^{{\mathfrak{w}}}\mathbb{G}(\varsigma)$ exists over time scale ${\mathfrak{T}}$, then

Proof. The proof is evident from Theorems 2.12 and 2.13. □

Theorem 3.2. For ${\it{m}} = [{\mathfrak{w}} + {\it{1}}]$ and for any $\varsigma \in {\mathfrak{T}}_{{\it{v}}^{{\it{n}}}}$, the C$\nabla$D and RL$\nabla$D satisfies:

for a fixed point $\alpha \in {\mathfrak{T}}$. Taylor's theorem defined in ^[27] proves this theorem.

Proof. Assume ld continuous function $\mathbb{G}$, then $\forall$ fixed $\alpha \in {\mathfrak{T}}$ and ${\it{m}} \in \mathbb{N}\cup\{0\}, {\it{m}} < {n}$. One can obtain

Taking the Riemann-Liouville derivative ${\it{D}}_{\alpha}^{{\mathfrak{w}}}$ in each side of Eq (3.1), Lemma 2.9 and Proposition 3.1 are used below:

From the above equation, we obtain

□

Proposition 3.3. If ${\mathfrak{w}} \in (0, 1),$ then ${\it{m}} = {\it{1}}$. Hence, from the Eq (3.3),

Case 1. If $\mathbb{G}(\alpha) \rightarrow 0$, as $\alpha \rightarrow 0$, then

Hence, C$\nabla$D and the Riemann-Liouville derivative coincide with each other.

Case 2. If ${\mathfrak{w}} \in \mathbb{N}$, by applying Eq (3.1) in Eq (3.2) and applying Lemma 2.9, one can get

$\therefore$ C$\nabla$D coincides with the nabla derivative.

4.   Existence and uniqueness of impulsive neutral fractional dynamic equation

A population dynamics model featuring a stop-start phenomenon can be used to compare the dynamic Eq (1.1) to that model. If we take into account a negative impact on that particular species, we can observe the population change, that the C$\nabla$D ${}^{C}{\it{D}}^{{\mathfrak{w}}}{\it{u}}(\varsigma)$ presents (at the initial stage of time), with respect to $\varsigma$ on

We investigate a scenario in specific times $\varsigma_{1}, \varsigma_{2}, \varsigma_{3}, \cdots$ such that

Impulse effects have an impact on people "momentarily," so there is a surge in the population ${\it{u}}(\varsigma)$, and ${\it{u}}(\varsigma_{{\it{k}}}^{+})$ and ${\it{u}}(\varsigma_{{\it{k}}}^{-})$ show the species population at the time $\varsigma_{{\it{k}}}$ before and after the impulsive effect.

Assume a collection of every ld continuous function ${\mathfrak{C}}(\mathscr{I}, \mathscr{R})$. Put $\mathscr{I}_o = [0, \varsigma_{1}]$ and $\mathscr{I}_{{\it{k}}} = [\varsigma_{{\it{k}}}, \varsigma_{{{\it{k}}+{{\it{1}}}}}]$ for each ${\it{k}} = {\it{1}}, {\it{2}}, \cdots, {\it{m}}$. Let

and

where ${\mathfrak{P}}{\mathfrak{C}}^{1}(\mathscr{I}, \mathscr{R})$ is collection of every function from $\mathscr{I}_{{\it{k}}}$ to $\mathscr{R}$, i.e., ld continuously $\nabla$ differentiable function.

The set ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$ is a Banach space

Definition 4.1. A function ${\it{u}} \in {\mathfrak{P}}{\mathfrak{C}}^{1}(\mathscr{I}, \mathscr{R})$ is a solution of the Eq (1.1), if ${\it{u}}$ satisfies the Eq (1.1) on $\mathscr{I}$ having

Lemma 4.2. Assume an ld continuous function $\mathscr{H}$: $\mathscr{I} \rightarrow \mathscr{R}$, such that solution (1.1) is

where the integral equation specifies

Proof. If $\varsigma \in \mathscr{I}_{o}$, then the solution of the Eq (4.1) is given by

For $\varsigma \in \mathscr{I}_{1}$, the problem

holds the solution

Again,

Applying Eq (4.5) in Eq (4.4), then

which follows that

Using the idea of mathematical induction and generalizing in this way for $\varsigma \in \mathscr{I}_{{\it{k}}}$, ${\it{k}} = {\it{1}}, {\it{2}}, \cdots, {\it{m}}$, one can say,

□

The following hypotheses are necessary in order to prove the existence and uniqueness of the solution to Eq (1.1):

$({\rm{A1}})$ ${\mathfrak{L}}$: $\mathscr{I}\times \mathscr{R}\times \mathscr{R}\rightarrow \mathscr{R}$ is ld continuous and there should be a constant $\mathcal{K} > 0$ and $0 < \mathscr{G} < 1$, which contents

$\theta_{i}, \zeta_{i} \in \mathscr{R}$ for ${\mathfrak{i}} = 1, 2.$

$({\rm{A2}})$ There exist constants $\mathcal{A} > 0, \mathcal{F} > 0$, and $0 < \mathcal{E} < 1$, such that

$({\rm{A3}})$ $\mathcal{I}_{{\it{k}}}(\varsigma, {\it{u}})$ is continuous $\forall$ ${\it{k}} = {\it{1}}, {\it{2}}, \cdots{\it{m}}$ and contents:

(ⅰ) There exists a "+" ve constant $\mathscr{M}_{{\it{k}}}$ for ${\it{k}} = {\it{1}}, {\it{2}}, \cdots, {\it{m}}$ such that

(ⅱ) There exists a "+" ve constatnt $\mathcal{L}_{{\it{k}}}$, for ${\it{k}} = {\it{1}}, {\it{2}}, {\it{3}}, \cdots, {\it{m}}$ such that

$({\rm{A4}})$ There must be a non "-" ve increasing function $\mu$: $\mathscr{R}^{+}\rightarrow \mathscr{R}^{+}$ such that

and a "+" ve constant $\mathscr{H}$ such that

$({\rm{A5}})$ For $\varsigma \in \mathscr{I}_{o}$ in a time scale interval, let the function ${\it{u}}(\varsigma)$ be

The Banach contraction theorem forms the basis of the following theorem.

Theorem 4.3. If all conditions (A1)–(A4) and

hold, then Eq (1.1) must contain a solution on $\mathscr{I}$.

Proof. Assume

Let $\varPi \subseteq {\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}_{{\it{k}}}, \mathscr{R})$ such that

and $\chi$: $\varPi \rightarrow \varPi$ such that

for $\varsigma \in \mathscr{I}_{o}$, and

for $\varsigma \in \mathscr{I}_{{\it{k}}}$, then ${\it{k}} = {\it{1}}, {\it{2}}, {\it{3}}, \cdots, {\it{m}}.$

Case 1. Let $\varsigma \in \mathscr{I}_{{\it{k}}}$, then ${\it{u}} \in \varPi$,

where ${\mathfrak{h}}\in \varPi$, $\varsigma \in \mathscr{I}$. By Eq (1.1), one can get

and

Again, taking the norm of ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$ in (4.6),

where

Using the condition of Case 1 and Proposition 2.4, we obtain

where

Case 2. If $\varsigma \in \mathscr{I}_{o}$, by a similar way, one can obtain

Thus, from (4.8), $||\chi_{{\it{u}}}||_{{\mathfrak{P}}{\mathfrak{C}}} \leq \omega$. Hence, $\chi(\varPi)$ is bounded. Also, for ${\it{u}}, {\it{v}} \in \varPi$,

where ${\mathfrak{i}} \in \varPi$, then ${\mathfrak{i}}(\varsigma) = {\mathfrak{L}}(\varsigma, {\it{v}}(\varsigma), {\mathfrak{i}}(\varsigma))$. For $\varsigma \in \mathscr{I}$, one can get

Taking the norm of ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$, (4.10) becomes

Using (4.11) in (4.9) and applying the Proposition 2.4,

Similarly, for $\varsigma \in \mathscr{I}_{o}$,

Thus, from (4.12) and (4.13), we obtain

where

Here, $\mathcal{U} < 1$, then $\chi$: $\varPi \rightarrow \varPi$ is a contraction operator. According to the Banach contraction theorem, it has a fixed point, which is the solution to Eq (1.1). □

Equation (1.1)'s adequate condition for a solution is based on the nonlinear alternative to Leray-Schauder's fixed point theorem.

Theorem 4.4. If (A1) through (A4) are true and there is a positive constant $\beta$, then

Therefore, there is at least one solution to Eq (1.1) in $\mathscr{I}$.

Proof. The following steps are used to demonstrate the theorem's proof.

Step 1. $\chi$: $\varPi \rightarrow \varPi$ is continuous.

Assume $\{{\it{u}}_{n}\}$ is a sequence of $\varPi$ such that ${\it{u}}_{n} \rightarrow {\it{u}}$, then $\varsigma \in \mathscr{I}_{{\it{k}}}, {\it{k}} = {\it{1}}, {\it{2}}, {\it{3}}, \cdots, {\it{m}}.$

where ${\mathfrak{h}}_{n} \in \varPi$, such that

and for $\varsigma \in \mathscr{I}_{{\it{k}}}$, we obtain

Taking the norm of ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$, (4.16) becomes

Using (4.17) in (4.15), we obtain

As $n\rightarrow \infty$, let ${\it{u}}_{n}\rightarrow {\it{u}}$ such that

As a result, $\chi$ is continuous.

Also, for $\varsigma \in \mathscr{I}_{o}$, the proof is similar.

Step 2. The operator $\chi$ map $\varPi$ to ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$.

Assume ${\it{x}}_{\it{1}}, {\it{x}}_{\it{2}} \in \mathscr{I}_{{\it{k}}}, {\it{k}} = {\it{1}}, {\it{2}}, \cdots, {\it{m}}$, such that ${\it{x}}_{\it{1}} < {\it{x}}_{\it{2}}$, then

Since $({\it{x}} -(\theta))^{{\mathfrak{w}}-{\it{1}}}$ is continuous, if ${{\it{x}}_{\it{1}}}\rightarrow {{\it{x}}_{\it{2}}}$, then

Thus, the operator $\chi$ is equicontinuous in $\mathscr{I}_{{\it{k}}}$. Since the result at ${\it{x}}_{\it{1}}, {\it{x}}_{\it{2}} \in \mathscr{I}_{o}$ is comparable, the evidence is left out.

Step 3. Let $\chi$ map $\varPi$ be a bounded set of ${\mathfrak{P}}{\mathfrak{C}}(\mathscr{I}, \mathscr{R})$.

From (4.7), it is clear that $||\chi({\it{p}})|| \leq \omega$ for $\omega \in \mathscr{R}$. As a consequence of Steps 1–3, using the Arzela-Ascoli theorem, one can discover that $\chi$ is entirely continuous.

Step 4. Let $\gamma \in (0, 1)$,

be bounded.

Also, by $\varsigma \in \mathscr{I}_{{\it{k}}}, {\it{k}} = {\it{1}}, {\it{2}}, {\it{3}}, \cdots, {\it{m}}$, one can obtain

Thus,

From Eq (4.14), we get a "+" ve constant $\beta$ such that $||{\it{u}}||_{{\mathfrak{P}}{\mathfrak{C}}} \neq \beta$. Consider a set

such that

is continuous and completely continuous.

Thus, no ${\it{u}} \in \partial (\psi)$ can be found such that ${\it{u}} = \gamma \chi ({\it{u}}), \gamma \in$ (0, 1). Hence, a nonlinear alternative of Leray Schauder's fixed point theorem gives that the answer to Eq (1.1) is a fixed point for $\chi$.

The outcome for $\varsigma \in \mathscr{I}_{o}$ is almost identical, thus it is not included. □

5.   Example

Example 5.1. Take into account a nonlocal initial condition over time scale in a neutral impulsive fractional dynamic equation

and we get

We set

It is evident that (5.2)'s right side is continuous for ${\it{u}}, {\it{v}} \in \mathscr{R}$ in relation to time scale. Again, $\forall$ $\varsigma \in [0, 1]\cap {\mathfrak{T}}$ and ${\mathfrak{h}}, {\mathfrak{i}} \in \mathscr{R}$. We obtain

then, we get

Next,

Thus, one can obtain

From the above data, we can say that the Eq (5.1) satisfies all the conditions of (A1)–(A4).

Again, for ${\it{m}} = 1$ we get

As a result, the requirements of Theorem 4.3 are met. Consequently, we came to the conclusion that the solution to Eq (5.1) is unique.

Below Table 1 represents the numerical approach for the theoretical results.

Figure 1 reveals a commendable correspondence between the numerical solution & exact solution across the entire interval.

6.   Conclusions

In this article, we examine both operators in the setting of time scales and analyze the C$\nabla$D and RL$\nabla$D. Additionally, the C$\nabla$D of the fractional dynamic equation including instantaneous impulses and a nonlocal initial condition are also examined. Later, the numerical technique is followed by an example based on all theoretical findings on the existence and uniqueness of the solution. A graph using MATLAB is also represented for the example.

Futher, in modeling the spread of infectious diseases like COVID-19, a dynamic equation in time scales can be used to capture the various stages of infection, transmission rates, and the impact of interventions over time. Mathematical models, often expressed as differential equations or agent-based models, can be adapted to include time scales that represent different temporal aspects of the disease dynamics.

Similarly, in cancer modeling, incorporating a dynamic equation in time scales allows for the consideration of the progression of the disease, the growth of tumors, and the response to treatments over time. This can lead to more accurate predictions and insights into the evolution of the disease and the effectiveness of different therapeutic interventions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2023/01/26285).

Conflict of interest

The authors declare that they have no conflicts of interest.