Dynamics of a Model of Polluted Lakes via Fractal-Fractional Operators with Two Different Numerical Algorithms

We employ Mittag–Leffler type kernels to solve a system of fractional differential equations using fractal-fractional (FF) operators with two fractal and fractional orders. Using the notion of FF-derivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a non-decreasing and compact mapping are used in order to prove the existence of a solution for the FF-model of polluted lake system. For this purpose, the Leray–Schauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on Adams–Bashforth and Newton polynomials methods. The effect of fractal-fractional differentiation is illustrated numerically. Moreover, the effect of the FF-derivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic.


Introduction
In the last century, pollution of waters has become a severe danger to the world we live in.The first step in preparing to conserve the natural environment is to monitor pollution levels.Monitoring pollution is possible to achieve with the use of mathematical analysis.Differential equations may be used to simulate environmental contamination, just as they can be used in many other fields.For example, Biazar et al. utilized in 2006 a set of differential equations to predict the pollution level in a series of lakes [1].In concrete, they have proposed a model of triple lakes connected by channels through compartment modeling.Some other scholars have investigated this concept using various methodologies.Yüzbaşi et al. [2] analyzed such levels of pollution under the collocation method in 2012.Later, Benhammouda et al. [3] utilized another method to solve the pollution model via a modified differential transform.Khader et al. [4] have also created a fractional case model and used the matrix properties in 2013.Recently, in 2019, Bildik and Deniz [5] considered an Atangana-Baleanu based model for approximating the solutions of a polluted lake system.After that, Ahmed and Khan turned to a similar model of lake pollution via different fractional methods [6].In 2020, Prakasha and Veeresha [7] solved such a system of polluted lakes via the so-called q-HATM method.More recently, in 2022, Shiri and Baleanu have done a research on the amount of pollution in a three-compartmental model and derived some analytical results [8].During these years, fractional models of real-world processes have been studied by many other researchers, showing the applicability of fractional operators in mathematical modeling: see, e.g., [9,10,11,12,13,14,15].Here we propose and study a mathematical model via a generalized family of derivatives equipped with two parameters.
Atangana introduced a new class of fractal-fractional notions, which brings together the two applicable areas of fractal and fractional calculi [16].The structure of these operators is a convolution of the power-law, exponential law, and modified Mittag-Leffler law with fractal derivatives, which establishes a connection between fractional and fractal mathematics.The fractal dimension and order are the two components of these operators and differential equations with fractal-fractional derivatives convert the putative system's order and dimension into a rational system.Because of this characteristic, conventional differential equations are naturally extended to systems with any order of derivatives and dimensions.The goal of these coupled operators is to look at distinct nonlocal boundary value problems (BVPs) or initial value problems (IVPs) that have fractal tendencies in nature.Many scholars provided results and discoveries in this area, demonstrating that fractal-fractional operators are more effective at describing real-world data and for mathematical modeling.Examples of such mathematical models include: fractal-fractional structures of dynamics of corona viruses [17], malaria transmission [18], dynamics of COVID-19 in Wuhan [19], transmission of AH1N1/09 virus [20], dynamics of Q fever [21], HIV [22], dynamics of CD4 + cells [23], tuberculosis disease [24], etc.
The incorporation of fractal-fractional (FF) operators with dual fractal and fractional orders in scientific research presents a promising avenue with multifaceted advantages.By leveraging two orders simultaneously, this approach allows for a more nuanced and refined representation of complex systems, capturing intricate patterns and irregularities that traditional methods might overlook.The synergy of fractal geometry and fractional calculus enhances the modeling and analysis of real-world phenomena, providing a more accurate reflection of the inherent self-similar structures and non-integer order dynamics.This not only refines our understanding of intricate processes but also facilitates the development of more robust mathematical models that can be applied across various disciplines.The utilization of FF operators holds the potential to revolutionize fields ranging from signal processing to image analysis, offering a versatile toolkit to address challenges that demand a deeper comprehension of intricate, multifractal behaviors.Embracing this innovative paradigm contributes to a more holistic and precise approach in scientific investigations, opening new frontiers for exploration and discovery.Here we conduct an analysis of a fractal-fractional model of polluted lakes in terms of various different characteristics.
The paper is organized as follows.In Section 2, we introduce a fractal-fractional system to model polluted lakes.Existence of a solution to the proposed system is proved in Section 3 by using the Leray-Schauder theorem.In Section 4, we employ the Banach principle for contrac-tions to demonstrate uniqueness of solution.Furthermore, using functional analysis, numerous requirements for different types of stability for the solution to the polluted lakes system model are explored in Section 5. To simulate our model, we use two different techniques: a fractional Adams-Bashforth approach (Section 6) and a second one based on Newton's polynomials (Section 7).The obtained theoretical results are then tested in Section 8 by applying our algorithms with some concrete data under various fractal and fractional order values in three different cases: linear, exponentially decaying and periodic input real models.We end with Section 9 of conclusion.
2 The FF-model for a polluted system of three lakes We model three lakes.Using three lakes in a system might be a good practical choice based on various factors such as land availability, cost, and efficiency.The decision of considering here three lakes is not purely mathematical, but involves environmental, economic, and logistical considerations.Mathematical modeling could help optimize the distribution and size of the lakes, but it is essential to balance these factors for a sustainable and effective solution.Therefore, we restrict ourselves to three lakes and their channels with a pollutant source.One can generalize our results to a finite number of lakes.
Each lake is treated as a compartment, a linking channel between two lakes being viewed as a pipe connecting the compartments.The direction of the flow across each channel or pipeline is shown by arrows.A contaminant c is considered in the first lake.By c(s) we denote the rate at which the contaminant/pollutant enters Lake 1 at time s.The major purpose is to determine the pollution levels in each lake at any given moment.To do so, we regard the concentration C i (s) of the pollutant in the lake i at time s, s ≥ 0, by where V i denotes the water volume at lake i, i ∈ {1, 2, 3}, assumed to be constant, and L i (s) specifies the quantity of pollution that is equally distributed over each lake at time s.We are interested to model the situation shown in Figure 1, where we use the symbol F ji to represent the flow rate entering the jth lake from the ith.Based on Figure 1, we derive the following conditions: Lake 2 Lake 3 Lake 1

Source of pollutant
Figure 1: Schematic of channels interconnecting the three lakes being modeled.
Lake 1 : Note that F 12 = 0 since there exists no pipe between the second and the first lakes.The flux F ji (s) of pollutant flowing from the ith lake to the jth lake at an arbitrary time s measures the flow rate of the concentration of pollutant.This index equals Based on the principle that the rate of change of the pollutant is given by the difference between the input rate and the output rate, we propose here the following fractal-fractional model for the dynamic behavior of the polluted lake system of three lakes via the generalized Mittag-Leffler kernel: subject to where FFML D (θ,σ) 0,s is the (θ, σ)-fractal-fractional derivative with Mittag-Leffler type kernel of fractional and fractal orders θ ∈ (0, 1] and σ ∈ (0, 1], respectively, as introduced by Atangana in [16]. Definition 1 (See [16]).Let f : (a, b) → [0, ∞) be a continuous map that is fractal differentiable of dimension σ.In this case, the Riemann-Liouville (θ, σ)-fractal-fractional derivative of f with the generalized Mittag-Leffler type kernel of order θ is given by where df (w) is the fractal derivative and AB(θ In what follows, we also use the corresponding notion of fractal-fractional integral. Definition 2 (See [16]).The (θ, σ)-fractal-fractional integral of a function f with generalized kernel is given by if it exists, where θ, σ > 0.

Existence
We begin by proving existence of solution to our problem (4)- (5).For that we use fixed point theory.To conduct our qualitative analysis, let us define the Banach space X = C 3 , where We rewrite the right-hand-side of the fractal-fractional polluted lake system (4) as In this case, the fractal-fractional polluted lake system (4) is transformed into the following system: In view of (9), we rewrite our tree-state system as the compact IVP where and By definition and by (10), we have Applying the fractal-fractional Atangana-Baleanu integral on (13), we get The extended representation of ( 14) is given by To derive a fixed-point problem, we now define the self-map F : X → X as (16) To prove existence of solution to our fractal-fractional polluted lake system (4), we make use of the following Leray-Schauder theorem.
Theorem 3 (Leray-Schauder fixed point theorem [25]).Let X be a Banach space, E ⊂ X a closed convex and bounded set, and O ⊂ E an open set with 0 ∈ O.Then, under the compact and continuous mapping F : Ō → E, either: Given that the polluted lake system models a real-world problem, its existence is subject to certain constraints.These constraints, denoted in Theorem 4 as (P1) and (P2), play a crucial role in shaping the dynamics and characteristics of the system.Indeed, (P1) and (P2) are indispensable to define and regulate the behavior of the polluted lake system within the confines of practicality and reality.Recognizing these constraints is essential for constructing a comprehensive understanding of the system and developing effective strategies.
then there exists a solution to the fractal-fractional polluted lake system (4).
Proof.First, consider F : X → X, which is formulated in (16), and assume for some r > 0. Clearly, as Q is continuous, thus F is also so.From (P1), we get Thus, F is uniformly bounded on X.Now, take s, v ∈ [0, S] such that s < v and K ∈ N r .By denoting sup we estimate We see that the right-hand side of ( 19) approaches to 0 independent of K, as v → s.Consequently, This gives the equicontinuity of F and, accordingly, the compactness of F on N r by the Arzelá-Ascoli thoerem.As Theorem 3 is fulfilled on F , we have one of (Y1) or (Y2).From (P2), we set for some ω > 0, such that From (P1) and ( 18), we have Suppose that there are K ∈ ∂Φ and 0 < µ < 1 such that K = µF (K).Then, by (20), we write which cannot hold true.Thus, (Y2) is not satisfied and F admits a fixed-point in Φ by Theorem 3.This proves the existence of a solution to the FF polluted lake model ( 4).

Uniqueness
As a first step to prove uniqueness of solution to our problem ( 4)-( 5), we begin by investigating a Lipschitz property of the fractal-fractional polluted lake system (4).
By invoking Lemma 5, we now prove uniqueness of solution to the FF-system (4).
Theorem 6.Let (C1) hold.If for j ∈ {1, 2, 3} and where α j > 0 are the Lipschitz constants introduced by (21), then the fractalfractional polluted lake system (4) possesses exactly one solution. and In this case, we estimate and so From (23), we can assert that the above inequality holds if which proves that the solution to the fractal-fractional polluted lake system (4) is unique.

Ulam-Hyers-Rassias stability
In this section, the stability of the solutions to the polluted lake system of three lakes is studied.Given the desire to establish robust mathematical foundations for the model, we consider four different notions of stability.More precisely, we prove stability for our fractal-fractional (FF) polluted lake system (4) with respect to Ulam-Hyers and Ulam-Hyers-Rassias notions and their respective generalizations.Stability analysis is pivotal in ensuring mathematical models' reliability and predictability, especially in real-world applications such as the polluted lake system.Ulam stability, Hyers stability, and their generalizations offer valuable frameworks for understanding the behavior of solutions to dynamic systems under perturbations.Given the intricate nature of fractal-fractional systems, the use of these stability notions allows us to ascertain the system's resilience to variations and disturbances, providing insights into the long-term behavior and reliability of the proposed model.By choosing stability in this context, we aim to enhance the credibility of the model and its applicability in addressing the complexities inherent in polluted lake systems.
Definition 7. The FF polluted lake system (4) is Ulam-Hyers stable if there exists a Q1 , a Q2 , a Q3 ∈ R + such that for all r j > 0, j = 1, 2, 3, and for all there exists (L 1 , L 2 , L 3 ) ∈ X satisfying the fractal-fractional polluted lake system (4) with Definition 8.The FF polluted lake system (4) Definition 10.The fractal-fractional polluted lake model (4) is Ulam-Hyers-Rassias stable with respect to there exists a solution (L 1 , L 2 , L 3 ) ∈ X of the FF-model of polluted lake system (4) such that Remark 11.If ℏ j (s) = 1, then Definition 10 reduces to the Ulam-Hyers criterion.
Definition 12.The FF polluted lake system (4) is generalized Ulam-Hyers-Rasias stable with respect to ℏ j if exists a (Qj ,ℏj This means that ( 26) is fulfilled.We prove ( 27) and (28) in a similar way.
We are now in a position to investigate the Ulam-Hyers stability for the FF-model of polluted lake system (4).
Proof.Let r 1 > 0 and L 1 * ∈ C be an arbitrary solution of (24).By Theorem 6, let L 1 ∈ C be the unique solution of the FF polluted lake system (4).Then L 1 (s) is defined as From the triangle inequality, Lemma 14 gives Hence, where We conclude that the FF-model of polluted lake system (4) is Ulam-Hyers stable.On the other hand, if we take then a Qj (0) = 0 and the proof is finished: ( 4) is generalized Ulam-Hyers stable.
Theorem 17.If (C1) and (C2) hold, then the FF-model of polluted lake system (4) is simultaneously stable in the sense of Definitions 7 and 8.
Proof.Let r 1 > 0, and L 1 * ∈ C satisfy (25).By Theorem 6, let L 1 ∈ C be the (unique) solution of the FF polluted lake system model (4).Then L 1 (s) becomes With the aid of the triangle inequality, Lemma 15 gives Accordingly, we obtain that . where As a consequence, the fractal-fractional polluted lake system (4) is stable in the sense of Definition 7. By defining r j = 1, j ∈ {1, 2, 3}, our FF polluted lake system model ( 4) is also stable in the sense of Definition 8.

Numerical algorithm via the Adams-Bashforth method
The Adams-Bashforth method is a robust numerical integration technique commonly used for solving most differential equations.Its higher-order accuracy and efficiency make it particularly suitable for approximating the solution of dynamic systems, such as those describing the behavior of polluted lake systems.By choosing the Adams-Bashforth technique, we aim to achieve accurate and stable numerical solutions for the fractal-fractional polluted lake system (4).
To do this, we apply the fractional Adams-Bashforth technique with two-step Lagrange polynomials.For that we redefine the fractal-fractional integral equations (15) at s k+1 .Precisely, we discretize the integral equations ( 15) for s = s k+1 as follows: The approximation of the above integrals are given by Next, we approximate w σ−1 Q j (w, L 1 (w), L 2 (w), L 3 (w)), j = 1, 2, 3, on [s ℓ , s ℓ+1 ] by applying twostep Lagrange interpolation polynomials under the step size h = s ℓ −s ℓ−1 .By direct computations, we obtain the following algorithm that yields numerical solutions to the FF-model of polluted lake system (4): where

Numerical algorithm via Newton's polynomials
Here we develop a different approximation algorithm (based on Newton's Polynomials) to compute numerically the solutions of our fractal-fractional polluted lake system (4).The use of Newton's polynomials in interpolation is motivated by their simplicity and applicability for approximating functions based on a set of given data points.In the context of modeling and analysis, Newton's polynomials offer a flexible approach to represent complex relationships within the polluted lake system.The polynomial interpolation technique enables us to construct a continuous function that approximates the behavior of the system, facilitating a more detailed and comprehensive understanding of its dynamics.To the best of our knowledge, the idea was first introduced in [26].Precisely, we follow [26] with the IVP (10) subject to the conditions ( 11) and (12).In this case, we have By discretizing the above equation at s = s k+1 = (k + 1)h, we get Approximating the above integral, we can write that (36) Now we approximate function Q * (s, K(s)) with the Newton polynomial Substituting (37) into (36), we obtain that Simplifying the above relations, we get and it follows that (38) On the other hand, by computing the above three integrals separately, one gets and By putting (39), (40), and (41) into (38), we obtain that Finally, we replace 42), and we get that Using the numerical scheme (43), the numerical solutions to the fractal-fractional polluted lake system (4) are given by and where Ψj (k, ℓ, θ) are defined in (44), j = 1, 2, 3.

Numerical simulations and discussion
Now we apply the Adams-Bashforth method (ABM) and Newton's polynomials method (NPM), proposed respectively in Sections 6 and 7, to examine and find numerical solutions L 1 , L 2 , L 3 of the proposed FF-model and to observe the applicability, accuracy, and exactness of the developed algorithms.To simulate the quantity of pollution in the modeled lakes, we coded the algorithms (33)-( 35) and ( 45)-(47) in MATLAB, version R2019A.
We consider the suggested FF-model in three cases: linear (Section 8.1), exponentially decaying (Section 8.2), and periodic (Section 8.3) input models.

Linear input model
In this case, we consider the model in which the Lake 1 has a contaminant with a linear concentration.Linear input states the steady behavior of the pollutant.At time zero, the pollutant concentration is zero but, as the time increases, the addition of pollutant is started and then is remained steadily.For example, when a factory starts production at time zero, waste discharge begins at a fixed rate and concentration.As a particular case, we chose c(s) = µs.Then, for µ = 100, from (4) we have In Figures 2 (a), (b), and (c), the behavior of the ABM approximations for each pair of the state functions Note that the parameter h is explicitly defined as the step size, distinct from the stability parameters ℏ j , j ∈ {1, 2, 3}, discussed in the stability Section 5.While here we emphasize h as the step size in a specific context, Ulam-Hyers-Rassias stability, as a theory, is primarily concerned with the stability properties of functional equations.Unlike the numerical solution of differential equations, the choice of step size is not a direct consideration in the realm of Ulam-Hyers-Rassias stability.This stability theory focuses on understanding how small variations in functional equations lead to proportionate changes in the solutions, and the concept of a step size does not play a prominent role in that context.In Table 1, we present some numerical results of the two numerical techniques, ABM and NPM, for the three state functions L 1 , L 2 and L 3 in the linear input case, under integer-order derivatives and step size h = 0.1.From the obtained numerical results, we can assert that the Adams-Bashforth approximations for the phase functions L 1 , L 2 , and L 3 strongly agree with the ones obtained by the Newton polynomials method for the time s up to 10 years.
In Figure 3, the comparison of the numerical results from ABM and NPM for the state functions L 1 , L 2 , and L 3 is shown graphically, for the time s ∈ [0, 60] and the linear input case.We observe that the results of ABM and NPM have a high agreement between them for each one of the state In Figure 4, we illustrate the behavior of the three state functions L 1 , L 2 , and L 3 when the ABM is applied under the fractal-fractional orders θ = σ = 0.85, 0.90, 0.95, 0.99.From these figures, we can observe that when the fractal-fractional order is getting closer to the integer case, then the effect of the pollution is increasing on each lake model at about the same rate.As an observation of these graphs, it can be said that the non-integer order operator has a positive effect on the pollution reduction in the lake pollution model.
A word is due about our choice of the values of the fractal-fractional orders.We considered fractional orders within the range of [0.85, 1] because within this interval we observed consistent behaviors for different fractional orders.Specifically, as the fractal-fractional order decreases, we noted a proportional reduction in the impact of pollution on each lake model at about the same rate.This consistent trend in behavior as the fractional order decreases led us to cut the interval at the value 0.85.This choice captures the essential aspects of the model's response to varying fractional orders and provides a meaningful representation of the system dynamics.

Exponentially decaying input model
When heavy dumping of pollutant is present, it makes sense to consider the exponentially decaying input model, i.e., the case when c(s) = re −ps .An example of this case occurs if every industry placed in a city collects and stores its wastage during some days and then dumps it to Lake 1 after that stored period.If we take r = 200 and p = 10, then system (4) becomes The graphical representation of the input function c(s) is illustrated in Figure 5 for the exponentially decaying input case c(s) = 200e −10s , s ∈ [0, 1].
In Figures 6 (a), (b), and (c), the behavior of the ABM approximations for each pair of the state functions L 1 − L 2 , L 1 − L 3 , and L 2 − L 3 , respectively, is shown, while in Figure 6 (d), the 3D view of L 1 − L 2 − L 3 under the integer-order derivative is graphically illustrated for the exponentially decaying input model with time s ∈ [0, 60] and step size h = 0.01.
In Table 2, we provide a comparison between the approximate solutions obtained using ABM and NPM for the exponentially decaying input case with time s ∈ [0, 10], step size h = 0.01,  2, we can conclude that the ABM approximations are also in good agreement with the NPM ones for the exponentially decaying input model.In Figure 7, we present a graphical comparison between ABM and NPM approximations for the state functions L 1 , L 2 , and L 3 in the exponentially decaying input case with time s ∈ [0, 60].From Figure 7, it can be concluded that the two introduced methods strongly agree with each other even in a large time domain of 60 years.
In Figure 8, we illustrate the numerical results of the three state variables L 1 , L 2 , and L 3 for the exponentially decaying input model when the ABM is applied under various fractal-fractional orders: θ = σ = 0.85, 0.90, 0.95, 0.99. Figure 8 shows that the non-integer fractal-fractional operators have an effect on decreasing the amount of pollution for each model when the time s increases, that is, the pollution is increasing harmoniously with the fractal-fractional order, getting closer to the integer-order case.

Periodic input model
As a last case of study, we consider a periodic input model in which the pollutant appears in the lake periodically.A factory that works during daytime only, can be an example of this case: it generates waste and dump it in the lakes during the day while at night the mixing of new pollutants stops.For a concrete case, we selected c(s) = a + τ sin(bs), where τ and b stands for the variations of amplitude and frequency, respectively.Also, a is considered as the average input of pollutant concentration.In such a case a = b = τ = 1, system (4) takes the following form: The graphical representation of the input function c(s) is illustrated in Figure 9 for the periodic input case c(s) = 1 + sin(s), s ∈ [0, 20].
In Figures 10 (a The tabular comparison between the numerical results obtained from the proposed techniques, ABM and NPM, for the three state functions L 1 , L 2 , and L 3 under the periodic input case, are reported in Table 3 for time s ∈ [0, 10], step size h = 0.1, and θ = σ = 1.From these results, we conclude that the solutions obtained by ABM and NPM highly agree with each other.In Figure 11, we illustrate our findings graphically, comparing the numerical results from ABM and NPM for each state function L 1 , L 2 , and L 3 , where the Lake 1 has a periodic pollutant input.Figure 11 shows that the two introduced techniques, ABM and NPM, strongly agree with each other for the time s ∈ [0, 60], step size h = 0.1, and θ = σ = 1.
In Figure 12, we illustrate the ABM approximations of the three state functions L 1 , L 2 , and L 3 under various fractal-fractional orders: θ = σ = 0.85, 0.90, 0.95, 0.99 for the periodic input case.Similar to cases of Sections 8.1 and 8.2, we observe that the non-integer order fractal-fractional operators have a great effect on decreasing the amount of contamination for each model while the time s increases.

Conclusion
We employed Mittag-Leffler type kernels to solve a system of fractional differential equations using fractal-fractional (FF) operators with two fractal and fractional orders.We derived equiv-    alent FF-integral equations from a compact initial value problem, and then proved existence and uniqueness results.A stability analysis was conducted in different versions.In the next sections, we examined and captured the behavior of the considered fractal-fractional operator model ( 4) with the help of two different numerical techniques: an Adams-Bashforth method (ABM) and a Newton polynomials method (NPM).From the obtained results, we conclude that the considered techniques, ABM and NPM, are in highly agreement and are very efficient to examine the system of fractional differential equations under fractal-fractional operators describing the dynamics of the pollution in the lakes.We also analyzed the considered model under various fractal-fractional orders and examined the effects of these non-integer orders on the behavior of each state variable L 1 (s), L 2 (s), and L 3 (s) for three specific input models: linear, exponentially decaying, and periodic.For each input model, we observed that when the fractal-fractional order gets closer to the classical integer-order case, then the effect of the pollution is increasing harmoniously for each lake model.As a conclusion of these observations, it can be said that the non-integer order operators have positive effects on the reduction of pollution in the lake pollution model.As future work, we plan to investigate different real-world models based on the techniques here developed.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

and L 2 −
L 3 , respectively, are given; while in Figure 2 (d), the 3D view of L 1 − L 2 − L 3 under integer-order derivatives are graphically illustrated for the linear input model with time s ∈ [0, 60] and step size h = 0.1.

Figure 3 :
Figure 3: Comparison between the ABM and NPM for (a) L 1 (s), (b) L 2 (s) and (c) L 3 (s) in the linear input model.
), (b), and (c), the graphical behavior of each pair of the state functions L 1 −L 2 , L 1 − L 3 , and L 2 − L 3 , respectively, is shown.In Figure10 (d), the 3D view of L 1 − L 2 − L 3 under the integer-order derivative is illustrated for the periodic input model with time s ∈ [0, 60] and step size h = 0.1.

Figure 5 :
Figure 5: Graphic of the exponentially decaying input.

Figure 7 :
Figure 7: Comparison between the ABM and NPM for (a) L 1 (s), (b) L 2 (s) and (c) L 3 in the exponentially decaying input model.

Figure 11 :Figure 12 :
Figure 11: Comparison between the ABM and NPM for (a) L 1 (s), (b) L 2 (s) and (c) L 3 in the periodic input model.

Table 1 :
Comparison between ABM and NPM for the linear input case.

Table 2 :
Numerical comparison between ABM and NPM for the exponentially decaying input case.