key: cord-0913842-m1o5me7p
authors: Partohaghighi, Mohammad; Yusuf, Abdullahi; Bayram, Mustafa
title: New Fractional Modelling, Analysis and Control of the Three Coupled Multiscale Non-Linear Buffering System
date: 2022-03-27
journal: Int J Appl Comput Math
DOI: 10.1007/s40819-022-01290-9
sha: 301193b3332c2ce1d6bf3a0b6bb2749044bc9d27
doc_id: 913842
cord_uid: m1o5me7p

This study aims to investigate the complicated dynamical [Formula: see text] buffering system using fractional operators which is not been investigated yet. We consider a new fractional mathematical model in the frame of fractional-order differential equations. In the proposed fractional-order model, we apply the Caputo-Fabrizio fractional operator with an exponential kernel. Then to solve the derived system of fractional equations, we suggest a quadratic numerical technique and prove its stability and convergence. Also, accurate control for the proposed system is considered. Behaviors of the approximate solutions for the considered model are provided by choosing different values of fractional orders along with integer order. Each figure manifests and compares the numerical solutions under selected orders. Figures, show how the results can be affected by changing the fractional orders.

In recent years, researchers have paid a considerable concentration to the non-integer differential and integral operators because these types of operators are effective instruments for the modelling of real-world phenoms with complicated dynamics [1] [2] [3] [4] [5] [6] . Studies on fractionalorder problems show that these types of operators can manifest intricate dynamical behaviours more perfectly than integer order systems. Oscillatory and complex behaviour of Caputo-Fabrizio fractional order HIV-1 infection model was reported in [7] . Effect of vaccination to control COVID-19 with fractal fractional operator can be read in [8] . Also, modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel can be seen in [9] . This is because fractional-order operators can identify the qualities of memory effects as a crucial perspective of numerous real-world phenoms. Modelling and simulations of the Definition 1.1 [39] For each f ∈ H and 0 < α < 1 the Caputo-Fabrizio(CF) fractional derivative and its related fractional integral operator are, respectively, presented by

By taking α = 1, Eq. (1) gives the integer-order integral yields to u(t) for α = 0. Now, we demonstrate some helpful relations of the CF definition [40, 41 ] -for f 1 and f 2 ∈ H 1 (0, T ) and c 1 , c 2 ∈ R, the CF derivative and integral are linear operators:

-If we consider u(t) = u c as a constant function, the CF operator is zero, as

-Because the CF operator (1) is the convolution integral of du(t) dt and ex p( −α 1 − α t), using the convolution theory for the Laplace transform, we get

This numerical study is prepared as follows. A new mathematical using fractional Caputo-Fabrizio operator is developed in Section 2. Also, in Sect. 3, a quadratic scheme is introduced and stability and convergent of this method are shown, successfully. Robust control for the non-linear dynamical buffering systems can be read in Sect. 4. Numerical experiments for the proposed model using different values of fractional orders and initial conditions can be observed in Sect. 5 and numerical experiments to show the effectiveness of the proposed approach for solving the considered fractional model are provided in Sect. 6. Finally, we provide the conclusion of this numerical investigation on the nonlinear dynamical HC O − 3 /C O 2 buffering system in Sect. 7.

Now we consider the integer order model of acid-base homeostatic HC O − 3 /C O 2 process reported in [42] as

In the above system l 1 depicts the cellular reproduction of H + and l 2 shows the H + lack. Moreover, we indicate the hydration and dehydration response measures by l 3 and l 4 , respectively, where the amounts have been arranged to reveal the carbonic anhydrase behaviour. Indeed, l 5 indicates the HC O − 3 treatment and/or completion. Also, acid secretion degree is represented by l 6 . The measure related to renal filtration for HC O − 3 is denoted by l 7 and

We depict the cellular reproduction of C O 2 with l 8 . Also, we illustrate the ventilation measure by l 9 and l 10 is minute amount ventilation. Indeed, A(t) indicates the concentration of the free hydrogen ions, B(t) denotes the condensation of bicarbonate and C(t) shows the concentration of carbon dioxide. In order to make the fractional model of the above system, we replace the classic derivative with the fractional Caputo-Fabrizio derivative. So, we have the following one

Subjected to the initial conditions A(0) = A 0 , B(0) = B 0 and C(0) = C 0 .

In order to get the approximate solutions of the proposed model (7), we suggest a quadratic numerical strategy. First, we convert the system (7) into its corresponded integral equation.

After that, a numerical scheme based on the trapezoidal method is presented to obtain the numerical solutions of the derived integral equation. To use the offered method in the framework of CF, we write Eq. (7) in the next form;

, is the vector function

meeting Lipschitz condition

and 0 = (A(0), B(0), C(0)) involves initial conditions. By implementing relation (1) on Eq. (7), we get

is an integer, and t s = sh, s = 0, 1, 2, · · · , S − 1, Also, we present the approximation of (t s ) via s . After that, ( (τ )) on subinterval [t k , t k+1 ] is approximated by the piecewise linear interpolation as:

To possess a approximate solution of Eq. (11), the next discretization for (1) can be a proper option:

Now, by replacing relation (12) into (13), we get (14) where the coefficient a s+1,k is in the following form:

We examine the convergence as well as stability of the scheme (14) using following theories. (14) is conditionally stable.

Proof We denote the perturbation of 0 and s (s = 0, 1, · · · , S − 1) via˜ 0 and˜ s , respectively. Thus, regarding the numerical technique (14), it results

Now, we replace Eq. (14) into Eq. (16), so we get

Applying triangle inequality as well ass Lipschitz condition, by Eq. (17), yields

Using Eq. (15), we obtain

where

Moreover,for a constant C φ such that for a adequately small h, we own

Thus

Finally, using Lemma 3.3 in [39] and using the Grönwall inequality, s+1 ≤ C 1 0 can be obtained, where in C 1 is a constant.

Proof We compute the difference between the exact solution (t m+1 ) and the approximate solution F m+1 (Eq. (14)) via

Now, we apply the triangle inequality and Lipschitz condition, along with Lemma 3.1 in [39] 

where C is a general constant. Using Eq. (15) and the parameters L, h, α meeting the inequal-

where φ(α, h) is taken into account from Eq. (21) and satisfies inequality (22) . At the end, using Lemma 3.3 in [39] and applying Grönwall inequality, yields

where C 1 is a generic constant.

We consider the following system as

Now we impose u A , u B and u C on the above model. So, we have

The purpose of the control is to overcome the chaotic performance of the model. We setĀ, B, andC as equilibrium points of the model.

Then by defining a control error we have the following relations 

As stated, the performance of chaos would be overcome in the model, thus points of equilibrium will be set at zero i.eĀ =B =C = 0, so we have 

From the above system, the control law is represented by the following relations: Proof We use the Lyapanov function to show the stability of the considered controller (34) . The proposed functions is as follows

Thus, for the above function we have the following derivative

Now, using Eqs. (33) and (34) , the next relations can be gained CF 0 

Now, by substituting Eq. (38) into Eq. (36), we obtain

So regarding A , B , C > 0 indicates that the Eq.(39) is negative or zero, therefore we prove that the considered controller is stable.

To see the performance of the offered method on the proposed fractional model 

In the current study, the complex behaviour of the non-linear buffering system was investigated in the frame of fractional concept for the first time. Firstly, fractional operator in the sense of CF was considered for modelling the non- orders. Also, accurate control is provided for the studied model. All in all, in this numerical investigation, accurate, flexible and successful performance of fractional Caputo-Fabrizio operator for the non-linear buffering system was done. Data availability statement Enquiries about data availability should be directed to the authors.

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