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# Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

1
Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia

2
Department of Computer Engineering, Biruni University, Istanbul, 34010, Turkey

3
Department of Mathematics, Science Faculty, Firat University, Elazig, 23119, Turkey

4
Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

5
Department of Physics, College of Khurma University College, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia

6
Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, 18000, Pakistan

* Corresponding Authors: Mustafa Inc. Email: ; K. H. Mahmoud. Email:

(This article belongs to the Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1669-1684. https://doi.org/10.32604/cmes.2022.022971

**Received** 02 April 2022; **Accepted** 06 June 2022; **Issue published** 27 October 2022

## Abstract

In this article, we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative (CFFD). The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii. Also, we enriched our work by establishing a stable result based on the Ulam-Hyers (U-H) concept. Also, the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method. We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders. The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier. Also, the suggested method results for the proposed system under the mentioned derivative are new. Furthermore, the adopted technique has some useful features, such as the lack of prior discrimination required by wavelet methods. our proposed method does not depend on auxiliary parameters like the homotopy method, which controls the method. Our proposed method is rapidly convergent and, in most cases, it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.## Keywords

Fractional calculus has gotten considerable attention in the last few decades. This is because of numerous applications in various fields of science and technology. Many real-world problems where hereditary properties and memory characteristics are involved can be comprehensively explained by using fractional calculus. For recent applications and interesting results, we refer to [1–4]. Keeping in mind the valuable uses of the said area, scientists have given much attention to investigating fractional order differential equations (FDEs) from various aspects. They developed the existence theory of solutions very well. Also, a large number of articles have been framed about numerical analysis and optimization theory for the said area. Also, researchers have developed stability results for various problems of FDEs. For the aforesaid area, we refer to [5–8]. Recently, various results related to fractional order chaotic systems have been investigated (see [9–12]).

In recent times, some new types of fractional differential operators have been introduced. The concerned definitions have been obtained, preserving the regular kernel instead of the singular one. In this regard, in 2015, Caputo et al. [4] introduced a new operator for fractional order derivatives, abbreviated here as CFFD. By using this new operator, researchers have developed numerous results, including existence theory and numerical solutions for various problems (see [13–16]).

Keeping the importance of FDEs in mind, various real-world problems have been investigated by using concepts of fractional calculus. Because fractional differential operators geometrically provide accumulation for a function, which includes its integer counter part as a special case. Also, in various cases, it has been found that fractional order derivative is more powerful than classical and describes the dynamics of various real world phenomena with more details (see [17–19]). Therefore, researchers have used FDEs in the study of dynamical problems of different natures. Among these real-world problems, Lorenz studied the Lorenz system first. The said problem has a chaotic solution for some parametric and initial values.

The said famous classical Lorenz system has been described in [20] given by

where the constants sigma, gamma and b are system parameters proportional to Prandtl, Rayleigh, and certain physical layer dimensions. The quantities involved are

Keeping in mind the importance of the said model, it has never been investigated till now by using CFFD.

Instead of ordinary calculus fractions, order derivatives and integrals are more practical in nature and preserve a greater degree of freedom. Using this type of operator, additional short and long-memory terms are better explained. Because power law singular kernels are used in Caputo and Reimann-Liouville operators. in numerical discretization, it causes difficulties. Therefore, by using those differential operators which involve exponential type kernels, the descriptions of some problems are more easily understood. Therefore, in this regard, the first one, which is increasingly used as CFFD, The Lorenz model has been investigated under various fractional order concepts by using different techniques. In most cases, researchers have investigated the approximate solution of the Lorenz model by using differential transform techniques [21], the homotopy perturbation method [22], and the Adomian decomposition method [23], etc. Also, authors [24] have studied the fractional order Lorenz system by using the homotopy analysis method. Further authors [25] investigated the numerical-analytical solution of nonlinear fractional-order Lorenz’s system. But in all the mentioned studies, the existence theory of the fractional order Lorenz system has not been investigated.

Also, to the best of our knowledge, the Lorenz system under CFFD for semi-analytical solutions by using Laplace Adomian decomposition has never been investigated. Therefore, we update the model (1) under CFFD as given in (2) as

where

It is a tedious job to deal with problems under the concept of fractional calculus for their exact or numerical solutions. Several algorithms, tools, and procedural theories have been established during the past few decades. A dynamical problem should be first treated for the criteria of its existence. Because, without knowing about its existence, we do not implement other techniques to compute numerical or semi-analytical results. For the existence theory, various tools have been developed. For instance, fixed point approaches, coincidence degree theories due to Mawhin and Schauder, etc., have been used very well. The most powerful one is the use of the fixed point approach to investigate a dynamical problem whether it has a solution or not (see [6]). On the other hand, numerous tools, including perturbation and decomposition techniques, have been used to approximate solutions to various dynamical problems (for details, see [26–30]). The mentioned tools have been used in excessive numbers for classical fractional order problems. Also, those problems involving the CFFD have been treated by using the homotopy and decomposition methods in various articles; for instance, see a few as [31–37].

Inspired by the aforesaid work, we are going to derive some adequate results for the existence of approximate solutions to the nonlinear system given in (2) under CFFD. We apply some fixed point approaches due to Krassnoselskii and Banach to establish the existence theory for the solution of the intended problem. Some stability results are also provided here in this work by using the U-H concept. Also, some approximations for the solution are established via a hybrid method due to the Laplace transform and Adomian decomposition. We also present some graphical representations of Lorenz equations using computational software such as Matlab.

Our work is organized as: We first provide some literature and refer to it in Section 1. In the Section 2, we recollect some elementary results. In Section 3, we provide our first portion of the main results. In Section 5, stability results are developed. In Section 4, we provide general algorithms of analytical results. In Section 5, some graphical presentations are provided. Finally, we will give a brief conclusion.

Definition 2.1. [4] Let

Definition 2.2. [4] For the function

Lemma 2.1. [4] Let

can be calculated as

Definition 2.3. [4] The Laplace transform of

Definition 2.4. Here we set

Theorem 2.5. [38] Let

Theorem 2.6. [38] If

1.

2.

3.

Theorem 2.7. Inview of conditions

Proof. Proof Keeping in mind the initial data of solution of the model (2), it is obvious that

therefore

3 Main Result for the Model (2)

This portion is devoted to the first part of our main results. Here we establish the existence criteria for our adopted model (2). We set the model (2) as

while

Further, we can write (5) as

where

We need the following hypothesis to be exist for onward analysis.

(A1) Subject constants

(A2) For constants

Using (6) and (7), we define two operators as

Theorem 3.1. Thank to hypothesis

Proof. Consider a closed bounded set as

Hence

For

Hence (9) implies that

Since at

Therefore

Theorem 3.2. Together with hypothesis

Proof. Let define

If

As a result,

Here we recollect basic notions for U-H stability from [1,2,8,15]

Definition 4.1. The integral Eq. (6) is U-H stable with

we have at most one solution

Also the integral Eq. (6) is generalized U-H stable if there exists a nondecreasing mapping

with

The given remark is needed.

Remark 1. Let

1.

2.

Remark 2. Let the perturbed problem be described as

Then the solution of (13) is computed as

Hence using Remark 1, (14) yields

where

Theorem 4.2. Inview of hypothesis

Moreover, the approximate solution of the model (2) is generalized U-H stable.

Proof. Let

From (16), upon simplification one has

Thus (6) is U-H stable. Setting

Obviously in (18), we see that

5 Algorithms for Approximate Solution of the Model (2)

We first develop a general algorithms for approximate solution to (2) as

Using initial condition, (19) yields

The solution we are computing can be expressed as

Also, the nonlinear terms can be decomposed as

Here few initial terms of Adomian polynomials are computed from (22) as

and so on.

Thus we calculate few terms (23) as

and so on. Using (21), (22) and (23) in (20), we have

Comparing terms on both sides of (24), we have

Applying inverse Laplace transform to both sides of (25), the following result can be obtained

Using

and so on. In this way the other terms are easy to compute. From (27), the approximate solution for each compartment of the model can be written as

Theorem 5.1. [34] If

Using Banach theorem

Also

1.

2.

5.1 Some Graphical Investigation

A phase portrait is a geometric description of a dynamical system’s paths in the phase plane. The collection of initial conditions is represented by a separate curve or point. Phase portraits are an immensely valuable tool in the study of dynamical systems. They are comprised of a structure of common state-space trajectories. This shows whether the selected parameter values have an attractor, repeller, or limit cycle. Phase portraits of a dynamical system can be used to study the directed characteristics of that system. In Fig. 1, the phase portraits of the approximate solution given (28) are shown for the first five terms. Here we use values of parameters from [20] to present the obtained five-term solution graphically.

A time series is a collection of data points that are indexed (listed or graphed) vs. time t in mathematics. It is a collection of points taken at evenly spaced intervals over a period of time. As a result, it is just a collection of discrete-time data. The time series includes sunspot counts and ocean tide heights, to name a few. Time series analysis depicts the behavior of state variables for each tiny value of time in the case of dynamical systems. Analyzing the time series data makes it simple to study the system’s stability and instability. In Fig. 2, we show the behavior of

5.3 Sensitivity Towards Initial Conditions

When a system is chaotic in its nature, it shows sensitive dependence on initial conditions. A very small change results in a great change in the dynamics of the system when it has chaotic behavior. Therefore, we present the dependence of our considered system on initial conditions. In Figs. 3a–3c, the sensitivity of

In the present work, we have derived some theoretical results based on some fixed point theorems due to Banach and Krassnoselskii for the existence and uniqueness of approximate solutions and their computation corresponding to the famous Lorenz nonlinear dynamical system. Sufficient conditions have been developed for the existence and uniqueness of solutions to the proposed model. Also, utilizing U-H and generalized U-H concepts, we have derived a few results for stability under some conditions for the considered system. Further, using a hybrid technique based on the Laplace transform and the Adomian decomposition method, we have also established an algorithm for approximate solutions. Some chaotic behaviors of the Lorenz system have been presented under the given fractional order by using five terms of approximate solution. Also, convergence and sensitivity of the model have been discussed. The proposed method has some features like being easy to implement, no need for prior discretization, and neither depends on auxiliary parameters like the homotopy analysis method. Also, the method is rapidly convergent in many cases. In future, we will investigate the aforesaid Lorenz model under piece-wise equations with fractional order derivative.

Funding Statement: The authors would like to acknowledge the financial support of Taif University Researchers Supporting Project No. (TURSP-2020/162), Taif University, Taif, Saudi Arabia. The authors extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant No. R.G.P.1/195/42.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*135*

*(2)*, 1669-1684. https://doi.org/10.32604/cmes.2022.022971

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*Comput. Model. Eng. Sci.*, vol. 135, no. 2, pp. 1669-1684. 2023. https://doi.org/10.32604/cmes.2022.022971