Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

231 
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Dynamic Effect of Green Building Concept for Sustainable Cities 

under Climate Change via Variable-Order Fractional Derivatives 

 

T.Saradhadevi1, A.Kalavathi2 
1Research Scholar, Department of Mathematics, Sri GVG,Visalakshi College for Women,Udumalpet,Tamilnadu, India, 

Email ID:search2saradha@gmail.com 

2Assistant Professor ,Department of Mathematics, Sri GVG,Visalakshi College for Women,Udumalpet,Tamilnadu, India, 

Email ID: kalavathigvg2018@gmail.com 

 

Article History: 

Received: 31-07-2024 

Revised: 09-09-2024 

Accepted: 18-09-2024 

Abstract:  

This study introduces an innovative fractional order model that delves into the core 

principles of sustainable constructions, incorporating Liouville-Caputo, Caputo-Fabrizio, 

and Atangana-Baleanu derivatives. It examines the increasing adoption of eco-friendly 

buildings in India as a crucial remedy against climate change.  By revealing innovative 

perspectives on fractional solutions, encompassing their validity and distinctiveness, the 

research conducts computational analyses to evaluate the impacts of Greenhouse Gas 

(GHG) elements. Visual representations demonstrate the dynamics of the model, 

emphasizing the influences of various fractional order parameters across different 

scenarios. 

Keywords: Fractional variable-order model, Liouville-Caputo, Caputo-Fabrizio, 

Atangana-Baleanu, Numerical simulations. 

 

1. Introduction 

In recent years, there has been a notable global increase in the utilization of sustainable construction 

methods as a key approach to tackling the issues related to climate change. Currently, buildings 

contribute to 7.85 gigatons (Gt) or 33% of all energy-linked CO2 emissions worldwide. Forecasts 

suggest that by 2030, these emissions could potentially rise to between 11 and 15.6 Gt. Specifically, 

developed nations observe buildings consuming over 41% of total energy usage. In India, the rapid 

growth in urbanization, population expansion, and the flourishing IT sector and its related industries 

are major factors contributing to the escalating energy needs. 

Green structures are created, constructed, and managed with the aim of minimizing their ecological 

impact while enhancing efficiency and functionality. This involves strategies like water preservation, 

reducing energy usage, and employing sustainable resources. In India, there has been a significant 

rise in the volume of environmentally friendly building space, with the Indian Green Building 

Council (IGBC) certifying over 7.6 billion square feet of green building space as of August 2023. 

This reflects a notable increase from the 3.6 billion square feet of certified green building space in 

2018. 

The expansion of sustainable construction in India is being primarily influenced by several factors, 

including governmental efforts, rising consumer awareness, and growing demand from multinational 

corporations. This growth is resulting in numerous favourable outcomes, such as the conservation of 



Communications on Applied Nonlinear Analysis 

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resources, decreased greenhouse gas emissions, enhanced indoor air quality, and the creation of 

employment opportunities. Recently, the Indian government has displayed a stronger dedication to 

encouraging eco-friendly building practices, while Indian consumers are growing more 

knowledgeable about the benefits of environmentally conscious constructions. As of October 21, 

2023, there are more than 3,088 building projects in India, encompassing a total area of over 1,315 

million square feet (122.2×10^6 m^2) of certified space. These initiatives have been approved by the 

Indian Green Building Council (IGBC), the primary green building assessment system in the nation. 

It is approximated that less than 10% of buildings in India conform to sustainable standards, a stark 

contrast to developed countries like the United States, where over 30% of buildings are considered 

environmentally friendly. 

A green building is characterized as one that utilizes less water, enhances energy efficiency, 

safeguards natural resources, minimizes waste, and provides healthier environments for occupants 

compared to a traditional building. It is commonly known as a 'high-performance' or sustainable 

structure [1]. Environmental issues are intertwined with economic and population growth. To 

achieve sustainability, India needs to effectively waste that contaminates the ecosystem [2]. Through 

life cycle analysis, eco-friendly buildings will preserve resources and mitigate climate change [3]. 

The idea of green buildings has developed in a surprising manner to achieve a more realistic 

sequence of events [4-6]. Sustainable buildings offer reduced energy consumption and minimized 

negative environmental impacts However, one of the causes of preventable health issues, increased 

consumption costs, and decreased productivity is the incapacity to adopt a "green" lifestyle [7]. 

While recent research studies [8, 9] have advised the utilization of innovative technology to enhance 

the state of buildings constructed from locally sourced materials, this recommendation has yet to be 

implemented. Concerning thermal comfort, dwellings featuring thatched roofs, bamboo cladding, 

and mud bricks tend to be more appealing to residents compared to traditional, unsustainable 

edifices. Enhancing greenery cover to implement nature-based strategies for mitigating heat can 

significantly reduce energy consumption, as well as the associated risks of mortality and heat-related 

ailments [10]. The rise in green building construction in urban areas is closely linked to the pressing 

need for economic development as cities strive to develop intelligent, eco-friendly, and sustainable 

environment [11-13]. The transition from integer-order differential equations to fractional 

differential equations represents a broadening of mathematical frameworks. Fractional calculus, 

widely utilized by scholars and mathematicians, serves as a powerful tool for modeling real-world 

scenarios [18-22, 33, 34]. The growing significance of variable-order fractional calculus is evident 

due to its diverse applications across science and engineering fields. These fractional operators with 

variable orders demonstrate intriguing practical uses [23-30]. 

In view of the above, this study is driven by the need to comprehend the dynamic effects of the green 

building concept for sustainable cities under climate change through the application of the variable-

order fractional derivative in the Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu senses. 

In section 2, the necessary preliminary information for the fractional model are provided. Section 3 

presents a detailed description of the model in the classical version of the "Green building concept 

for sustainable cities" and its corresponding fractional order version. The uniqueness and existence 

of fractional solutions are explained in section 5. Section 6 provides detailed numerical results and 



Communications on Applied Nonlinear Analysis 

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simulations for the suggested model on Liouville Caputo, Caputo-Fabrizio, and Atangana-Baleanu 

derivatives. All obtained results of the work are concluded in the final section. 

2.  Prelminaries 

This section provides some basic definitions of variable order fractional derivatives which are used 

in subsequent sections. 

Definition 2.1:The Liouville–Caputo (LC) fractional derivative with variable-order ν(t) is defined as 

[32] 

𝐷𝑡
ν(t) 

0
𝐿𝐶 𝑓(𝑡) = 

1

⎾1−ν(t)
∫ (𝑡 − 𝑢)−ν(t)𝑡

0
𝑓(𝑢)𝑑𝑢 , 0 < ν(t) ≤ 1 

Definition 2.2: The Caputo–Fabrizio (CF) derivative with variable-order ν(t) in Liouville–Caputo 

sense is defined as follows 

𝐷𝑡
ν(t)

0
𝐶𝐹 𝑓(𝑡) =

(2−ν(t))𝑀(ν(t))

2(1−ν(t))
∫ exp [

−ν(t)

(1−ν(t))

𝑡

0
(𝑡 − 𝑢)]𝑓′(𝑢)𝑑𝑢  , 0 < ν(t) <1 

where  𝑀(ν(t)) =
2

2−ν(t)
 is a normalization function.  

Definition 2.3: The Atangana–Baleanu (AB) fractional derivative with variable-order ν(t) in 

Liouville–Caputo sense is defined as follows [17]  

𝐷𝑡
ν(t)

0
𝐴𝐵 𝑓(𝑡) =

𝐵(ν(t))

(1−ν(t))
∫ 𝐸ν(t) [

−ν(t)

(1−ν(t))

𝑡

0
(𝑡 − 𝑢)ν(t)]𝑓′(𝑢)𝑑𝑢 , 0 < ν(t) ≤ 1 

where 𝐵(ν(t)) = 1 − ν(t) +
ν(t)

⎾ν(t)
 is a normalization function. 

Remark: When ν(t) is a constant, then we retrieve the constant-order fractional derivative in 

Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu sense. 

3.   Model Formulation in Classical and Fractional Sense 

3.1 Classical model of Green Building concept for sustainable cities under climate change 

A mathematical model [15] has been developed to examine the effects of global warming excessive 

greenhouse gas emissions on sustainable urban. This model considers four key variables: G(t) 

represents the quantity of Green buildings at any given time t, C(t) signifies the level of GHGs at 

time t, I(t) denotes the energy generated by green buildings at time t, and H(t) reflects the human 

populations in a specific region at time t. 

𝑑𝐺

𝑑𝑡
=  𝛬 + 𝑝1𝐺(𝑡)𝐶(𝑡) − 𝑝2𝐺(𝑡)   

𝑑𝐶

𝑑𝑡
=  𝛺 − 𝑝1𝐺(𝑡)𝐶(𝑡) + 𝑝3𝐻(𝑡)𝐶(𝑡) − 𝑝4𝐶(𝑡)                                                     (1) 

𝑑𝐼

𝑑𝑡
= 𝑝5𝐺(𝑡)𝐼(𝑡) − 𝑝6𝐻(𝑡)𝐼(𝑡) − 𝑝7𝐼(𝑡)                             

𝑑𝐻

𝑑𝑡
=  𝛹 + 𝑝6𝐻(𝑡)𝐼(𝑡) − 𝑝8𝐻(𝑡)  



Communications on Applied Nonlinear Analysis 

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With initial conditions 𝐺(0) = 𝐺0 ≥ 0 ;  𝐶(0) = 𝐶0 > 0; 𝐼(0) = 𝐼0 ; 𝐻(0) = 𝐻0 ≥ 0 

Where 𝛬 refers the amount of green building , 𝑝1 is the rate at which GHGs are absorbed by green 

buildings, 𝑝2 is the reduction rate of green building, GHGs are measured by  𝛺 , 𝑝3 is the rate of 

human originated GHGs, 𝑝4 represents rate of reduction of GHGs, an ingredient production rate by 

green buildings is measured by 𝑝5, 𝑝6 is the adoption rate of ingredients by human communities, the 

reduction rate of ingredients is defined as  𝑝7, 𝛹 is the amount of human community and  𝑝8 is the 

reduction rate of human communities. 

3.2 Fractional version of the classical model 

By substituting the classical derivative with the operator  
   𝑑ν(t)𝑓(𝑡)

𝑑𝑡
 , the fractional model of system 

(1) is obtained. 

𝑑ν(t)𝐺

𝑑𝑡
=  𝛬 + 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) − 𝑝2
ν(t)𝐺(𝑡)  

𝑑ν(t)𝐶

𝑑𝑡
=  𝛺 − 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) + 𝑝3
ν(t)𝐻(𝑡)𝐶(𝑡) − 𝑝4

ν(t)𝐶(𝑡)     

𝑑ν(t)𝐼

𝑑𝑡
= 𝑝5

ν(t)𝐺(𝑡)𝐼(𝑡) − 𝑝6
ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝7

ν(t)𝐼(𝑡)                                (2)                      

𝑑ν(t)𝐻

𝑑𝑡
=  𝛹 + 𝑝6

ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝8
ν(t)𝐻(𝑡)     

With 𝐺(0) = 𝐺0 ≥ 0 ;  𝐶(0) = 𝐶0 > 0; (0) = 𝐼0 ; 𝐻(0) = 𝐻0 ≥ 0 

The equilibria of the above fractional-order model can be obtained from 
   𝑑ν(t)𝐺(𝑡)

𝑑𝑡
 =0  

   𝑑ν(t)𝐶(𝑡)

𝑑𝑡
 = 0, 

   𝑑ν(t)𝐼(𝑡)

𝑑𝑡
=0 and 

   𝑑ν(t)𝐻(𝑡)

𝑑𝑡
=0. 

It was observed that the system (1) has two equilibria, one of them is trivial equilibrium  

𝐸𝑇𝐸 =(𝐺0 , 𝐶0, 𝐼0, 𝐻0) and the other is Global equilibrium 𝐸𝐺𝐸 = (𝐺∗, 𝐶∗, 𝐼∗, 𝐻∗).  Where  

𝐺∗ =  α =
𝛬𝑝3

𝜈(𝑡)
𝑝7

𝜈(𝑡)
+ 𝛬𝑝4

𝜈(𝑡)
𝑝6

𝜈(𝑡)

[(𝑝2
𝜈(𝑡)

𝑝3
𝜈(𝑡)

𝑝7
𝜈(𝑡)

+ 𝑝2
𝜈(𝑡)

𝑝4
𝜈(𝑡)

𝑝6
𝜈(𝑡)

) + (𝛬𝑝3
𝜈(𝑡)

𝑝5
𝜈(𝑡)

− 𝛬𝑝1
𝜈(𝑡)

𝑝6
𝜈(𝑡)

) − 𝑝1
𝜈(𝑡)

𝛺𝑝6
𝜈(𝑡)

]
 

𝐶∗ = 
𝛺𝑝6

𝜈(𝑡)

(𝑝3
𝜈(𝑡)

𝑝7
𝜈(𝑡)

+ 𝑝4
𝜈(𝑡)

𝑝6
𝜈(𝑡)

) − α(𝑝3
𝜈(𝑡)

𝑝5
𝜈(𝑡)

+ 𝑝1
𝜈(𝑡)

𝑝6
𝜈(𝑡)

)
 

𝐼∗ =
𝑝5

𝜈(𝑡)
𝑝8

𝜈(𝑡)
α−𝑝7

𝜈(𝑡)
𝑝8

𝜈(𝑡)
−Ψ𝑝6

𝜈(𝑡)

𝑝6
𝜈(𝑡)

(𝑝5
𝜈(𝑡)

 α−𝑝7
𝜈(𝑡)

)
  

𝐻∗ =
𝑝5

𝜈(𝑡)
α − 𝑝7

𝜈(𝑡)

𝑝6
𝜈(𝑡)

 

The eigen values are the solutions of the characteristic equation det(𝐴𝑖 − 𝜆𝐼) = 0. 

Where the matrix 𝐴𝑖 and the unit matrix is I with the eigen values calculated at 𝐸𝑇𝐸 and 𝐸𝐺𝐸 . For 

further details of the results can be found in [15] 



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Since the parameters are dimensionless, the fractional models within LC, CF and AB sense will be 

the same and it will not be necessary to investigate again. 

 The Jacobian matrix of the system (2) is as follows 

J= 

[
 
 
 
 
𝑝1

ν(t)𝐶 − 𝑝2
ν(t) 𝑝1

ν(t)𝐺 0 0

−𝑝1
ν(t)𝐶 −𝑝1

ν(t)𝐺 + 𝑝3
ν(t)𝐻 − 𝑝4

ν(t) 0 𝑝3
ν(t)𝐶

𝑝5
ν(t)𝐼 0 𝑝5

ν(t)𝐺 − 𝑝6
ν(t)𝐻 − 𝑝7

ν(t) −𝑝6
ν(t)𝐼

0 0 𝑝6
ν(t)𝐻 𝑝6

ν(t)𝐼 − 𝑝8
ν(t)𝐻]

 
 
 
 

 

4. Existence and Uniqueness of Fractional solutions 

4.1 Existence and Uniqueness of Fractional solutions by the Liouville-Caputo model 

In this section, we establish the existence and uniqueness of solutions of the Liouville- Caputo 

model. 

 Let us construct the system (2) as 

𝐷𝑡
𝜈(𝑡)

0
𝐿𝐶 [𝐺(𝑡)] = 𝐹1(𝑡, 𝐺) = 𝛬 + 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) − 𝑝2
ν(t)𝐺(𝑡)  

𝐷𝑡
𝜈(𝑡)

0
𝐿𝐶 [𝐶(𝑡)] = 𝐹2(𝑡, 𝐶) = 𝛺 − 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) +  𝑝3
ν(t)𝐻(𝑡)𝐶(𝑡) − 𝑝4

ν(t)𝐶(𝑡) 

𝐷𝑡
𝜈(𝑡)

0
𝐿𝐶 [𝐼(𝑡)] = 𝐹3(𝑡, 𝐼) =  𝑝5

ν(t)𝐺(𝑡)𝐼(𝑡) − 𝑝6
ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝7

ν(t)𝐼(𝑡)                        (3)                                      

𝐷𝑡
𝜈(𝑡)

0
𝐿𝐶 [𝐻(𝑡)] = 𝐹4(𝑡, 𝐻) = 𝛹 + 𝑝6

ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝8
ν(t)𝐻(𝑡)  

By using Liouville-Caputo fractional integral operator to the above system, we get 

𝐺(𝑡) − 𝐺(0) = 
1

⎾𝜈(𝑡)
∫ (𝑡 − 𝑘)𝜈(𝑡)−1𝑡

0
𝐹1(𝐾, 𝐺(𝑡))𝑑𝑘  

𝐶(𝑡) − 𝐶(0) = 
1

⎾𝜈(𝑡)
∫ (𝑡 − 𝑘)𝜈(𝑡)−1𝑡

0
𝐹2(𝐾, 𝐶(𝑡))𝑑𝑘             

𝐼(𝑡) − 𝐼(0) = 
1

⎾𝜈(𝑡)
∫ (𝑡 − 𝑘)𝜈(𝑡)−1𝑡

0
𝐹3(𝐾, 𝐼(𝑡))𝑑𝑘                             

𝐻(𝑡) − 𝐻(0) = 
1

⎾𝜈(𝑡)
∫ (𝑡 − 𝑘)𝜈(𝑡)−1𝑡

0
𝐹4(𝐾,𝐻(𝑡))𝑑𝑘      

We will show that the kernel 𝐹𝑖 for 𝑖 = 1,2,3,4 follows the Lipschitz condition and contraction. 

Theorem 4.1.1 

The kernel 𝐹𝑖(𝐾, 𝐺)  for  𝑖 = 1,2,3,4  satisfies Lipschitz condition and contraction if the following 

inequality 0 ≤ 𝑟𝑖 < 1 holds. 

Proof 

Consider two functions 𝐺 𝑎𝑛𝑑 �̅� 

‖𝐹1(𝑡, 𝐺) − 𝐹1(𝑡, �̅�)‖ 

≤ 𝑝1
ν(t)‖𝐶‖‖𝐺 − �̅�‖ − 𝑝2

ν(t)‖𝐺 − �̅�‖ ≤ [𝑝1
ν(t)𝑢2 − 𝑝2

ν(t)]‖𝐺 − �̅�‖ 

≤ 𝑟1‖𝐺 − �̅�‖                                                                                             (4) 



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where ‖𝐺(𝑡)‖ ≤ 𝑢1 , ‖𝐶(𝑡)‖ ≤ 𝑢2 , ‖𝐼(𝑡)‖ ≤ 𝑢3,   ‖𝐻(𝑡)‖ ≤ 𝑢4 and 𝑟1 = 𝑝1
ν(t)𝑢2 − 𝑝2

ν(t) are 

positive constants .As a result, the Lipschitz condition is met for 𝑟1and if 0 ≤ 𝑟1 < 1,then 𝑟1follows 

contraction. Similarly, it can be exihibited and demonstrated in the other equations as follows  

‖𝐹2(𝑡, 𝐶) − 𝐹2(𝑡, 𝐶̅)‖  ≤  𝑟2‖𝐶 − 𝐶̅‖ 

‖𝐹3(𝑡, 𝐼) − 𝐹3(𝑡, 𝐼)̅‖  ≤  𝑟3‖𝐼 − 𝐼‖̅ 

‖𝐹4(𝑡, 𝐻) − 𝐹3(𝑡, �̅�)‖ ≤  𝑟4‖𝐻 − �̅�‖ 

Therefore  𝐹𝑖 satisfies Lipschitz condition. Also, if  0 ≤ 𝑟𝑖 < 1, then the kernels follows contractions. 

From system (3), the recurrent form can be written as follows 

𝛷1𝑛 = 𝐺𝑛(𝑡) − 𝐺𝑛−1(𝑡)     =
1

⎾𝜈(𝑡)
∫(𝑡 − 𝑘)𝜈(𝑡)−1

𝑡

0

[𝐹1(𝐾, 𝐺𝑛−1) − 𝐹1(𝐾, 𝐺𝑛−2)]𝑑𝑘 

𝛷2𝑛 = 𝐶𝑛(𝑡) − 𝐶𝑛−1(𝑡)    =
1

⎾𝜈(𝑡)
∫(𝑡 − 𝑘)𝜈(𝑡)−1

𝑡

0

[𝐹2(𝐾, 𝐶𝑛−1) − 𝐹2(𝐾, 𝐶𝑛−2)]𝑑𝑘 

𝛷3𝑛 = 𝐼𝑛(𝑡) − 𝐼𝑛−1(𝑡)     =
1

⎾𝜈(𝑡)
∫ (𝑡 − 𝑘)𝜈(𝑡)−1𝑡

0
[𝐹3(𝐾, 𝐼𝑛−1) − 𝐹3(𝐾, 𝐼𝑛−2)]𝑑𝑘 

𝛷4𝑛 = 𝐻𝑛(𝑡) − 𝐻𝑛−1(𝑡)  =
1

⎾𝜈(𝑡)
∫(𝑡 − 𝑘)𝜈(𝑡)−1

𝑡

0

[𝐹4(𝐾, 𝐻𝑛−1) − 𝐹4(𝐾,𝐻𝑛−2)]𝑑𝑘 

Using the initial conditions 

𝐺(𝑡) = 𝐺(0),𝐶(𝑡) = 𝐶(0),  𝐼(𝑡) = 𝐼(0), 𝐻(𝑡) = 𝐻(0) for the above equation and taking norm, we 

get  

‖𝛷1𝑛(𝑡)‖ = ‖𝐺𝑛(𝑡) − 𝐺𝑛−1(𝑡)‖ = ‖
1

⎾𝜈(𝑡)
∫(𝑡 − 𝑘)𝜈(𝑡)−1

𝑡

0

[𝐹1(𝐾, 𝐺𝑛−1) − 𝐹1(𝐾, 𝐺𝑛−2)]𝑑𝑘‖ 

Now using Lipschitz condition in the above equation, we obtain 

‖𝛷1𝑛(𝑡)‖ ≤
𝑟1

⎾𝜈(𝑡)
∫‖𝛷1(𝑛−1)(𝑘)‖𝑑𝑘

𝑡

0

 

Similarly, 

‖𝛷2𝑛(𝑡)‖ ≤
𝑟2

⎾𝜈(𝑡)
∫‖𝛷2(𝑛−1)(𝑘)‖𝑑𝑘

𝑡

0

 

‖𝛷3𝑛(𝑡)‖ ≤
𝑟3

⎾𝜈(𝑡)
∫ ‖𝛷3(𝑛−1)(𝑘)‖𝑑𝑘

𝑡

0
           

‖𝛷4𝑛(𝑡)‖ ≤
𝑟4

⎾𝜈(𝑡)
∫ ‖𝛷4(𝑛−1)(𝑘)‖𝑑𝑘

𝑡

0
                                              (5)  



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which implies that it can be written as 

𝐺𝑛(𝑡) = ∑ 𝛷1𝑖(𝑡)
𝑛
𝑖=1  ; 𝐶𝑛(𝑡) = ∑ 𝛷2𝑖

𝑛
𝑖=1 (𝑡), 𝐼𝑛(𝑡) = ∑ 𝛷3𝑖(𝑡)

𝑛
𝑖=1 , 𝐻𝑛(𝑡) = ∑ 𝛷4𝑖(𝑡)

𝑛
𝑖=1  

Theorem 4.1.2 

The Liouville -Caputo model (3) has system of solutions if there exists t >1 such that 
𝑟𝑖𝑡

⎾𝜈(𝑡)
≤ 1 for 

i=1, 2, 3, 4 

Proof 

Consider, 

‖𝛷1𝑛(𝑡)‖ ≤
𝑟1

⎾𝜈(𝑡)
∫‖𝛷1(𝑛−1)(𝑘)‖𝑑𝑘

𝑡

0

 

Replacing n by n-1 in the above inequality 

‖𝛷1(𝑛−1)(𝑡)‖ ≤
𝑟1

⎾𝜈(𝑡)
∫ ‖𝛷1(𝑛−2)(𝑘)‖𝑑𝑘

𝑡

0
   ≤ [

𝑟1

⎾𝜈(𝑡)
]
2

∫ ‖𝛷1(𝑛−2)(𝑘)‖𝑑𝑘
𝑡

0
 

Again, replacing n by n-2 in the given inequality  

‖𝛷1(𝑛−2)(𝑡)‖ ≤  [
𝑟1

⎾𝜈(𝑡)
]
3

∫‖𝛷1(𝑛−3)(𝑘)‖𝑑𝑘

𝑡

0

 

On substituting in this way and use the initial condition  

We obtain 

‖𝛷1𝑛(𝑡)‖ ≤ ‖𝐺𝑛(0)‖ [
𝑟1𝑡

⎾𝜈(𝑡)
]
𝑛

 

Similarly, we get ν(t) 

‖𝛷2𝑛(𝑡)‖ ≤ ‖𝐶𝑛(0)‖ [
𝑟2𝑡

⎾𝜈(𝑡)
]
𝑛

 

‖𝛷3𝑛(𝑡)‖ ≤ ‖𝐼𝑛(0)‖ [
𝑟3𝑡

⎾𝜈(𝑡)
]
𝑛

 

‖𝛷4𝑛(𝑡)‖ ≤ ‖𝐻𝑛(0)‖ [
𝑟4𝑡

⎾𝜈(𝑡)
]
𝑛

 

This result proved the existence and continuity of solutions 

To show that G(t) , C(t) , I(t)  and H(t) are the solutions of (3) ,we consider the following equations 

𝐺(𝑡) − 𝐺(0) = 𝐺𝑛(𝑡) − 𝑅1𝑛(𝑡) 

𝐶(𝑡) − 𝐶(0) = 𝐶𝑛(𝑡) − 𝑅2𝑛(𝑡)                                                                                                            

𝐼(𝑡) − 𝐼(0) = 𝐼𝑛(𝑡) − 𝑅3𝑛(𝑡)                                                                                         (6)                   



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𝐻(𝑡) − 𝐻(0) = 𝐻𝑛(𝑡) − 𝑅4𝑛(𝑡)            

‖𝑅1𝑛(𝑡)‖ = ‖
1

⎾𝜈(𝑡)
∫[𝐹1(𝐾, 𝐺𝑛) − 𝐹1(𝐾, 𝐺𝑛−1)]

𝑡

0

𝑑𝑘‖ 

                ≤
1

⎾𝜈(𝑡)
∫ ‖[𝐹1(𝐾, 𝐺𝑛) − 𝐹1(𝐾, 𝐺𝑛−1)]‖

𝑡

0
𝑑𝑘 

                ≤
1

⎾𝜈(𝑡)
𝑟1‖𝐺𝑛 − 𝐺𝑛−1‖𝑡          

Applying the above process recursively, 

‖𝑅1𝑛(𝑡)‖ = [
𝑟1𝑡

⎾𝜈(𝑡)
]
𝑛+1

. 𝑀 

where M is the Lipschitz constant 

when 𝑛 → ∞, ‖𝑅1𝑛(𝑡)‖ → 0 

similarly we prove for  

‖𝑅2𝑛(𝑡)‖ → 0, ‖𝑅3𝑛(𝑡)‖ → 0 𝑎𝑛𝑑‖𝑅4𝑛(𝑡)‖ → 0  as 𝑛 → ∞ 

Hence the proof. 

Theorem 4.1.3 

If the condition  [1 −
𝑟𝑖𝑡

⎾𝜈(𝑡)
] ≥ 0 , for 𝑖 = 1,2,3,4  holds then Caputo model have unique solution. 

Proof  

To establish the uniqueness for a solution of the system (3), consider the different set of solutions for 

the system (3), say 𝐺 ̅, 𝐶 ̅  𝐼 ̅ 𝑎𝑛𝑑   𝐻 ̅̅ ̅ . Then as an outcome of the first equation of (3), we write 

𝐺(𝑡) − �̅�(𝑡) =
1

⎾𝜈(𝑡)
∫[𝐹1(𝐾, 𝐺) − 𝐹1(𝐾, �̅�)]

𝑡

0

𝑑𝑘 

Using the norm of above equation 

‖𝐺(𝑡) − �̅�(𝑡)‖ = ‖
1

⎾𝜈(𝑡)
∫[𝐹1(𝐾, 𝐺) − 𝐹1(𝐾, �̅�)]

𝑡

0

𝑑𝑘‖ 

Now by applying Lipschitz condition, 

‖𝐺(𝑡) − �̅�(𝑡)‖ =
1

⎾𝜈(𝑡)
𝑟1𝑡‖𝐺(𝑡) − �̅�(𝑡)‖          

Consequently, 

‖𝐺(𝑡) − �̅�(𝑡)‖ −
1

⎾𝜈(𝑡)
𝑟1𝑡‖𝐺(𝑡) − �̅�(𝑡)‖ ≤  0 



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‖𝐺(𝑡) − �̅�(𝑡)‖[1 −
1

⎾𝜈(𝑡)
𝑟1𝑡] ≤ 0                                                  (7)  

Since  [1 −
1

⎾𝜈(𝑡)
𝑟𝑖𝑡] > 0 ,  

equation (7) becomes the form 

  ‖𝐺(𝑡) − �̅�(𝑡)‖ = 0 

Therefore,  𝐺(𝑡) = �̅�(𝑡) 

similarly, we prove 

𝐶(𝑡) = 𝐶̅(𝑡) , 𝐼(𝑡) = 𝐼(̅𝑡) and 𝐻(𝑡) = �̅�(𝑡) 

4.2 Existence and Uniqueness of Fractional solutions by the Caputo-Fabrizio model 

Let us construct the system (2) in the sense of Caputo-Fabrizio, we have 

𝐷𝑡
𝜈(𝑡)

0
𝐶𝐹 [𝐺(𝑡)] = 𝐹1(𝑡, 𝐺) = 𝛬 + 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) − 𝑝2
ν(t)𝐺(𝑡)  

𝐷𝑡
𝜈(𝑡)

0
𝐶𝐹 [𝐶(𝑡)] = 𝐹2(𝑡, 𝐶) = 𝛺 − 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) +  𝑝3
ν(t)𝐻(𝑡)𝐶(𝑡) − 𝑝4

ν(t)𝐶(𝑡)    (8) 

𝐷𝑡
𝜈(𝑡)

0
𝐶𝐹 [𝐼(𝑡)] = 𝐹3(𝑡, 𝐼) = 𝑝5

ν(t)𝐺(𝑡)𝐼(𝑡) − 𝑝6
ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝7

ν(t)𝐼(𝑡)                           

𝐷𝑡
𝜈(𝑡)

0
𝐶𝐹 [𝐻(𝑡)] = 𝐹4(𝑡, 𝐻) = 𝛹 + 𝑝6

ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝8
ν(t)𝐻(𝑡) 

The Caputo-Fabrizio integral form of the above system is 

𝐺(𝑡) − 𝐺(0) = 
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝐹1(𝑡, 𝐺) +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ 𝐹1(𝜏, 𝐺)𝑑𝜏

𝑡

0
 

𝐶(𝑡) − 𝐶(0) =
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝐹2(𝑡, 𝑋) +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ 𝐹2(𝜏, 𝐶)𝑑𝜏

𝑡

0
                                                                        

𝐼(𝑡) − 𝐼(0) = 
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝐹3(𝑡, 𝐼) +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ 𝐹3(𝜏, 𝐼)𝑑𝜏

𝑡

0
                         (9) 

𝐻(𝑡) − 𝐻(0) = 
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝐹3(𝑡, 𝐻) +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ 𝐹4(𝜏, 𝐻)𝑑𝜏

𝑡

0
 

Here we have to prove the kernel 𝐹𝑖 for 𝑖 = 1,2,3,4 follows the Lipschitz condition and a contraction. 

Theorem 4.2.1 

The kernel 𝐹𝑖(𝜏, 𝐺), for  𝑖 = 1,2,3,4 satisfies the Lipschitz condition and a contraction if the 

following inequality 0 ≤ 𝜌𝑖 < 1 holds. 

Proof  

This theorem is proved as similar as theorem 4.1.1 

The recurrent form of (9) for the first equation is 

𝜓1𝑛 = 𝐺𝑛(𝑡) − 𝐺𝑛−1(𝑡) 



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          =
1 − 𝜈(𝑡)

𝑀(𝜈(𝑡))
[𝐹1(𝑡, 𝐺𝑛−1) − 𝐹1(𝑡, 𝐺𝑛−2)] +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫[𝐹1(𝜏, 𝐺𝑛−1) − 𝐹1(𝜏, 𝐺𝑛−2)]𝑑𝜏

𝑡

0

 

Similarly  𝜓2𝑛 and 𝜓3𝑛 are also be derived 

Using the initial condition and taking norm, we get 

‖𝜓1𝑛‖ ≤
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
‖[𝐹1(𝑡, 𝐺𝑛−1) − 𝐹1(𝑡, 𝐺𝑛−2)]‖ +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ ‖[𝐹1(𝜏, 𝐺𝑛−1) − 𝐹1(𝜏, 𝐺𝑛−2)]‖

𝑡

0
𝑑𝜏 

Since 𝜌1satisfies Lipschitz condition 

‖𝜓1𝑛(𝑡)‖ ≤
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1‖[𝜓1(𝑛−1)(𝑡)]‖ + 

𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 ∫ ‖𝜓1(𝑛−1)(𝜏)‖

𝑡

0
𝑑𝜏                                        (10)  

Similarly ‖𝜓2𝑛(𝑡)‖ , ‖𝜓3𝑛(𝑡)‖  and ‖𝜓4𝑛(𝑡)‖can also be obtained. Therefore, 

𝐺𝑛(𝑡) = ∑ 𝜓1𝑖(𝑡) ,
𝑛
𝑖=1   𝐶𝑛(𝑡) = ∑ 𝜓2𝑖(𝑡) ,

𝑛
𝑖=1  𝐼𝑛(𝑡) = ∑ 𝜓3𝑖(𝑡) 

𝑛
𝑖=1 , 𝐻𝑛(𝑡) = ∑ 𝜓4𝑖(𝑡) 

𝑛
𝑖=1  

Theorem  4.2.2  

The Caputo Fabrizio fractional derivative model (8) has system of solutions if there exists  𝑣 > 1 

such that     

  [
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌𝑖 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌𝑖 𝑣] ≤ 1 , for i=1,2,3,4 

Proof 

Operating (10) recursively and using the initial conditions, we have 

‖𝜓1𝑛(𝑡)‖ ≤ ‖𝐺𝑛(0)‖ [
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1𝑣]

𝑛

                  

Similarly, we have for ‖𝜓2𝑛(𝑡)‖, ‖𝜓3𝑛(𝑡)‖ and ‖𝜓4𝑛(𝑡)‖this result proved the existence and 

continuity of solution. 

To show that G(𝑡), 𝐶(𝑡), 𝐼(𝑡)  and 𝐻(𝑡) are solutions of (8), 

consider, 𝐺(𝑡) − 𝐺(0) = 𝐺𝑛(𝑡) − 𝐷1𝑛(𝑡) 

               𝐶(𝑡) − 𝐶(0) = 𝐶𝑛(𝑡) − 𝐷2𝑛(𝑡)                 

               𝐼(𝑡) − 𝐼(0) = 𝐼𝑛(𝑡) − 𝐷3𝑛(𝑡) 

              𝐻(𝑡) − 𝐻(0) = 𝐻𝑛(𝑡) − 𝐷3𝑛(𝑡) 

Now, 

‖𝐷1𝑛(𝑡)‖ ≤
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
‖[𝐹1(𝑡, 𝐺𝑛) − 𝐹1(𝑡, 𝐺𝑛−1)]‖    +

𝜈(𝑡)

𝑀(𝜈(𝑡))
∫ ‖[𝐹1(𝜏, 𝐺𝑛) − 𝐹1(𝜏, 𝐺𝑛−1)]‖

𝑡

0
𝑑𝜏 

                ≤  
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1‖𝐺𝑛 − 𝐺𝑛−1‖ +

𝜈(𝑡)

𝑀(𝜈(𝑡))
 𝜌1  ‖𝐺𝑛 − 𝐺𝑛−1‖𝑣 

              ≤ [
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
 𝜌1𝑣] ‖𝐺𝑛 − 𝐺𝑛−1‖ 

Applying the above process recursively 



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‖𝐷1𝑛(𝑡)‖ ≤ [
1 − 𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
 𝜌1𝑣]

𝑛+1

. 𝑆 

where S is the Lipschitz constant, when 𝑛 → ∞ , ‖𝐷1𝑛‖ → 0    

Similarly, we prove for‖𝐷2𝑛‖ → 0 , ‖𝐷3𝑛‖ → 0 and  ‖𝐷4𝑛‖ → 0 as 𝑛 → ∞ 

Hence the proof 

Theorem 4.2.3 

If the condition [1 − [
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌𝑖 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
  𝜌𝑖𝑣]] ≥ 0  , for i=1,2,3,4  holds then the Caputo-Fabrizio 

fractional derivative model have unique solutions. 

Proof 

Suppose the system (8) has another solution �̅�, 𝐶̅, 𝐼,̅ 𝑎𝑛𝑑 �̅� 

 𝐺(𝑡) − �̅�(𝑡) =
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
[𝐹1(𝑡, 𝐺) − 𝐹1(𝑡, �̅�)]  +

𝜈(𝑡)

𝑀(𝜈(𝑡))
 ∫ [𝐹1(𝜏, 𝐺) − 𝐹1(𝜏, �̅�)]

𝑡

0
𝑑𝜏 

Using norm and applying  Lipschitz condition 

‖𝐺(𝑡) − �̅�(𝑡)‖ ≤ [
1 − 𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
  𝜌1𝑣] ‖𝐺(𝑡) − �̅�(𝑡)‖ 

Consequently, we have 

‖𝐺(𝑡) − �̅�(𝑡)‖ [1 − [
1 − 𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌1 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
  𝜌1𝑣]] ≤ 0 

Since [1 − [
1−𝜈(𝑡)

𝑀(𝜈(𝑡))
𝜌𝑖 +

𝜈(𝑡)

𝑀(𝜈(𝑡))
  𝜌𝑖𝑣]] > 0,we have 

‖𝐺(𝑡) − �̅�(𝑡)‖ = 0 

Therefore      𝐺(𝑡) = �̅�(𝑡) 

Similarly we prove  

𝐶(𝑡) = 𝐶̅(𝑡) , 𝐼(𝑡) = 𝐼(̅𝑡) and 𝐻(𝑡) = �̅�(𝑡) 

Hence the proof 

4.3 Existence and Uniqueness of solutions for the Atangana-Baleanu fractional model 

Let us construct (2) in Atangana Baleanu fractional derivative in Caputo sense 

𝐷𝑡
𝜈(𝑡)

0
𝐴𝐵 [𝐺(𝑡)] = 𝐹1(𝑡, 𝐺) = 𝛬 + 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) − 𝑝2
ν(t)𝐺(𝑡)  

𝐷𝑡
𝜈(𝑡)

0
𝐴𝐵 [𝐶(𝑡)] = 𝐹2(𝑡, 𝐶)  = 𝛺 − 𝑝1

ν(t)𝐺(𝑡)𝐶(𝑡) +  𝑝3
ν(t)𝐻(𝑡)𝐶(𝑡) − 𝑝4

ν(t)𝐶(𝑡) 

𝐷𝑡
𝜈(𝑡)

0
𝐴𝐵 [𝐼(𝑡)] = 𝐹3(𝑡, 𝐼) =  𝑝5

ν(t)𝐺(𝑡)𝐼(𝑡) − 𝑝6
ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝7

ν(t)𝐼(𝑡) 

𝐷𝑡
𝜈(𝑡)

0
𝐴𝐵 [𝐻(𝑡)] = 𝐹4(𝑡, 𝐻)   = 𝛹 + 𝑝6

ν(t)𝐻(𝑡)𝐼(𝑡) − 𝑝8
ν(t)𝐻(𝑡)                                                 (11)       



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The Atangana-Baleanu integral form of the above system is 

𝐺(𝑡) − 𝐺(0) = 
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝐹1(𝑡, 𝐺) +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
  ∫ (𝑡 − 𝛾)𝜈(𝑡)−1𝐹1(𝛾, 𝐺)𝑑𝛾

𝑡

0
 

𝐶(𝑡) − 𝐶(0) =
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝐹2(𝑡, 𝐶) +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 ∫ (𝑡 − 𝛾)𝜈(𝑡)−1𝐹2(𝛾, 𝐶)𝑑𝛾

𝑡

0
                                (12)                                           

 𝐼(𝑡) − 𝐼(0) = 
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝐹3(𝑡, 𝐼) +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫ (𝑡 − 𝛾)𝜈(𝑡)−1𝐹3(𝛾, 𝐼)𝑑𝛾

𝑡

0
    

𝐻(𝑡) − 𝐻(0) = 
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝐹4(𝑡, 𝐻) +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫ (𝑡 − 𝛾)𝜈(𝑡)−1𝐹4(𝛾, 𝐻)𝑑𝛾

𝑡

0
 

Now to prove the kernel 𝐹𝑖 for 𝑖 = 1,2,3,4 follows the Lipschitz condition and contraction. 

Theorem 4.3.1 

 The kernel 𝐹𝑖(𝛾, 𝐺), for 𝑖 = 1,2,3,4 satisfies the Lipschitz condition and contraction if 0 ≤ 𝛿𝑖 < 1 

holds. 

Proof  

The proof is similar to the proof of 4.1.1. Now the recurrent form of (12) is   𝜃1𝑛 = 𝐺𝑛(𝑡) − 𝐺𝑛−1(𝑡)       

 =
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
[𝐹1(𝑡, 𝐺𝑛−1) − 𝐹1(𝑡, 𝐺𝑛−2)] +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫ (𝑡 − 𝛾)𝜈(𝑡)−1[𝐹1(𝛾, 𝐺𝑛−1) −

𝑡

0

 𝐹1(𝛾, 𝐺𝑛−2)]𝑑𝛾         

Similarly  𝜃2𝑛 , 𝜃3𝑛 and 𝜃4𝑛are also be derived. Using the initial condition and taking norm, we get 

‖𝜃1𝑛‖ ≤
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
‖[𝐹1(𝑡, 𝐺𝑛−1) − 𝐹1(𝑡, 𝐺𝑛−2)]‖ +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫ (𝑡 − 𝛾)𝜈(𝑡)−1‖[𝐹1(𝛾, 𝐺𝑛−1) −

𝑡

0

𝐹1(𝛾, 𝐺𝑛−2)]‖ 𝑑𝛾 

Since 𝛿1 satisfies Lipschitz condition 

‖𝜃1𝑛‖ ≤
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1‖𝜃1(𝑛−1)‖  +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
𝛿1 ∫ (𝑡 − 𝛾)𝜈(𝑡)−1‖𝜃1(𝑛−1)(𝛾)‖

𝑡

0
𝑑𝛾 

Similarly for‖𝜃2𝑛‖, ‖𝜃3𝑛‖ and  ‖𝜃4𝑛‖ 

which implies that it can be written as 

𝐺𝑛(𝑡) = ∑ 𝜃1𝑖(𝑡) ,
𝑛
𝑖=1 𝐶𝑛(𝑡) = ∑ 𝜃2𝑖(𝑡) ,

𝑛
𝑖=1 𝐼𝑛(𝑡) = ∑ 𝜃3𝑖(𝑡) 

𝑛
𝑖=1 ,𝐻𝑛(𝑡) = ∑ 𝜃4𝑖(𝑡) 

𝑛
𝑖=1                (13)      

Theorem 4.3.2 

The Atangana- Baleanu derivative model (11) have system of solutions, if there exists  𝜇 >1such that 

 [
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿𝑖 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
𝛿𝑖𝜇] ≤ 1 𝑓𝑜𝑟  𝑖 = 1,2,3,4 

Proof  Consider, 

‖𝜃1𝑛‖ ≤ ‖𝐺𝑛(0)‖ [
1 − 𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿𝑖 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
𝛿𝑖𝜇]

𝑛

 

Similarly,‖𝜃2𝑛‖ , ‖𝜃3𝑛‖ and  ‖𝜃4𝑛‖ can also be obtained 



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These results proved the existence and continuity of solution. 

Now to show that  𝐺(𝑡), 𝐶(𝑡), 𝐼(𝑡) and 𝐻(𝑡) are solutions of (11) 

Consider 

𝐺(𝑡) − 𝐺(0) = 𝐺𝑛(𝑡) − 𝐸1𝑛(𝑡) 

𝐶(𝑡) − 𝐶(0) = 𝐶𝑛(𝑡) − 𝐸2𝑛(𝑡)                 

𝐼(𝑡) − 𝐼(0) = 𝐼𝑛(𝑡) − 𝐸3𝑛(𝑡) 

𝐻(𝑡) − 𝐻(0) = 𝐻𝑛(𝑡) − 𝐸4𝑛(𝑡) 

Now 

‖𝐸1𝑛(𝑡)‖ ≤
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
‖[𝐹1(𝑡, 𝐺𝑛) − 𝐹1(𝑡, 𝐺𝑛−1)]‖  +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫ (𝑡 − 𝛾)𝛼(𝑡)−1‖[𝐹1(𝛾, 𝐺𝑛) −

𝑡

0

𝐹1(𝛾, 𝐺𝑛−1)]‖ 𝑑𝛾 

  ≤  
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1‖𝐺𝑛 − 𝐺𝑛−1‖ +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
  𝛿1‖𝐺𝑛 − 𝐺𝑛−1‖𝜇 

  ≤ [
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿1𝜇] ‖𝐺𝑛 − 𝐺𝑛−1‖ 

Applying the above process recursively 

‖𝐸1𝑛(𝑡)‖ ≤ [
1 − 𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿1𝜇]

𝑛+1

.𝑊 

where W is the Lipschitz constant 

when 𝑛 → ∞ , ‖𝐸1𝑛‖ → 0    

Similarly we prove for ‖𝐸2𝑛‖ → 0   , ‖𝐸3𝑛‖ → 0    and ‖𝐸4𝑛‖ → 0    as 𝑛 → ∞ 

 Hence the proof. 

Theorem 4.3.3 

If the condition [1 − [
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿𝑖 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿𝑖𝜇]] ≥ 0   For i=1,2,3,4  holds then the Atangana-

Baleanu fractional derivative model have unique solutions. 

Proof 

Suppose the system (16) has another solution 𝐺 ̅, 𝐶 ̅ , 𝐼 ̅  and 𝐻 ̅̅ ̅then 

𝐺(𝑡) − �̅�(𝑡) =
1 − 𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
[𝐹1(𝑡, 𝐺) − 𝐹1(𝑡, �̅�)]

+
𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
∫(𝑡 − 𝛾)𝛼(𝑡)−1[𝐹1(𝛾, 𝐺) − 𝐹1(𝛾, �̅�)]𝑑𝛾

𝑡

0

 

Using norm and apply Lipschitz condition 



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‖𝐺(𝑡) − �̅�(𝑡)‖ ≤ ([
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿1𝜇] ‖𝐺(𝑡) − �̅�(𝑡)‖) 

Consequently we have  

[1 − [
1 − 𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿1 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿1𝜇]] ‖𝐺(𝑡) − �̅�(𝑡)‖ ≤ 0 

Since[1 − [
1−𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))
𝛿𝑖 +

𝜈(𝑡)

𝐴𝐵(𝜈(𝑡))⎾𝜈(𝑡)
 𝛿𝑖𝜇]] > 0,we have 

‖𝐺(𝑡) − �̅�(𝑡)‖ = 0 

 Therefore,         𝐺(𝑡) = �̅�(𝑡) 

Similarly ,we prove  

𝐶(𝑡) = 𝐶̅(𝑡) , 𝐼(𝑡) = 𝐼(̅𝑡) and 𝐻(𝑡) = 𝐻(𝑡) 

Hence the proof 

5. Numerical Scheme 

In this section Numerical scheme [31] is considered in the sense of Liouville-Caputo, Caputo-

Fabrizio and Atangana –Baleanu fractional derivatives. 

Let us consider our fractional model as 

𝐷𝑡
𝛼

0
∗ 𝑢(𝑡) =𝑓(𝑡, 𝑢(𝑡))      

Where * denotes LC, CF and AB terms and 𝑢(𝑡) = (𝐺(𝑡), 𝐶(𝑡), 𝐼(𝑡), 𝐻(𝑡)). 

Now we use the numerical scheme [31] represented for Liouville-Caputo (14), Caputo-Fabrizio(15) 

and Atangana-Baleanu(16) fractional derivatives in (2) 

𝑢𝑛+1(𝑡) = 𝑢(0) +
1

⎾ν(t)
∑

(

 
 
 

ℎν(t)𝑓(𝑡𝑚 ,𝑢𝑚)

ν(t)(ν(t)+1)
 

((𝑛 − 𝑚 +  2 +  2𝛼) −
ℎν(t)𝑓(𝑡𝑚−1 ,𝑢𝑚−1)

ν(t)(ν(t)+1)

(
(𝑛 + 1 −  𝑚)ν(t)+1 −

(𝑛 − 𝑚)ν(t)(𝑛 − 𝑚 + 1 + ν(t)
)

)

 
 
 

𝑛
𝑚=0                        (14)                               

  (𝑢𝑛+1) = (𝑢𝑛) + 

  [
(2−ν(t))(1−ν(t))

2
+

3ℎ

4
ν(t)(2 − ν(t))] 𝑓(𝑡𝑛, 𝑢𝑛) − [

(2−ν(t))(1−ν(t))

2
+

ℎ

4
ν(t) (2 − ν(t))

] 𝑓(𝑡𝑛−1, 𝑢𝑛−1)   

𝑢𝑛+1(𝑡) = 𝑢(0) +
⎾ν(t)(1−ν(t))

⎾ν(t)(1−ν(t))+ν(t)
𝑓(𝑡𝑛, 𝑢𝑛)                                                                               (15) 

 



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 +
1

(ν(t)+1)((1−ν(t))⎾ν(t))+ν(t)
∑

(

 
 
 
 
 

ℎν(t)𝑓(𝑡𝑚 , 𝑢𝑚)

(
(𝑛 + 1 − 𝑚)ν(t) (𝑛 − 𝑚 + 2 + ν(t))

− (𝑛 − 𝑚)ν(t)(𝑛 − 𝑚 + 2 + 2𝛼(𝑡)
)

−ℎν(t)𝑓(𝑡𝑚−1 , 𝑢𝑚−1)

(
(𝑛 + 1 −  𝑚)ν(t)+1

− (𝑛 − 𝑚)ν(t)  (𝑛 − 𝑚 + 1 + ν(t))
)

)

 
 
 
 
 

𝑛
𝑚=0                   (16) 

6. Results and Discussions 

The primary aim of our model is to effectively address the adverse effects of greenhouse gases 

(GHGs) by integrating them into green building systems using a fractional model. Existence and 

uniqueness of the fractional operators are established.  Numerical or computer simulations are 

considered the most appropriate method to accurately represent the solution. We employed 

MATLAB R2023a programming language to conduct numerical simulations of the proposed 

fractional model. These simulations aim to demonstrate the dynamic behavior of green building 

dynamics in order to mitigate GHG emissions and gain a comprehensive understanding of the impact 

on human communities. 

Human communities are significant contributors to unwanted GHG emissions. The rise in these 

emissions due to various human activities poses a threat to the well-being of all organisms. The 

continuous increase in the human population leads to higher levels of these harmful gases. In cases 

of rapid emission of these gases by human societies, green constructions utilize their advanced 

technologies to tackle the issue. Consequently, green buildings offer a range of advantages while 

effectively addressing the problem of GHGs. By incorporating sustainable practices, energy-efficient 

designs, and innovative technologies, these buildings help to create a more environmentally friendly 

and sustainable built environment. These energy-efficient structures also provide benefits to human 

societies. 

The fractional order model is ideal for illustrating the memory effect/history of increasing GHGs on 

green building systems to enhance their efficiency. 

Case 1(Variable-order case): The rate of green building projects varies due to a multitude of 

factors. These include the progression of eco-friendly building initiatives, government policies, 

public awareness, and technological advancements. Additionally, economic incentives, market 

demand, industry standards, access to resources, regulatory environment, collaboration, and public 

perception play crucial roles. Variable order study of the green building concept allows for a more 

comprehensive understanding of its dynamics and complexities.  

The variable order of  𝐺(𝑡) can be represented as ν(t)= L/(1+exp(k(t-t0)) ,where L is the highest 

percentage value of buildings that are deemed to be green, k is the logistic growth rate and t0 is the 

time at which the growth is halfway between its starting and ending values. Various factors, 

including building materials, energy sources, energy efficiency measures, occupant behavior, and 

maintenance practices, may have an impact on the function of greenhouse gas emission rates in green 

buildings.  



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𝐶(𝑡) takes the variable order as a linear function ν(t)= R0-𝜆t, where R0 represents the starting point 

of greenhouse gas emissions and 𝜆 is the decreasing rate of GHGs in green buildings. The usage of 

ingredients in green buildings 𝐼(𝑡)  can be represented in variable order as ν(t)=S0+𝜇t, where S0 

symbolizes the initial adoption of eco-friendly components and 𝜇 denotes the speed of growth in 

usage. These components might include things like renewable energy sources and other elements that 

enhance a building's green features. When new technologies are created, building codes are updated, 

and societal priorities change over time, the function may change. 

The growth rate of a human community in green buildings is determined by various factors, 

including population dynamics, urbanization trends, economic conditions, and the availability of 

green buildings .Here the variable order of 𝐻(𝑡) can be represented as ν(t)=𝜗+α 

exp((ω)(t)(M)),where 𝜗 denotes the baseline growth factor, α represents a coefficient scaling of the 

exponential growth, ω indicates the decay rate coefficient of the exponential growth and M 

represents a coefficient affecting the speed at which the growth function approaches zero as t 

increases. 

 

Fig 1: Comparison graph of 𝐺(𝑡) via LC,CF and AB 

Fig 1 illustrates the growth rate of green buildings with a variable order represented by 

ν(t)=0.90/(1+exp(0.1(t-10)). Here,0.90 represents buildings with the highest percentage of green 

features,0.1 represents the rate of logistic growth and 10 indicates the growth reaches a point midway 

between its initial and final values. It is observed that in all cases of LC,CF and AB the  growth rate 

of green buildings rises accordingly when time increases. 



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Fig 2: Comparison graph of 𝐶(𝑡) via LC, CF and AB 

Fig 2 depicts the change of GHGs rate with a variable order ν(t)= 0.1-0.5t.where 0.1 signifies the 

initial stage of greenhouse gas emissions.0.5 indicates the rate at which greenhouse gas emission 

decreases. By Operating the above function using LC,CF and AB, AB gives declination of GHGs in 

Green buildings more appropriately as in [15] than LC and CF. Gradually, as green technologies and 

practices are adopted and refined, the rate of greenhouse gas emissions declines. 

 

Fig 3: Comparison graph of 𝐼(𝑡)  via LC, CF and AB 

Fig 3 displays the growth rate of ingredients by the variable order ν(t) = 0.5 + 0.05t. Here, 0.5 

represents the initial usage of green ingredients and 0.05 represents the rate of increase in usage. 

Using LC, CF and AB for observing the considered function, the growth rate of ingredients gradually 

rises over time. Also CF, AB have better memory effect than LC. 



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                                                    Fig 4: Comparison graph of 𝐻(𝑡) via LC,CF and AB 

Fig 4 provides a graph of the variable order function ν(t)=1+0.1exp(-(0.1)(t)(0.50)),where 1 

represents the fundamental growth element, 0.1 denotes the growth and decay rate coefficient of 

human community and 0.50 indicates how quickly the growth function approaches zero as time t 

increases. Here the growth rate of human community in green buildings progressively rises over time 

in all cases of LC, CF and AB. 

 

           (a)                 (b) 



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                   (c)                     (d) 

Fig5.Numerical simulation of Green Buildings for various values of ν=0.65,0.75,0.85 and 0.95 in 

Liouville-Caputo sense 

 

      (a)           (b) 



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     (c)          (d) 

Fig 6.Numerical simulation of Green Buildings for various values of ν=0.65,0.75,0.85 and 0.95 in 

Caputo-Fabrizio sense. 

 

                    (a)                      (b) 



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       (c)          (d) 

Fig 7.Numerical simulation of Green Buildings for various values of ν=0.65,0.75,0.85 and 0.95 in 

Atangana-Baleanu sense. 

Case 2(Fractional order case):Fig[5] shows that when ν increases, the growth rate of green 

buildings absorbing greenhouse gases also increases drastically, but in Fig 6 and 7, it increases more 

gradually over time, which is more in line with real world circumstances.In Fig[5-7], GHG emissions 

decrease gradually over time as ν increases. According to Fig 6 and 7, green buildings can produce 

ingredients proportionally with increasing v due to their high absorption of greenhouse gases, 

whereas in Fig 5, as ν close to 1, the production of ingredients reaches a peak and then falls rapidly, 

this is because of LC's singularity behavior. Fig 6 and 7 show that the human community will benefit 

from receiving the ingredients, hence it rises proportionally in cases of CF and AB, whereas in Fig 5, 

we see that for small values of ν , the human community remains stable, which violates the real-time 

scenario. Hence CF and AB approaches give better results than the LC approach. 

Conclusion 

This study developed a fractional variable-order model using Liouville-Caputo, Caputo-Fabrizio, and 

Atangana-Baleanu derivatives to explore sustainable building practices in India, focusing on their 

impact on greenhouse gas (GHG) emissions. Numerical simulations showed that green buildings can 

effectively reduce GHGs, with the Caputo-Fabrizio and Atangana-Baleanu approaches providing 

more realistic and accurate results compared to the Liouville-Caputo method. The findings highlight 

the potential of green buildings in fostering sustainable communities, where gradual GHG absorption 

and the growth of eco-friendly components align with real-world scenarios. 

References 

[1] Mohammad Qasim, Shalilka Mehta, Sandeep Salhotra, “Cost Optimization of a  Building Using Sustainable   

Building Concept,”  IJITEE, Vol-8,Issue-8,2019. 



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