G. Sowmya, Gireesha B.J., Muhammad Ijaz Khan, Shaher Momani and Tasawar Hayat
The purpose of this study is to conduct a numerical computation to analyse the thermal attribute and heat transfer phenomenon of a fully wetted porous fin of a longitudinal…
Abstract
Purpose
The purpose of this study is to conduct a numerical computation to analyse the thermal attribute and heat transfer phenomenon of a fully wetted porous fin of a longitudinal profile. The fin considered is that of a functionally graded material (FGM). Based on the spatial dependency of thermal conductivity, three cases such as linear, quadratic and exponential FGMs are analysed.
Design/methodology/approach
The governing equations are nondimensionalised and solved by applying Runge-Kutta-Fehlberg fourth-fifth order technique.
Findings
The parametric investigation is executed to access the significance of the pertinent parameters on the thermal feature of the fin and heat transmit rate. The outcomes are portrayed in a graphical form.
Originality/value
No such study has yet been published in the literature.
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Iqbal M. Batiha, Adel Ouannas, Ramzi Albadarneh, Abeer A. Al-Nana and Shaher Momani
This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional…
Abstract
Purpose
This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.
Design/methodology/approach
In order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.
Findings
In accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.
Originality/value
The existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.
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Sunil Kumar, R.P. Chauhan, Shaher Momani and Samir Hadid
This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely…
Abstract
Purpose
This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.
Design/methodology/approach
The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.
Findings
Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.
Originality/value
This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.
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Sunil Kumar, Surath Ghosh, Shaher Momani and S. Hadid
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species…
Abstract
Purpose
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.
Design/methodology/approach
This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.
Findings
This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Research limitations/implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Practical implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Social implications
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
Originality/value
The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.
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Sana Abu‐Gurra, Vedat Suat Ertürk and Shaher Momani
The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method…
Abstract
Purpose
The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators.
Design/methodology/approach
The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators.
Findings
The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth‐order Runge‐Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results.
Originality/value
The paper demostrates that a hybrid method of differential transform method, Laplace transform and Padé approximations provides approximate solutions of the oscillatory systems.
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Vedat Suat Erturk and Shaher Momani
The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have constant…
Abstract
Purpose
The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have constant size over the period of the epidemic.
Design/methodology/approach
The differential transform method (DTM) is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations of these problems.
Findings
Results obtained using the scheme presented here agree well with the results obtained by the Adomian decomposition and power series methods. Some plots are presented to show the reliability and simplicity of the method.
Originality/value
This paper is believed to represent a new application for DTM on solving systems of nonlinear ordinary differential equations.
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Shaher Momani and Vedat Suat Ertürk
This paper sets out to study a system of fourth‐order obstacle boundary value problems associated with obstacle, unilateral and contact problems.
Abstract
Purpose
This paper sets out to study a system of fourth‐order obstacle boundary value problems associated with obstacle, unilateral and contact problems.
Design/methodology/approach
Differential transform method was used to solve the system.
Findings
It is demonstrated that the proposed scheme validates for this type of problems.
Originality/value
It is the first time, to the best of one's knowledge, that the method is applied to obstacle boundary value problems. Also, the technique implemented in this study can be used for this type of physical length sensitive problems.
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Vedat Suat Erturk, Ahmet Yıldırım, Shaher Momanic and Yasir Khan
The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the…
Abstract
Purpose
The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.
Design/methodology/approach
The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.
Findings
The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.
Originality/value
Illustrative examples are included to demonstrate the validity and applicability of the technique.