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B S, Srilatha P, Khan U, R NK, Ben Ahmed S, Kumar R. Numerical study of thermal and solutal advancements in ZnO-SAE50 nanolubricant flow past a convergent/divergent channel with the effects of thermophoretic particle deposition. NANOSCALE ADVANCES 2023; 5:6647-6658. [PMID: 38024299 PMCID: PMC10662241 DOI: 10.1039/d3na00816a] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/24/2023] [Accepted: 10/31/2023] [Indexed: 12/01/2023]
Abstract
The present research focuses on the significance of thermophoretic particle deposition on a ZnO-SAE50 nanolubricant flow in a stretchable/shrinkable convergent/divergent channel in the presence of an applied magnetic field and nonlinear heat radiation. A parameter in the governing differential equations and wall boundary conditions defines the physical mechanism of the model. The Galerkin finite element method, combined with similarity transformation, is adopted to solve the governing equations. The Levenberg-Marquardt backpropagating algorithm of an artificial neural network model forecasts heat and mass transfer properties. The results reveal that by stretching/shrinking the walls enough, the classical flow and heat properties are significantly affected. The stretching of the convergent or divergent channel is observed to increase the velocity profiles, whilst shrinking results in backflow regions. In terms of the temperature field, stretching causes more heat to be produced in the flow; nevertheless, the thermal layer is decreased and cooling is attained by channel shrinkage, which may have important technical implications.
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Affiliation(s)
- Shilpa B
- Department of Mathematics, Dayananda Sagar College of Engineering Bengaluru Karnataka India
| | - Pudhari Srilatha
- Department of Mathematics, Institute of Aeronautical Engineering Hyderabad India
| | - Umair Khan
- Department of Computer Science and Mathematics, Lebanese American University Byblos Lebanon
| | - Naveen Kumar R
- Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham Bengaluru India
| | - Samia Ben Ahmed
- Department of Chemistry, College of Sciences, King Khalid University Abha P.O. Box 9004 Saudi Arabia
| | - Raman Kumar
- Department of Mechanical Engineering, University Centre for Research & Development, Chandigarh University Mohali Punjab 140413 India
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Srilatha P, J M, Khan U, Kumar RN, Gowda RJP, Ben Ahmed S, Kumar R. Heat transfer analysis in magnetohydrodynamic nanofluid flow induced by a rotating rough disk with non-Fourier heat flux: aspects of modified Maxwell-Bruggeman and Krieger-Dougherty models. NANOSCALE ADVANCES 2023; 5:5941-5951. [PMID: 37881708 PMCID: PMC10597558 DOI: 10.1039/d3na00711a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/31/2023] [Accepted: 09/20/2023] [Indexed: 10/27/2023]
Abstract
Non-Newtonian fluids have unique heat transfer properties compared to Newtonian fluids. The present study examines the flow of a Maxwell nanofluid across a rotating rough disk under the effect of a magnetic field. Furthermore, the Cattaneo-Christov heat flux model is adopted to explore heat transport features. In addition, a comparison of fluid flow without and with aggregation is performed. Using similarity variables, the governing partial differential equations are transformed into a system of ordinary differential equations, and this system is then solved by employing the Runge-Kutta Fehlberg fourth-fifth order method to obtain the numerical solution. Graphical depictions are used to examine the notable effects of various parameters on velocity and thermal profiles. The results reveal that an increase in the value of Deborah number decreases the velocity profile. An increase in the thermal relaxation time parameter decreases the thermal profile. An artificial neural network is employed to calculate the rate of heat transfer and surface drag force. The R values for skin friction and Nusselt number were computed. The results demonstrate that artificial neural networks accurately predicted skin friction and Nusselt number values.
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Affiliation(s)
- Pudhari Srilatha
- Department of Mathematics, Institute of Aeronautical Engineering Hyderabad-500043 India
| | - Madhu J
- Department of Studies in Mathematics, Davangere University Davangere 577007 India
| | - Umair Khan
- Department of Computer Science and Mathematics, Lebanese American University Byblos Lebanon
| | - R Naveen Kumar
- Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham Bengaluru India
| | - R J Punith Gowda
- Department of Mathematics, Bapuji Institute of Engineering & Technology Davanagere 577004 India
| | - Samia Ben Ahmed
- Department of Chemistry, College of Sciences, King Khalid University P.O. Box 9004 Abha Saudi Arabia
| | - Raman Kumar
- Department of Mechanical Engineering, University Centre for Research & Development, Chandigarh University Mohali-140413 Punjab India
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Roy NC, Ghosh A. Viscoelastic hybrid nanofluid flow over a vertical plate with sinusoidal surface temperature variations. Heliyon 2023; 9:e15703. [PMID: 37180905 PMCID: PMC10173623 DOI: 10.1016/j.heliyon.2023.e15703] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2022] [Revised: 04/11/2023] [Accepted: 04/19/2023] [Indexed: 05/16/2023] Open
Abstract
Natural convection of a viscoelastic hybrid nanofluid along a vertically heated plate with sinusoidal surface temperature variations is investigated. The current investigation explores the non-similar boundary layer flow patterns and heat transfer of second-grade viscoelastic flow of hybrid nanofluid. Effects of magnetic field and thermal radiation are considered. The governing dimensional equations are converted into a non-dimensional form taking suitable transformations. Resulting equations are solved with the aid of finite difference method. It is discovered that the momentum boundary layer lessens while the thermal boundary layer grows for higher radiation parameters, surface temperature parameters, Eckert numbers, magnetic field parameters and amount of nanoparticles. For larger Deborah numbers (De1), shear stress (τ) and heat transfer rate (q) accelerate, but momentum and thermal boundary decline near the leading edge of the vertical plate. However, the effects of Deborah number (De2) show opposite results. Increase in magnetic field parameters causes a reduction in shear stress. The higher volume fraction of nanoparticles (φ1, φ2) enhances q as it was expected. Moreover, τ and q were increased with larger surface temperature parameters and decrease with higher Eckert numbers. This is because higher surface temperature boost up the fluid temperature, but higher Eckert numbers admit the fluid to spread over the surface. An increase in the amplitude of surface temperature oscillation enhances the shear stress and heat transfer rate.
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Qayyum M, Ahmad E, Afzal S, Sajid T, Jamshed W, Musa A, Tag El Din ESM, Iqbal A. Fractional analysis of unsteady squeezing flow of Casson fluid via homotopy perturbation method. Sci Rep 2022; 12:18406. [PMID: 36319834 PMCID: PMC9626585 DOI: 10.1038/s41598-022-23239-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2022] [Accepted: 10/27/2022] [Indexed: 01/24/2023] Open
Abstract
The objective of this article is to model and analyze unsteady squeezing flow of fractional MHD Casson fluid through a porous channel. Casson fluid model is significant in understanding the properties of non-Newtonian fluids such as blood flows, printing inks, sauces and toothpaste etc. This study provides important results as unsteady flow of Casson fluid in fractional sense with aforementioned effects has not been captured in existing literature. After applying similarity transformations along with fractional calculus a highly non-linear fractional-order differential equation is obtained. Modeled equation is then solved along with no-slip boundary conditions through a hybrid of Laplace transform with homotopy perturbation algorithm. For validity purposes, solution and errors at various values in fractional domain are compared with existing results. LHPM results are better in terms of accuracy than other available results in literature. Effects of fractional parameter on the velocity profile, skin friction and behaviors of involved fluid parameters is the focal point of this study. Comprehensive, quantitative and graphical analysis is performed for investigating the effects of pertinent fluid parameters on the velocity profile and skin friction. Analysis revealed that fractional parameter depicts similar effect in case of positive and negative squeeze number. Also, skin friction decreases with an increasing fractional parameter. Moreover, in fractional environment Casson parameter has shown similar effect on the velocity profile in case of positive and negative squeeze number.
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Affiliation(s)
- Mubashir Qayyum
- grid.444797.d0000 0004 0371 6725Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan
| | - Efaza Ahmad
- grid.444797.d0000 0004 0371 6725Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan
| | - Sidra Afzal
- grid.444797.d0000 0004 0371 6725Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan
| | - Tanveer Sajid
- grid.509787.40000 0004 4910 5540Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad, 44000 Pakistan
| | - Wasim Jamshed
- grid.509787.40000 0004 4910 5540Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad, 44000 Pakistan
| | - Awad Musa
- grid.449553.a0000 0004 0441 5588Department of Physics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Aflaj, 11912 Saudi Arabia ,grid.440840.c0000 0000 8887 0449Department of Physics, College of Science, Sudan University of Science and Technology, Khartoum, Sudan
| | - El Sayed M. Tag El Din
- grid.440865.b0000 0004 0377 3762Electrical Engineering, Faculty of Engineering and Technology, Future University in Egypt, New Cairo, 11835 Egypt
| | - Amjad Iqbal
- grid.6979.10000 0001 2335 3149Department of Materials Technologies, Faculty of Materials Engineering, Silesian University of Technology, 44-100 Gliwice, Poland ,grid.8051.c0000 0000 9511 4342CEMMPRE—Centre for Mechanical Engineering Materials and Processes, Department of Mechanical Engineering, University of Coimbra, Rua Luı’s Reis Santos, 3030-788 Coimbra, Portugal
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Saeed M, Abbas T, Mahmood ul Hasan Q, Ahmad B, Khan SU, Rajhi W, Torchani A, Aichouni M, Ezeddini S. Heat and mass transfer inspection for slip flow of radiative Maxwell fluid when role of thermal conductivity and viscosity is variable: A Reynolds viscosity model. J INDIAN CHEM SOC 2022. [DOI: 10.1016/j.jics.2022.100709] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Raza A, Thumma T, Khan SU, Boujelbene M, Boudjemline A, Chaudhry IA, Elbadawi I. Thermal mechanism of carbon nanotubes with Newtonian heating and slip effects: A Prabhakar fractional model. J INDIAN CHEM SOC 2022. [DOI: 10.1016/j.jics.2022.100731] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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Closed-form solution of oscillating Maxwell nano-fluid with heat and mass transfer. Sci Rep 2022; 12:12205. [PMID: 35842498 PMCID: PMC9288494 DOI: 10.1038/s41598-022-16503-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2022] [Accepted: 07/11/2022] [Indexed: 11/08/2022] Open
Abstract
The primary goal of this article is to analyze the oscillating behavior of Maxwell Nano-fluid with regard to heat and mass transfer. Due to high thermal conductivity of engine oil is taken as a base fluid and graphene Nano-particles are introduced in it. Coupled partial differential equations are used to model the governing equations. To evaluate the given differential equations, certain dimensionless factors and Laplace transformations are used. The analytical solution is obtained for temperature, concentration and velocity. The temperature and concentration gradient are also finds to analyze the rate of heat and mass transfer. As a special case, the solution for Newtonian fluid is discussed. Finally, the behaviors of various physical factors are studied graphically for both sine and cosine oscillation and give physical meanings to the parameters.
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8
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Regularity and Travelling Wave Profiles for a Porous Eyring–Powell Fluid with Darcy–Forchheimer Law. Symmetry (Basel) 2022. [DOI: 10.3390/sym14071451] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
The goal of this study is to provide analytical and numerical assessments to a fluid flow based on an Eyring–Powell viscosity term and a Darcy–Forchheimer law in a porous media. The analysis provides results about regularity, existence and uniqueness of solutions. Travelling wave solutions are explored, supported by the Geometric Perturbation Theory to build profiles in the proximity of the equation critical points. Finally, a numerical routine is provided as a baseline for the validity of the analytical approach presented for low Reynolds numbers typical in a porous medium.
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Magnetic Dipole and Thermophoretic Particle Deposition Impact on Bioconvective Oldroyd-B Fluid Flow over a Stretching Surface with Cattaneo–Christov Heat Flux. NANOMATERIALS 2022; 12:nano12132181. [PMID: 35808017 PMCID: PMC9268314 DOI: 10.3390/nano12132181] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Revised: 06/16/2022] [Accepted: 06/23/2022] [Indexed: 01/25/2023]
Abstract
This study emphasizes the performance of two-dimensional electrically non-conducting Oldroyd-B fluid flowing across a stretching sheet with thermophoretic particle deposition. The heat and mass transfer mechanisms are elaborated in the presence of a magnetic dipole, which acts as an external magnetic field. The fluid possesses magnetic characteristics due to the presence of ferrite particles. The gyrotactic microorganisms are considered to keep the suspended ferromagnetic particles stable. Cattaneo–Christov heat flux is cogitated instead of the conventional Fourier law. Further, to strengthen the heat transfer and mass transfer processes, thermal stratification and chemical reaction are employed. Appropriate similarity transformations are applied to convert highly nonlinear coupled partial differential equations into non-linear ordinary differential equations (ODEs). To numerically solve these ODEs, an excellent MATLAB bvp4c approach is used. The physical behavior of important parameters and their graphical representations are thoroughly examined. The tables are presented to address the thermophoretic particle velocity deposition, rate of heat flux, and motile microorganisms’ density number. The results show that the rate of heat transfer decreases as the value of the thermal relaxation time parameter surges. Furthermore, when the thermophoretic coefficient increases, the velocity of thermophoretic deposition decreases.
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Entropy Optimization on Axisymmetric Darcy–Forchheimer Powell–Eyring Nanofluid over a Horizontally Stretching Cylinder with Viscous Dissipation Effect. COATINGS 2022. [DOI: 10.3390/coatings12060749] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The effect of entropy optimization on an axisymmetric Darcy–Forchheimer Powell–Eyring nanofluid flow caused by a horizontally permeable stretching cylinder, as well as non-linear thermal radiation, was investigated in this research work. The leading equations of the current problem were changed into ODEs by exhausting appropriate transformations. To deduce the reduced system, the numerical method bvp4c was used. The outcome of non-dimensional relevant factors on velocity, entropy, concentration, temperature, Bejan number, drag force, and Nusselt number is discussed and demonstrated using graphs and tables. It is perceived that, with a higher value of volume fraction parameter, the skin friction falls down. Likewise, it is found that the Nusselt number drops with enhancing the value of the volume fraction. Moreover, the result reveals that the entropy generation increases as the volume fraction, curvature parameter, and Brinkman number increase.
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11
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Khan Z, Ul Haq S, Ali F, Andualem M. Free convection flow of second grade dusty fluid between two parallel plates using Fick's and Fourier's laws: a fractional model. Sci Rep 2022; 12:3448. [PMID: 35236870 PMCID: PMC8891311 DOI: 10.1038/s41598-022-06153-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2021] [Accepted: 01/17/2022] [Indexed: 11/09/2022] Open
Abstract
The paper aims to investigate the channel flow of second grade visco-elastic fluid generated due to an oscillating wall. The effect of heat and mass transfer has been taken into account. The phenomenon has been modelled in terms of PDEs. The constitutive equations are fractionalized by using the definition of the Caputo fractional operator with Fick’s and Fourier’s Laws. The system of fractional PDEs is non-dimensionalized by using appropriate dimensionless variables. The closed-form solutions of thermal and concentration boundary layers are obtained by using the Laplace and finite Fourier-Sine transforms, while the momentum equation is solved by a numerical approach by Zakian using \documentclass[12pt]{minimal}
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\begin{document}$$\textit{PYTHON}$$\end{document}PYTHON. Furthermore, the parametric influence of various embedded physical parameters on momentum, temperature, and concentration distributions is depicted through various graphs. It is observed that the fractional approach is more convenient and realistic as compared to the classical approach. It is worth noting that the increasing values of \documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document}M, \documentclass[12pt]{minimal}
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\begin{document}$$Sc$$\end{document}Sc and \documentclass[12pt]{minimal}
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\begin{document}$$Re$$\end{document}Re retard the boundary layer profile. For instance, this behaviour of \documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document}M is significant where boundary control is necessary. That is, in the case of resonance, the physical solution may be obtained by adding the effect of MHD. The Reynolds number is useful in characterising the transport properties of a fluid or a particle travelling through a fluid. The Reynolds number is one of the main controlling parameters in all viscous flow. It determines whether the fluid flow is laminar or turbulent. The evolution of the rate of heat, mass transfer, and skin friction on the left plate with various physical parameters are presented in tables. These quantities are of high interest for engineers. Keeping in mind the effect of various parameters on these engineering quantities, they make their feasibility reports.
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Affiliation(s)
- Zahid Khan
- Department of Mathematics, Islamia College Peshawar, Peshawar, 25000, Khyber PakhtunKhwa, Pakistan
| | - Sami Ul Haq
- Department of Mathematics, Islamia College Peshawar, Peshawar, 25000, Khyber PakhtunKhwa, Pakistan
| | - Farhad Ali
- Department of Mathematics, City University of Science & Information Technology, Peshawar, 25000, Pakistan.
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The Impact of Cattaneo–Christov Double Diffusion on Oldroyd-B Fluid Flow over a Stretching Sheet with Thermophoretic Particle Deposition and Relaxation Chemical Reaction. INVENTIONS 2021. [DOI: 10.3390/inventions6040095] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The current study focuses on the characteristics of flow, heat, and mass transfer in the context of their applications. There has been a lot of interest in the use of non-Newtonian fluids in biological and technical disciplines. Having such a substantial interest in non-Newtonian fluids, our goal is to explore the flow of Oldroyd-B liquid over a stretching sheet by considering Cattaneo–Christov double diffusion and heat source/sink. Furthermore, the relaxation chemical reaction and thermophoretic particle deposition are considered in the modelling. The equations that represent the indicated flow are changed to ordinary differential equations (ODEs) by choosing relevant similarity variables. The reduced equations are solved using the Runge–Kutta–Fehlberg fourth–fifth order technique (RKF-45) and a shooting scheme. Physical descriptions are strategized and argued using graphical representations to provide a clear understanding of the behaviour of dimensionless parameters on dimensionless velocity, concentration, and temperature profiles. The results reveal that the rising values of the rotation parameter lead to a decline in the fluid velocity. The rise in values of relaxation time parameters of temperature and concentration decreases the thermal and concentration profiles, respectively. The increase in values of the heat source/sink parameter advances the thermal profile. The rise in values of the thermophoretic and chemical reaction rate parameters declines the concentration profile.
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Arif M, Kumam P, Kumam W, Akgul A, Sutthibutpong T. Analysis of newly developed fractal-fractional derivative with power law kernel for MHD couple stress fluid in channel embedded in a porous medium. Sci Rep 2021; 11:20858. [PMID: 34675245 PMCID: PMC8531019 DOI: 10.1038/s41598-021-00163-3] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Accepted: 10/05/2021] [Indexed: 11/09/2022] Open
Abstract
Fractal-fractional derivative is a new class of fractional derivative with power Law kernel which has many applications in real world problems. This operator is used for the first time in such kind of fluid flow. The big advantage of this operator is that one can formulate models describing much better the systems with memory effects. Furthermore, in real world there are many problems where it is necessary to know that how much information the system carries. To explain the memory in a system fractal-fractional derivatives with power law kernel is analyzed in the present work. Keeping these motivation in mind in the present paper new concept of fractal-fractional derivative for the modeling of couple stress fluid (CSF) with the combined effect of heat and mass transfer have been used. The magnetohydrodynamics (MHD) flow of CSF is taken in channel with porous media in the presence of external pressure. The constant motion of the left plate generates the CSF motion while the right plate is kept stationary. The non-dimensional fractal-fractional model of couple stress fluid in Riemann-Liouville sense with power law is solved numerically by using the implicit finite difference method. The obtained solutions for the present problem have been shown through graphs. The effects of various parameters are shown through graphs on velocity, temperature and concentration fields. The velocity, temperature and concentration profiles of the MHD CSF in channel with porous media decreases for the greater values of both fractional parameter [Formula: see text] and fractal parameter [Formula: see text] respectively. From the graphical results it can be noticed that the fractal-fractional solutions are more general as compared to classical and fractional solutions of CSF motion in channel. Furthermore, the fractal-fractional model of CSF explains good memory effect on the dynamics of couple stress fluid in channel as compared to fractional model of CSF. Finally, the skin friction, Nusselt number and Sherwood number are evaluated and presented in tabular form.
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Affiliation(s)
- Muhammad Arif
- Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand
- Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand
| | - Poom Kumam
- Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.
- Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand.
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan.
| | - Wiyada Kumam
- Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, 12110, Thailand
| | - Ali Akgul
- Department of Mathematics, Faculty of Arts and Sciences, Siirt University, TR-56100, Siirt, Turkey
| | - Thana Sutthibutpong
- Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand
- Theoretical and Computational Physics Group, Department of Physics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand
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