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Shafiq M, Abbas M, Emadifar H, Alzaidi ASM, Nazir T, Aini Abdullah F. Numerical investigation of the fractional diffusion wave equation with exponential kernel via cubic B-Spline approach. PLoS One 2023; 18:e0295525. [PMID: 38100449 PMCID: PMC10723738 DOI: 10.1371/journal.pone.0295525] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2023] [Accepted: 11/21/2023] [Indexed: 12/17/2023] Open
Abstract
Splines are piecewise polynomials that are as smooth as they can be without forming a single polynomial. They are linked at specific points known as knots. Splines are useful for a variety of problems in numerical analysis and applied mathematics because they are simple to store and manipulate on a computer. These include, for example, numerical quadrature, function approximation, data fitting, etc. In this study, cubic B-spline (CBS) functions are used to numerically solve the time fractional diffusion wave equation (TFDWE) with Caputo-Fabrizio derivative. To discretize the spatial and temporal derivatives, CBS with θ-weighted scheme and the finite difference approach are utilized, respectively. Convergence analysis and stability of the presented method are analyzed. Some examples are used to validate the suggested scheme, and they show that it is feasible and fairly accurate.
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Affiliation(s)
- Madiha Shafiq
- Department of Mathematics, University of Sargodha, Sargodha, Pakistan
| | - Muhammad Abbas
- Department of Mathematics, University of Sargodha, Sargodha, Pakistan
| | - Homan Emadifar
- Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, Tamil Nadu, India
- Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
| | - Ahmed SM Alzaidi
- Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia
| | - Tahir Nazir
- Department of Mathematics, University of Sargodha, Sargodha, Pakistan
| | - Farah Aini Abdullah
- School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia
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Joshi H, Jha BK, Yavuz M. Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:213-240. [PMID: 36650763 DOI: 10.3934/mbe.2023010] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
In this paper, we construct the SV1V2EIR model to reveal the impact of two-dose vaccination on COVID-19 by using Caputo fractional derivative. The feasibility region of the proposed model and equilibrium points is derived. The basic reproduction number of the model is derived by using the next-generation matrix method. The local and global stability analysis is performed for both the disease-free and endemic equilibrium states. The present model is validated using real data reported for COVID-19 cumulative cases for the Republic of India from 1 January 2022 to 30 April 2022. Next, we conduct the sensitivity analysis to examine the effects of model parameters that affect the basic reproduction number. The Laplace Adomian decomposition method (LADM) is implemented to obtain an approximate solution. Finally, the graphical results are presented to examine the impact of the first dose of vaccine, the second dose of vaccine, disease transmission rate, and Caputo fractional derivatives to support our theoretical results.
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Affiliation(s)
- Hardik Joshi
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
| | - Brajesh Kumar Jha
- Department of Mathematics, School of Technology, Pandit Deendayal Energy University, Gandhinagar 382007, India
| | - Mehmet Yavuz
- Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya 42090, Türkiye
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Omame A, Abbas M, Abdel-Aty AH. Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives. CHAOS, SOLITONS, AND FRACTALS 2022; 162:112427. [PMID: 35844899 PMCID: PMC9271450 DOI: 10.1016/j.chaos.2022.112427] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Revised: 06/07/2022] [Accepted: 07/05/2022] [Indexed: 05/16/2023]
Abstract
A new non-integer order mathematical model for SARS-CoV-2, Dengue and HIV co-dynamics is designed and studied. The impact of SARS-CoV-2 infection on the dynamics of dengue and HIV is analyzed using the tools of fractional calculus. The existence and uniqueness of solution of the proposed model are established employing well known Banach contraction principle. The Ulam-Hyers and generalized Ulam-Hyers stability of the model is also presented. We have applied the Laplace Adomian decomposition method to investigate the model with the help of three different fractional derivatives, namely: Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives. Stability analyses of the iterative schemes are also performed. The model fitting using the three fractional derivatives was carried out using real data from Argentina. Simulations were performed with each non-integer derivative and the results thus obtained are compared. Furthermore, it was concluded that efforts to keep the spread of SARS-CoV-2 low will have a significant impact in reducing the co-infections of SARS-CoV-2 and dengue or SARS-COV-2 and HIV. We also highlighted the impact of three different fractional derivatives in analyzing complex models dealing with the co-dynamics of different diseases.
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Affiliation(s)
- Andrew Omame
- Department of Mathematics, Federal University of Technology, Owerri, Nigeria
- Abdus Salam School of Mathematical Sciences, Government College University, Katchery Road, Lahore 54000, Pakistan
| | - Mujahid Abbas
- Abdus Salam School of Mathematical Sciences, Government College University, Katchery Road, Lahore 54000, Pakistan
- Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
| | - Abdel-Haleem Abdel-Aty
- Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
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Kumar P, Govindaraj V, Erturk VS, Mohamed MS. Effects of greenhouse gases and hypoxia on the population of aquatic species: a fractional mathematical model. ADVANCES IN CONTINUOUS AND DISCRETE MODELS 2022; 2022:31. [PMID: 35450200 PMCID: PMC9010246 DOI: 10.1186/s13662-022-03679-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/22/2021] [Accepted: 12/30/2021] [Indexed: 11/10/2022]
Abstract
Study of ecosystems has always been an interesting topic in the view of real-world dynamics. In this paper, we propose a fractional-order nonlinear mathematical model to describe the prelude of deteriorating quality of water cause of greenhouse gases on the population of aquatic animals. In the proposed system, we recall that greenhouse gases raise the temperature of water, and because of this reason, the dissolved oxygen level goes down, and also the rate of circulation of disintegrated oxygen by the aquatic animals rises, which causes a decrement in the density of aquatic species. We use a generalized form of the Caputo fractional derivative to describe the dynamics of the proposed problem. We also investigate equilibrium points of the given fractional-order model and discuss the asymptotic stability of the equilibria of the proposed autonomous model. We recall some important results to prove the existence of a unique solution of the model. For finding the numerical solution of the established fractional-order system, we apply a generalized predictor-corrector technique in the sense of proposed derivative and also justify the stability of the method. To express the novelty of the simulated results, we perform a number of graphs at various fractional-order cases. The given study is fully novel and useful for understanding the proposed real-world phenomena.
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Affiliation(s)
- Pushpendra Kumar
- Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609 India
| | - V Govindaraj
- Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609 India
| | - Vedat Suat Erturk
- Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Atakum, 55200 Samsun Turkey
| | - Mohamed S Mohamed
- Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif, 21944 Saudi Arabia
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Haq I, Khan A, Ahmad S, Ali A, Rahman MU. Modeling and analysis of a fractional anthroponotic cutaneous leishmania model with Atangana-Baleanu derivative. Comput Methods Biomech Biomed Engin 2022; 25:1722-1743. [PMID: 35344457 DOI: 10.1080/10255842.2022.2035372] [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] [Indexed: 12/23/2022]
Abstract
Very recently, Atangana and Baleanu defined a novel arbitrary order derivative having a kernel of non-locality and non-singularity, known as AB derivative. We analyze a non-integer order Anthroponotic Leshmania Cutaneous (ACL) problem exploiting this novel AB derivative. We derive equilibria of the model and compute its threshold quantity, i.e. the so-called reproduction number. Conditions for the local stability of the no-disease as well as the disease endemic states are derived in terms of the threshold quantity. The qualitative analysis for solution of the proposed problem have derived with the aid of the theory of fixed point. We use the predictor corrector numerical approach to solve the proposed fractional order model for approximate solution. We also provide, the numerical simulations for each of the compartment of considered model at different fractional orders along with comparison with integer order to elaborate the importance of modern derivative. The fractional investigation shows that the non-integer order derivative is more realistic about the inner dynamics of the Leishmania model lying between integer order.
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Affiliation(s)
- Ikramul Haq
- Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan
| | - Amir Khan
- Department of Mathematics, University of Swat, Khyber Pakhtunkhawa, Pakistan
| | - Saeed Ahmad
- Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan
| | - Amir Ali
- Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan
| | - Mati Ur Rahman
- Department of Mathematics, Shanghai Jiao Tong University, Shanghai, P.R. China
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New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator. FRACTAL AND FRACTIONAL 2022. [DOI: 10.3390/fractalfract6030171] [Citation(s) in RCA: 12] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.
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Liu X, Arfan M, Ur Rahman M, Fatima B. Analysis of SIQR type mathematical model under Atangana-Baleanu fractional differential operator. Comput Methods Biomech Biomed Engin 2022; 26:98-112. [PMID: 35271386 DOI: 10.1080/10255842.2022.2047954] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
In the given manuscript, the fractional mathematical model for the current pandemic of COVID-19 is investigated. The model is composed of four agents of susceptible (S), infectious (I), quarantined (Q) and recovered (R) cases respectively. The fractional operator of Atangana-Baleanu-Caputo (ABC) is applied to the considered model for the fractional dynamics. The basic reproduction number is computed for the stability analysis. The techniques of existence and uniqueness of the solution are established with the help of fixed point theory. The concept of stability is also derived using the Ulam-Hyers stability technique. With the help of the fractional order numerical method of Adams-Bashforth, we find the approximate solution of the said model. The obtained scheme is simulated on different fractional orders along with the comparison of integer orders. Varying the numerical values for the contact rate ζ, different simulations are performed to check the effect of it on the dynamics of COVID-19.
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Affiliation(s)
- Xuan Liu
- Department of Mathematics, Hanshan Normal University, Chaozhou, China
| | - Muhammad Arfan
- Department of Mathematics, Government Degree College Gulabad Dir (L), Khyber Pakhtunkhwa, Pakistan
| | - Mati Ur Rahman
- School of Mathematical Science, Shanghai Jiao Tong University, Shanghai, P.R. China
| | - Bibi Fatima
- Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, Pakistan
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Khan T, Ullah R, Zaman G, Alzabut J. A mathematical model for the dynamics of SARS-CoV-2 virus using the Caputo-Fabrizio operator. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:6095-6116. [PMID: 34517525 DOI: 10.3934/mbe.2021305] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The pandemic of SARS-CoV-2 virus remains a pressing issue with unpredictable characteristics which spread worldwide through human interactions. The current study is focusing on the investigation and analysis of a fractional-order epidemic model that discusses the temporal dynamics of the SARS-CoV-2 virus in a community. It is well known that symptomatic and asymptomatic individuals have a major effect on the dynamics of the SARS-CoV-2 virus therefore, we divide the total population into susceptible, asymptomatic, symptomatic, and recovered groups of the population. Further, we assume that the vaccine confers permanent immunity because multiple vaccinations have commenced across the globe. The new fractional-order model for the transmission dynamics of SARS-CoV-2 virus is formulated via the Caputo-Fabrizio fractional-order approach with the maintenance of dimension during the process of fractionalization. The theory of fixed point will be used to show that the proposed model possesses a unique solution whereas the well-posedness (bounded-ness and positivity) of the fractional-order model solutions are discussed. The steady states of the model are analyzed and the sensitivity analysis of the basic reproductive number is explored. Moreover to parameterize the model a real data of SARS-CoV-2 virus reported in the Sultanate of Oman from January 1st, 2021 to May 23rd, 2021 are used. We then perform the large scale numerical findings to show the validity of the analytical work.
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Affiliation(s)
- Tahir Khan
- Department of Mathematics, University of Malakand Chakdara, Dir (L), Pakhtunkhwa, Pakistan
- Department of Computing, Muscat College, Muscat Oman
| | - Roman Ullah
- Department of Computing, Muscat College, Muscat Oman
| | - Gul Zaman
- Department of Mathematics, University of Malakand Chakdara, Dir (L), Pakhtunkhwa, Pakistan
| | - Jehad Alzabut
- Department of Mathematics and General Sciences, Prince Sultan, University, Riyadh, Saudi Arabia
- Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Turkey
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