A ROBUST COMPUTATIONAL DYNAMICS OF FRACTIONAL-ORDER SMOKING MODEL WITH RELAPSE HABIT

The social habit of smoking has affected the whole world in a social manner. It is the main cause of diseases like cancers, asthma, bad breath, etc., and a source of spreading of infectious diseases like COVID-19. This work is related to an existing smoking model with relapse habit converted in fractional order. First, formulation of fractional-order smoking model is presented and then the dynamics of proposed problem is analyzed. Fixed-point theory via Banach contraction and Schauder theorems is used to derive the existence and uniqueness of the model. At last, the adaptive predictor–corrector algorithm and Runge–Kutta fourth-order (RK4) strategy are used to perform simulation. To bolster the validity of the theoretical results, a set of numerical simulations are performed. A good agreement between hypothetical and numerical results is demonstrated via numerical simulations using MATLAB software.

Download Full-text

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

Georgian Mathematical Journal ◽

10.1515/gmj-2021-2102 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Alberto Cabada ◽

Om Kalthoum Wanassi

Keyword(s):

Existence And Uniqueness ◽

Fixed Point Theory ◽

Mixed Boundary ◽

Point Theory ◽

Uniqueness Of Solutions ◽

Nonlinear Part ◽

Banach Contraction ◽

Fractional Differential ◽

Exhaustive Study ◽

The Banach Contraction Principle

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

Download Full-text

An efficient numerical simulation and mathematical modeling for the prevention of tuberculosis

International Journal of Biomathematics ◽

10.1142/s1793524522500152 ◽

2021 ◽

Author(s):

Zain Ul Abadin Zafar ◽

Samina Younas ◽

Sumera Zaib ◽

Cemil Tunç

Keyword(s):

Fixed Point Theory ◽

Dynamical Behavior ◽

Point Theory ◽

Model Parameters ◽

Fractional Operator ◽

Significant Preference ◽

Relevant Model ◽

Clinical Associations ◽

Predictor Corrector ◽

Fractional Orders

The main purpose of this research is to use a fractional-mathematical model including Atangana–Baleanu derivatives to explore the clinical associations and dynamical behavior of the tuberculosis. Herein, we used a lately introduced fractional operator having Mittag-Leffler kernel. The existence and inimitability problems to the relevant model were examined through the fixed-point theory. To verify the significance of the arbitrary fractional-order derivative, numerical outcomes were explored from the biological and mathematical viewpoints using the values of model parameters. The graphical simulations show the comparison of the predictor–corrector method (PCM) and Caputo method (CM) for different fractional orders and the results indicated the significant preference of PCM over CM.

Download Full-text

FRACTIONAL ORDER ANALYSIS OF HBV AND HCV CO-INFECTION UNDER ABC DERIVATIVE

Fractals ◽

10.1142/s0218348x22400369 ◽

2021 ◽

Author(s):

HUSSAM ALRABAIAH ◽

MATI UR RAHMAN ◽

IBRAHIM MAHARIQ ◽

SAMIA BUSHNAQ ◽

MUHAMMAD ARFAN

Keyword(s):

Mathematical Model ◽

Fixed Point ◽

Approximate Solution ◽

Fractional Order ◽

Fixed Point Theory ◽

Point Theory ◽

Comparative Result ◽

Order Analysis ◽

The Stability ◽

Fractional Orders

In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.

Download Full-text

Erratum: Abdou, A. A. N. and Khamsi, M.A. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·). Mathematics 2020, 8, 76

Mathematics ◽

10.3390/math8040578 ◽

2020 ◽

Vol 8 (4) ◽

pp. 578

Author(s):

Afrah A. N. Abdou ◽

Mohamed Amine Khamsi

Keyword(s):

Fixed Point ◽

Fixed Points ◽

Banach Spaces ◽

Variable Exponent ◽

Fixed Point Theory ◽

Point Theory ◽

Sequence Spaces ◽

Vector Spaces ◽

Nonexpansive Maps ◽

Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces lp(·). We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

Download Full-text

Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative

Symmetry ◽

10.3390/sym11050672 ◽

2019 ◽

Vol 11 (5) ◽

pp. 672 ◽

Cited By ~ 2

Author(s):

Mouffak Benchohra ◽

Soufyane Bouriah ◽

Juan J. Nieto

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fractional Derivative ◽

Fixed Point Theory ◽

Point Theory ◽

Uniqueness Of Solutions ◽

Krasnoselskii’S Fixed Point Theorem ◽

Banach Contraction ◽

Fractional Differential ◽

Terminal Value Problem

We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Banach contraction principle and Krasnoselskii’s fixed point theorem. We illustrate our main findings, with a particular case example included to show the applicability of our outcomes.

Download Full-text

Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel

Advances in Mechanical Engineering ◽

10.1177/1687814017690069 ◽

2017 ◽

Vol 9 (2) ◽

pp. 168781401769006 ◽

Cited By ~ 20

Author(s):

Devendra Kumar ◽

Jagdev Singh ◽

Maysaa Al Qurashi ◽

Dumitru Baleanu

Keyword(s):

Numerical Simulation ◽

Fixed Point ◽

Fractional Derivative ◽

Fractional Order ◽

Existence And Uniqueness ◽

Fixed Point Theory ◽

Point Theory ◽

Logistic Equation ◽

Iterative Technique ◽

Uniqueness Of The Solution

In this work, we aim to analyze the logistic equation with a new derivative of fractional order termed in Caputo–Fabrizio sense. The logistic equation describes the population growth of species. The existence of the solution is shown with the help of the fixed-point theory. A deep analysis of the existence and uniqueness of the solution is discussed. The numerical simulation is conducted with the help of the iterative technique. Some numerical simulations are also given graphically to observe the effects of the fractional order derivative on the growth of population.

Download Full-text

Numerical behavior of a fractional order dynamical model of RNA silencing

International Journal of Scientific World ◽

10.14419/ijsw.v4i2.6474 ◽

2016 ◽

Vol 4 (2) ◽

pp. 52

Author(s):

A.M.A. El-Sayed ◽

M. Khalil ◽

A.A.M. Arafa ◽

Amaal Sayed

Keyword(s):

Differential Equations ◽

Numerical Simulations ◽

Detailed Analysis ◽

Fractional Order ◽

Rna Silencing ◽

Fractional Derivatives ◽

Numerical Solutions ◽

The Stability ◽

Predictor Corrector ◽

With Memory

A class of fractional-order differential models of RNA silencing with memory is presented in this paper. We also carry out a detailed analysis on the stability of equilibrium and we show that the model established in this paper possesses non-negative solutions. Numerical solutions are obtained using a predictor-corrector method to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. Numerical simulations are presented to illustrate the results. Also, the numerical simulations show that, modeling the phenomena of RNA silencing by fractional ordinary differential equations (FODE) has more advantages than classical integer-order modeling.

Download Full-text

Multi-valued F-contractions and the solutions of certain functional and integral equations

Filomat ◽

10.2298/fil1307259s ◽

2013 ◽

Vol 27 (7) ◽

pp. 1259-1268 ◽

Cited By ~ 69

Author(s):

Margherita Sgroi ◽

Calogero Vetro

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Integral Equations ◽

Metric Spaces ◽

Fixed Point Theory ◽

Point Theory ◽

Contractive Condition ◽

Ordered Metric Spaces ◽

Complete Metric Spaces ◽

Banach Contraction

Wardowski [Fixed Point Theory Appl., 2012:94] introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric spaces. An example and two applications, for the solution of certain functional and integral equations, are given to illustrate the usability of the obtained results.

Download Full-text

A Class of Different Fractional-Order Chaotic (Hyperchaotic) Complex Duffing-Van Der Pol Models and Their Circuits Implementations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4052569 ◽

2021 ◽

Author(s):

Gamal Mahmoud ◽

Tarek Abed-Elhameed ◽

Motaz Elbadry

Keyword(s):

Fixed Points ◽

Lyapunov Exponents ◽

Fractional Order ◽

Transfer Functions ◽

Viscoelastic Beam ◽

Electronic Circuits ◽

Van Der Pol ◽

Shape Model ◽

Predictor Corrector ◽

Good Agreement

Abstract In this paper, we introduce three versions of fractional-order chaotic (or hyperchaotic) complex Duffing-van der Pol models. The dynamics of these models including their fixed points and their stability is investigated. Using the predictor-corrector method and Lyapunov exponents we calculate numerically the intervals of their parameters at which chaotic, hyperchaotic solutions and solutions that approach fixed points exist. These models appear in several applications in physics and engineering, e.g., viscoelastic beam and electronic circuits. The electronic circuits of these models with different fractional-order are proposed. We determine the approximate transfer functions for novel values of fractional-order and find the equivalent tree shape model (TSM). This TSM is used to build circuits simulations of our models}. A good agreement is found between both numerical and simulations results. Other circuits diagrams can be similarly designed for other fractional-order models.

Download Full-text

A new modeling and existence–uniqueness analysis for Babesiosis disease of fractional order

Modern Physics Letters B ◽

10.1142/s0217984921504431 ◽

2021 ◽

pp. 2150443

Author(s):

Rajarama Mohan Jena ◽

Snehashish Chakraverty ◽

Mehmet Yavuz ◽

Thabet Abdeljawad

Keyword(s):

Fixed Point ◽

Fractional Derivative ◽

Numerical Simulations ◽

Arbitrary Order ◽

Fixed Point Theory ◽

Point Theory ◽

Qualitative Properties ◽

Diseased Animal ◽

Transform Method ◽

Homotopy Perturbation

In this study, we consider the dynamics of the Babesiosis transmission on bovine populations and ticks. The most important role in the transmission of the parasite is the ticks from the Ixodidae family. The vector tick takes factors (merozoites in erythrocytes) from the diseased animal while sucking the blood. To model and investigate the transmissions of this parasite and address this important issue, we have considered the disease in a fractional epidemiological model. This paper, therefore, discusses the mechanisms of transmission of Babesiosis defined in the fractional derivative sense. The Caputo–Fabrizio (CF) derivative is considered to study the propagation mechanisms of Babesiosis. First, the important characteristics of the model have been presented, and then the transmission of the Babesiosis model defined in CF is discussed. The application of fixed-point theory is used to derive the concept of the qualitative properties of the mentioned model. The solution is obtained by using the Homotopy perturbation Elzaki transform method (HPETM). Numerical simulations are performed, and the effects of the arbitrary-order derivatives are investigated graphically.

Download Full-text