Numerical investigation of fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis

2021 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Jagdev Singh ◽  
Ahmed M. Alshehri ◽  
Sarvesh Dubey ◽  
Devendra Kumar
Keyword(s):  
Fractional Order ◽  
Computational Methods ◽  
Fixed Point Theory ◽  
Time Derivative ◽  
Point Theory ◽  
Power Series Method ◽  
Transform Method ◽  
Toxic Phytoplankton ◽  
Wide Range ◽  
Sumudu Transform

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

scholarly journals Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform

2021 ◽  
Vol 5 (3) ◽  
pp. 94
Author(s):  
Saima Rashid ◽  
Zakia Hammouch ◽  
Hassen Aydi ◽  
Abdulaziz Garba Ahmad ◽  
Abdullah M. Alsharif
Keyword(s):  
Fractional Order ◽  
Fixed Point Theory ◽  
Series Representation ◽  
Point Theory ◽  
Hybrid Technique ◽  
Transform Method ◽  
Fractional Integral Operators ◽  
Generalized Fractional Integral ◽  
In Series ◽  
Strongly Agree

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena.

scholarly journals Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel

2022 ◽  
Vol 7 (1) ◽  
pp. 756-783
Author(s):  
Muhammad Farman ◽  
◽  
Ali Akgül ◽  
Kottakkaran Sooppy Nisar ◽  
Dilshad Ahmad ◽  
...  
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Fixed Point Theory ◽  
Control Strategies ◽  
Point Theory ◽  
Fractional Operators ◽  
Order Model ◽  
Epidemiological Analysis ◽  
Fractional Order Model ◽  
Sumudu Transform

<abstract> <p>This paper derived fractional derivatives with Atangana-Baleanu, Atangana-Toufik scheme and fractal fractional Atangana-Baleanu sense for the COVID-19 model. These are advanced techniques that provide effective results to analyze the COVID-19 outbreak. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model COVID-19 model. We also proved the property of boundedness and positivity for the fractional-order model. The Atangana-Baleanu technique and Fractal fractional operator are used with the Sumudu transform to find reliable results for fractional order COVID-19 Model. The generalized Mittag-Leffler law is also used to construct the solution with the different fractional operators. Numerical simulations are performed for the developed scheme in the range of fractional order values to explain the effects of COVID-19 at different fractional values and justify the theoretical outcomes, which will be helpful to understand the outbreak of COVID-19 and for control strategies.</p> </abstract>

Vibration Suppression of an Elastically Supported Beam With Closely Spaced Natural Frequencies

2020 ◽  
Author(s):  
Haizhou Liu ◽  
Hao Gao
Keyword(s):  
Natural Frequencies ◽  
Vibration Suppression ◽  
Fixed Point Theory ◽  
Dynamic Performance ◽  
Point Theory ◽  
Primary System ◽  
Computationally Efficient ◽  
Wide Range ◽  
Univariate Optimization ◽  
Elastically Supported

Abstract Vibration suppression of distributed parameter systems is of great interest and has a wide range of applications. The dynamic performance of a primary system can be improved by adding dynamic vibration absorbers (DVA). Although the relevant topics have been studied for decades, the trade-off between capability of suppressing multiple resonant peaks and complexity of absorbers has not been well addressed. In this paper, the vibration suppression problem of a uniform Euler-Bernoulli beam with closely spaced natural frequencies is investigated. To achieve desired vibration reduction, a two-DOF DVA is connected to the beam through a pair of a spring and a dashpot. By introducing a virtual ground spring, the parameters of the absorber are determined via extended fixed point theory. The proposed method only requires univariate optimization and is computationally efficient. Numerical examples conducted verify the viability of the proposed method and the effectiveness of a two-DOF DVA in suppressing double resonances.

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

Fractals ◽  
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.

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

2017 ◽  
Vol 9 (2) ◽  
pp. 168781401769006 ◽  
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.

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

Fractals ◽  
2021 ◽  
Author(s):  
ANWAR ZEB ◽  
SUNIL KUMAR ◽  
TAREQ SAEED
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Computational Dynamics ◽  
The Social ◽  
Banach Contraction ◽  
Bad Breath ◽  
Predictor Corrector ◽  
Good Agreement

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.

scholarly journals Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

2015 ◽  
Vol 2 (4) ◽  
pp. 140511 ◽  
Author(s):  
Brajesh K. Singh ◽  
Vineet K. Srivastava
Keyword(s):  
Diffusion Equation ◽  
Fractional Order ◽  
Series Solution ◽  
Approximate Solutions ◽  
Test Problems ◽  
Mathematical Tool ◽  
Transform Method ◽  
Wide Range ◽  
Differential Transform ◽  
Powerful Mathematical Tool

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

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

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.

scholarly journals A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi
Keyword(s):  
Fractional Order ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Point Theory ◽  
Euler Method ◽  
Transmission Model ◽  
Order Model ◽  
Influenza Transmission ◽  
The Stability ◽  
Disease Free

Abstract We study the SEIR epidemic model for the spread of AH1N1 influenza using the Caputo–Fabrizio fractional-order derivative. The reproduction number of system and equilibrium points are calculated, and the stability of the disease-free equilibrium point is investigated. We prove the existence of solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. In the numerical section, we present a simulation to examine the system, in which we calculate equilibrium points of the system and examine the behavior of the resulting functions at the equilibrium points. By calculating the results of the model for different fractional order, we examine the effect of the derivative order on the behavior of the resulting functions and obtained numerical values. We also calculate the results of the integer-order model and examine their differences with the results of the fractional-order model.

scholarly journals On Convergence Analysis and Analytical Solutions of the Conformable Fractional Fitzhugh–Nagumo Model Using the Conformable Sumudu Decomposition Method

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 243
Author(s):  
Suliman Alfaqeih ◽  
Emine Mısırlı
Keyword(s):  
Convergence Analysis ◽  
Decomposition Method ◽  
Fixed Point Theory ◽  
Adomian Decomposition Method ◽  
Nerve Impulse ◽  
Point Theory ◽  
Easy Method ◽  
Adomian Decomposition ◽  
Sumudu Transform ◽  
Mathematica Software

The current article studied a nonlinear transmission of the nerve impulse model, the Fitzhugh–Nagumo (FN) model, in the conformable fractional form with an efficient analytical approach based on a combination of conformable Sumudu transform and the Adomian decomposition method. Convergence analysis and error analysis were also carried out based on the Banach fixed point theory. We also provided some examples to support our results. The results obtained revealed that the presented approach is very fantastic, effective, reliable, and is an easy method to handle specific problems in various fields of applied sciences and engineering. The Mathematica software carried out all the computations and graphics in this paper.

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