scholarly journals Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Oyoon Abdul Razzaq ◽  
Sankar Parsad Mondal ◽  
Qammar Rubbab
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
Equilibrium Points ◽  
System Of Equations ◽  
Biological Parameters ◽  
Kutta Method ◽  
Interacting Species ◽  
Runge Kutta Method ◽  
Novel Approach ◽  
Predator Model

Abstract The key objective of this paper is to study the imprecise biological complexities in the interaction of two species pertaining to harvesting threshold. It is explained by taking the prey–predator model with imprecise biological parameters and fractional order generalized Hukuhara (fgH) differentiability. In this vain, different possible systems of the model are constructed, according to the increasing and decreasing behavior of population growth. Feasibility and stability analyses of equilibrium points of the stated models are also discussed by means of variational matrix with Routh–Hurwitz conditions. In addition, the numerical elaborations are carried out by taking parametric expansion of fuzzy fractional Laplace transform (FFLT). This significantly helps the researchers in using a novel approach to analyze the constant solutions of the dynamical systems in the presence of fractional index. This would allow the avoidance of any intricacy that occurs while solving fractional order derivatives. Furthermore, this attempt also provides numerical and pictorial results, obtained through some well-known methods, namely fifth-forth Runge–Kutta method (FFRK), Grunwald–Letnikov’s definition (GL) and Adams–Bashforth method (ABM) that are deemed appropriate to scrutinize the dynamics of the system of equations.

scholarly journals Calculation of turbulent diffusion jets under effects of gravity and moving surrounding air

2001 ◽  
Vol 23 (2) ◽  
pp. 87-94
Author(s):  
Bui Van Ga ◽  
Nhan Hong Quang ◽  
Jean Marc Vignon
Keyword(s):  
Velocity Field ◽  
Turbulent Diffusion ◽  
Concentration Field ◽  
Integral Model ◽  
System Of Equations ◽  
Kutta Method ◽  
Air Velocity ◽  
Runge Kutta Method ◽  
Jet Axis ◽  
General Conditions

The basis theory for the turbulent diffusion of jet and flame has been presented previously [1, 2]. But that one applies only in quiet surrounding air with the effects of buoyancy neglected. In the present paper, the theory is developed further by establishing an integral model for a jet in more general conditions with variable inclined angles, under effects of gravity and surrounding air velocity in any direction compared to the jet axis. The system of equations is closed by turbulence k-E model and is solved by 4th order Runge-Kutta method. In the first stage, the model is applied to predict the velocity field, the concentration field and with development of a 0.3 m diameter jet.

scholarly journals Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shaher Momani ◽  
Asad Freihat ◽  
Mohammed AL-Smadi
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

scholarly journals ON ONSET OF CHAOTIC CONVECTION IN COUPLE-STRESS FLUIDS

2014 ◽  
Vol 19 (3) ◽  
pp. 359-370 ◽  
Author(s):  
Jadallah M. Jawdat ◽  
Ishak Hashim ◽  
Beer S. Bhadauria ◽  
Shaher Momani
Keyword(s):  
Dynamical Systems ◽  
Nonlinear System ◽  
Couple Stress ◽  
Fluid Layer ◽  
Kutta Method ◽  
Fluid Field ◽  
Runge Kutta Method ◽  
Chaotic Convection ◽  
Low Dimensional ◽  
Couple Stress Fluids

The effect of couple-stress fluid field on chaotic convection in a fluid layer heated from below was studied in this paper based on the theory of dynamical systems. A low-dimensional, Lorenz-like model was obtained using Galerkin truncated approximations. The fourth-order Runge–Kutta method was employed to solve the nonlinear system. The results show that inhibition of chaotic convection depends strongly on the couple-stress parameter.

scholarly journals On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

2020 ◽  
Vol 28 (4) ◽  
pp. 327-345
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Law ◽  
Numerical Experiments ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Quadratic Invariant ◽  
Quadratic Invariants ◽  
Runge Kutta ◽  
Autonomous Dynamical Systems

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

Comparison of Numerical Methods of the SEIR Epidemic Model of Fractional Order

2014 ◽  
Vol 69 (1-2) ◽  
pp. 81-89 ◽  
Author(s):  
Anwar Zeb ◽  
Madad Khan ◽  
Gul Zaman ◽  
Shaher Momani ◽  
Vedat Suat Ertürk
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Nonnegative Solution ◽  
Incidence Rates ◽  
Kutta Method ◽  
Transform Method ◽  
Seir Model ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model by taking into account both standard and bilinear incidence rates of fractional order. First, the nonnegative solution of the SEIR model of fractional order is presented. Then, the multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. Finally, the obtained results are compared with those obtained by the fourth-order Runge-Kutta method and non-standard finite difference (NSFD) method in the integer case.

scholarly journals Stability of a fractional order harvested prey–predator model in the existence of infection in prey

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Subhashis Das ◽  
◽  
Sanat Mahato ◽  
Prasenjit Mahato
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Type I ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Different Types ◽  
Predator Model ◽  
Predictor Corrector ◽  
Uniqueness Criteria ◽  
Type Ii Functional Response

The growing relationship between prey and their predator is one of the important aspects in the field of ecology and mathematical biology. On the other hand, the utility of fractional calculus in different types of mathematical modelling have been applied extensively. In this paper, a fractional order prey–predator model is developed with the consideration of Holling type-I and Holling type-II functional response of the predator. As infection spreads through prey, the prey population is divided into two parts. In addition, we exploit the effect of harvesting to control the excessive spread of the infection. The existence and uniqueness criteria, the boundedness of the solution of the proposed model are investigated. A number of five possible equilibrium points of the proposed model are determined along with the feasibility conditions for each equilibrium points. The local stability at these equilibrium points and global stability at interior equilibrium point are investigated. Numerical simulation is presented with the help of modified Predictor-corrector method in MATLAB software to understand the dynamics of the proposed model.

scholarly journals Stresses of feller buncher on thinnings during technological moves

2018 ◽  
Author(s):  
В.А. Александров ◽  
А.В. Александров ◽  
Г.Ш. Гасымов
Keyword(s):  
Mathematical Model ◽  
Dynamic System ◽  
Dynamic Load ◽  
Mathematical Description ◽  
Lagrange Equation ◽  
Process Equipment ◽  
System Of Equations ◽  
Programming Environment ◽  
Kutta Method ◽  
Runge Kutta Method

Разработана математическая модель динамической системы: «Валочно- пакетирующая машина – предмет труда – дерево». Математическое описание составлено в форме уравнения Лагранжа 2-го рода. Система уравнений решена методом Рунге–Кутта, с помощью среды программирования MathCAD. Апробация модели осуществлена на примере серийно выпускаемых валочно-пакетирующих машин ЛП – 19А в режимах разгона и стопорения. Установлено, что динамическая нагрузка на технологическое оборудование при технологических переездах сопоставима с нагрузкой при обработке (пакетировании) дерева. A mathematical model of a dynamic system has been developed: «The feller buncher – subject of labor – the tree». The mathematical description is made up in the form of the Lagrange equation of the 2nd kind. The system of equations is solved by the Runge–Kutta method, using the MathCAD programming environment. Approbation of the model is carried out on the example of the commercially available LP-19A feller buncher machines in acceleration and stopping modes. Concluded that the dynamic load on the process equipment during technological moves is comparable to the load during the processing (packaging) of the tree.

Numerical simulation of a class of space fractional bistable systems based on the Fourier spectral method

2021 ◽  
pp. 146134842110637
Author(s):  
Haitao Liu ◽  
Wang Yulan ◽  
Li Cao ◽  
Wei Zhang
Keyword(s):  
Spectral Method ◽  
Fractional Order ◽  
Fourth Order ◽  
Numerical Approach ◽  
Bistable System ◽  
Fourier Spectral Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Bistable Systems ◽  
Runge Kutta

Nonlinear vibration arises everywhere in a bistable system. The bistable system has been widely applied in physics, biology, and chemistry. In this article, in order to numerically simulate a class of space fractional-order bistable system, we introduce a numerical approach based on the modified Fourier spectral method and fourth-order Runge-Kutta method. The fourth-order Runge-Kutta method is used in time, and the Fourier spectrum is used in space to approximate the solution of the space fractional-order bistable system. Numerical experiments are given to illustrate the effectiveness of this method.

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