scholarly journals Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Salih Djilali ◽  
Behzad Ghanbari
Keyword(s):  
Fractional Derivative ◽  
Dynamical Behavior ◽  
Time Derivative ◽  
Interaction Model ◽  
Integration Rule ◽  
Existence And Stability ◽  
Natural Result ◽  
Biological Environment ◽  
Fractional Time Derivative ◽  
The Impact

AbstractThe behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

scholarly journals Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Majid Bagheri ◽  
Ali Khani
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Kdv Equation ◽  
Time Derivative ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Fractional Time Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

scholarly journals Integral Balance Methods for Stokes’ First Equation Described by the Left Generalized Fractional Derivative

Physics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 154-166 ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Heat Balance ◽  
Time Derivative ◽  
Double Integral ◽  
Derivative Operator ◽  
Integral Methods ◽  
Fractional Time Derivative ◽  
The Heat Balance ◽  
Generalized Fractional Derivative

In this paper, the integral balance methods of the Stokes’ first equation have been presented. The approximate solution of the fractional Stokes’ first equation using the heat balance integral method has been proposed. The approximate solution of the fractional Stokes’ first equation using the double integral methods has been proposed. The generalized fractional time derivative operator has been used. The graphical representations of the cubic profile and the quadratic profile for the Stokes’ first problem have been provided. The impacts of the orders of the generalized fractional derivative in the Stokes’ first problem have been investigated. The exponent of the assumed profile for the Stokes’ first equation has been discussed.

On the Approximate Solution of Caputo-Riesz-Feller Fractional Diffusion Equation

2020 ◽  
pp. 224-244
Author(s):  
Samir Shamseldeen ◽  
Ahmed Elsaid ◽  
Seham Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Fractional Time Derivative ◽  
Complex Exponential ◽  
Fractional Equation

In this work, a space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative is introduced. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. Also, a very useful Riesz-Feller fractional derivative is proved; the property is essential in applying iterative methods specially for complex exponential and/or real trigonometric functions. The analytic series solution of the problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to highlight the effect of changing the fractional derivative parameters on the behavior of the obtained solutions. The results in this work are originally extracted from the author's work.

scholarly journals Natural Transform along with HPM Technique for Solving Fractional ADE

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
N. Pareek ◽  
A. Gupta ◽  
G. Agarwal ◽  
D. L. Suthar
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Time Derivative ◽  
Space Time ◽  
Physical Parameters ◽  
Homotopy Perturbation ◽  
Advection Dispersion ◽  
The Impact ◽  
Fractional Space

The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.

scholarly journals Semianalytic Solution of Space-Time Fractional Diffusion Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
A. Elsaid ◽  
S. Shamseldeen ◽  
S. Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Time Fractional Diffusion Equation ◽  
Fractional Time Derivative

We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

scholarly journals Bio-nano interface: The impact of biological environment on nanomaterials and their delivery properties

2017 ◽  
Vol 263 ◽  
pp. 211-222 ◽  
Author(s):  
Kaimin Cai ◽  
Andrew Z. Wang ◽  
Lichen Yin ◽  
Jianjun Cheng
Keyword(s):  
Biological Environment ◽  
The Impact

Blind Identification of Impact Forces From Multiple Remote Vibratory Measurements

2000 ◽  
Author(s):  
J. Antunes ◽  
P. Izquierdo ◽  
M. Paulino
Keyword(s):  
Explicit Knowledge ◽  
Transfer Functions ◽  
Dynamical Behavior ◽  
Practical Interest ◽  
Operating Conditions ◽  
Blind Identification ◽  
Minimum Entropy ◽  
Response Measurement ◽  
Identification Techniques ◽  
The Impact

Abstract Structures and mechanical components are often subjected to impulsive forces. There is a need for identification techniques which enable monitoring of such loads under operating conditions. For safety reasons and convenience, force identification must often be based on response motions sensed at accessible locations, remote from the impact points. In our previous work we presented techniques for the experimental identification of both isolated impacts and complex rattling forces on a beam, generated at a single and also at several impacting supports. The system dynamical behavior was modeled using traveling flexural beam waves. Although successful, these techniques obviously assume a good understanding of the system dynamic parameters. This is not always the case, a fact that highlights the practical interest of blind identification techniques. This relatively recent field, connected with higher-order statistics, avoids any explicit knowledge of the system transfer functions or impulse responses. Our previous work, based on a single response measurement, is extended in this paper to include several simultaneous responses. We develop a multi-trace version of Wiggins minimum-entropy blind deconvolution algorithm. From numerical simulations and experiments, it is shown that the robustness to noise contamination is increased by using multiple response data. These results suggest that blind identification techniques will prove very useful in practical situations.

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