Axisymmetric self-similar finite-time singularity solution of the Euler equationsAdvances in Continuous and Discrete Models - Tập 2023 - Trang 1-15 - 2023
Rodrigo Cádiz, Diego Martínez-Argüello, Sergio Rica
Self-similar finite-time singularity solutions of the axisymmetric Euler equations in an infinite system with a swirl are provided. Using the Elgindi approximation of the Biot–Savart kernel for the velocity in terms of vorticity, we show that an axisymmetric incompressible and inviscid flow presents a self-similar finite-time singularity of second specie, with a critical exponent ν. Contrary to the recent findings by Hou and collaborators, the current singularity solution occurs at the origin of the coordinate system, not at the system’s boundaries or on an annular rim at a finite distance. Finally, assisted by a numerical calculation, we sketch an approximate solution and find the respective values of ν. These solutions may be a starting point for rigorous mathematical proofs.
A computational method based on the generalized Lucas polynomials for fractional optimal control problemsAdvances in Continuous and Discrete Models - Tập 2022 - Trang 1-29 - 2022
Sh. Karami, A. Fakharzadeh Jahromi, M. H. Heydari
Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, being generated in an easy way, having exponential rates of convergence, having fewer terms and reducing computational errors in comparison with some others, and producing most important basic polynomials. In this regard, this paper deals with a new indirect numerical method to solve fractional optimal control problems based on the generalized Lucas polynomials. Through the way, the left and right Caputo fractional derivatives operational matrices for these polynomials are derived. Based on the Pontryagin maximum principle, the necessary optimality conditions for this problem reduce into a two-point boundary value problem. The main and efficient characteristic behind the proposed method is to convert the problem under consideration into a system of algebraic equations which reduces many computational costs and CPU time. To demonstrate the efficiency, applicability, and simplicity of the proposed method, several examples are solved, and the obtained results are compared with those obtained with other methods.
Stability of thermoelastic Timoshenko beam with suspenders and time-varying feedbackAdvances in Continuous and Discrete Models - Tập 2023 - Trang 1-19 - 2023
Soh Edwin Mukiawa, Cyril Dennis Enyi, Salim A. Messaoudi
This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We also gave some examples to illustrate our theoretical finding.
Effects of greenhouse gases and hypoxia on the population of aquatic species: a fractional mathematical modelAdvances in Continuous and Discrete Models - Tập 2022 Số 1 - 2022
Pushpendra Kumar, V. Govindaraj, Vedat Suat Ertürk, Mohamed S. Mohamed
AbstractStudy 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.
Euler equations for isentropic gas dynamics with general pressure lawAdvances in Continuous and Discrete Models - Tập 2022 Số 1 - Trang 1-14 - 2022
Ibrahim, Muhammad, Din, Anwarud, Yusuf, Abdullahi, Lv, Yu-Pei, Jahanshahi, Hadi, Aly, Ayman A.
In this work, we explore the limiting behavior of Riemann solutions to the Euler equations in isentropic gas dynamics with general pressure law. We demonstrate that in the distributional sense the delta wave of zero-pressure gas dynamics is formed by a limit solution. Finally, to present the concentration phenomena, we also offer some numerical outcomes.
A delayed plant disease model with Caputo fractional derivativesAdvances in Continuous and Discrete Models - Tập 2022 - Trang 1-22 - 2022
Pushpendra Kumar, Dumitru Baleanu, Vedat Suat Erturk, Mustafa Inc, V. Govindaraj
We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.
Mittag–Leffler stability, control, and synchronization for chaotic generalized fractional-order systemsAdvances in Continuous and Discrete Models - Tập 2022 Số 1
Tarek M. Abed‐Elhameed, Tarek Aboelenen
AbstractIn this paper, we investigate the generalized fractional system (GFS) with order lying in $(1, 2)$
(
1
,
2
)
. We present stability analysis of GFS by two methods. First, the stability analysis of that system using the Gronwall–Bellman (G–B) Lemma, the Mittag–Leffler (M–L) function, and the Laplace transform is introduced. Secondly, by the Lyapunov direct method, we study the M–L stability of our system with order lying in $(1, 2)$
(
1
,
2
)
. Using the modified predictor–corrector method, the solutions of GFSs are calculated and they are more complicated than the classical fractional one. Based on linear feedback control, we investigate a theorem to control the chaotic GFSs with order lying in $(1, 2)$
(
1
,
2
)
. We present an example to verify the validity of control theorem. We state and prove a theorem to calculate the analytical formula of controllers that are used to achieve synchronization between two different chaotic GFSs. An example to study the synchronization for systems with orders lying in $(1, 2)$
(
1
,
2
)
is given. We found an agreement between analytical results and numerical simulations.
Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciencesAdvances in Continuous and Discrete Models - Tập 2023 - Trang 1-22 - 2023
Roumen Anguelov, Jean M.-S. Lubuma
We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems (viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.
Analysis of a stochastic predator–prey system with fear effect and Lévy noiseAdvances in Continuous and Discrete Models - Tập 2022 - Trang 1-24 - 2022
Renxiu Xue, Yuanfu Shao, Minjuan Cui
This paper studies a stochastic predator–prey model with Beddington–DeAngelis functional response, fear effect, and Lévy noise, where the fear is of prey induced by predator. First, we use Itô’s formula to prove the existence and uniqueness of a global positive solution and its moment boundedness. Next, sufficient conditions for the persistence and extinction of both species are given. We further investigate the stability in distribution of our system. Finally, we verify our analytical results by exhaustive numerical simulations.
Investigation of a time-fractional COVID-19 mathematical model with singular kernelAdvances in Continuous and Discrete Models - Tập 2022 - Trang 1-19 - 2022
Adnan, Amir Ali, Mati ur Rahmamn, Zahir Shah, Poom Kumam
We investigate the fractional dynamics of a coronavirus mathematical model under a Caputo derivative. The Laplace–Adomian decomposition and Homotopy perturbation techniques are applied to attain the approximate series solutions of the considered system. The existence and uniqueness solution of the system are presented by using the Banach fixed-point theorem. Ulam–Hyers-type stability is investigated for the proposed model. The obtained approximations are compared with numerical simulations of the proposed model as well as associated real data for numerous fractional-orders. The results reveal a good comparison between the numerical simulations versus approximations of the considered model. Further, one can see good agreements are obtained as compared to the classical integer order.
