Showing papers in "International Journal of Differential Equations in 2017"
TL;DR: In this article, a fractional order model is proposed to describe the dynamics of human immunodeficiency virus (HIV) infection, where the infection transmission process is modeled by a specific functional response.
Abstract: We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results.
50 citations
TL;DR: In this article, an alternative approximate analytic solution to time fractional partial differential equations with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense.
Abstract: This paper deals with an alternative approximate analytic solution to time fractional partial differential equations (TFPDEs) with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense. The proposed series solutions are found to converge to exact solution rapidly. To confirm the efficiency and validity of FRDTM, the computation of three test problems of TFPDEs with proportional delay was presented. The scheme seems to be very reliable, effective, and efficient powerful technique for solving various types of physical models arising in science and engineering.
42 citations
TL;DR: In this paper, a bioeconomic equilibrium model for two fishermen who catch three species taking into consideration the fact that the prices of fish populations vary according to the quantity harvested; these species compete with each other for space or food; the natural growth of each species is modeled using a logistic law.
Abstract: Bioeconomic modeling of the exploitation of biological resources such as fisheries has gained importance in recent years. In this work we propose to define and study a bioeconomic equilibrium model for two fishermen who catch three species taking into consideration the fact that the prices of fish populations vary according to the quantity harvested; these species compete with each other for space or food; the natural growth of each species is modeled using a logistic law. The main purpose of this work is to define the fishing effort that maximizes the profit of each fisherman, but all of them have to respect two constraints: the first one is the sustainable management of the resources and the second one is the preservation of the biodiversity. The existence of the steady states and their stability are studied using eigenvalue analysis. The problem of determining the equilibrium point that maximizes the profit of each fisherman leads to Nash equilibrium problem; to solve this problem we transform it into a linear complementarity problem (LCP); then we prove that the obtained problem (LCP) admits a unique solution that represents the Nash equilibrium point of our problem. We close our paper with some numerical simulations.
28 citations
TL;DR: In this article, an effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented, which is an excellent mathematical tool for finding the numerical solutions.
Abstract: An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.
21 citations
Abstract: A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out.
19 citations
TL;DR: In this article, the development and numerical analysis of mathematical models used to describe complex biological systems in the framework of Integrated Pest Management (IPM) has been discussed, and a comprehensive parametric study of an IPM-based impulsive control scheme is carried out via path-following techniques.
Abstract: The paper is concerned with the development and numerical analysis of mathematical models used to describe complex biological systems in the framework of Integrated Pest Management (IPM). Established in the late 1950s, IPM is a pest management paradigm that involves the combination of different pest control methods in ways that complement one another, so as to reduce excessive use of pesticides and minimize environmental impact. Since the introduction of the IPM concept, a rich set of mathematical models has emerged, and the present work discusses the development in this area in recent years. Furthermore, a comprehensive parametric study of an IPM-based impulsive control scheme is carried out via path-following techniques. The analysis addresses practical questions, such as how to determine the parameter values of the system yielding an optimal pest control, in terms of operation costs and environmental damage. The numerical study concludes with an exploration of the dynamical features of the impulsive model, which reveals the presence of codimension-1 bifurcations of limit cycles, hysteretic effects, and period-doubling cascades, which is a precursor to the onset of chaos.
15 citations
TL;DR: In this paper, a block hybrid trigonometrically fitted Runge-Kutta-Nystrom method (BHTRKNM) is presented, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems.
Abstract: In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nystrom method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
12 citations
TL;DR: In this article, the oscillation constant for Euler type half-linear second-order differential equation with multi-different periodic coefficients was derived explicitly for the case of Euler types.
Abstract: We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.
11 citations
TL;DR: In this article, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to the singularly perturbed ODE.
Abstract: In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.
10 citations
TL;DR: In this article, the behavior of a predator-prey model with switching and stage-structure for predator was studied and bounded positive solution, equilibria, and stabilities were determined for the system of delay differential equation.
Abstract: This paper studies the behavior of a predator-prey model with switching and stage-structure for predator. Bounded positive solution, equilibria, and stabilities are determined for the system of delay differential equation. By choosing the delay as a bifurcation parameter, it is shown that the positive equilibrium can be destabilized through a Hopf bifurcation. Some numerical simulations are also given to illustrate our results.
8 citations
TL;DR: In this paper, a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form, and it was shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease.
Abstract: We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form , and it is shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease. Numerical simulations are performed using a nonlinear incidence rate to estimate the basic reproduction number and illustrate our analytical findings.
TL;DR: In this paper, the authors proposed the morbidity of the multivariable grey prediction MGM model based on the Gerschgorin Panel Theorem, and analyzed the factors that affect the morbidness of the multi-variable grey model.
Abstract: This paper proposes the morbidity of the multivariable grey prediction MGM model. Based on the morbidity of the differential equations, properties of matrix, and Gerschgorin Panel Theorem, we analyze the factors that affect the morbidity of the multivariable grey model and give a criterion to justify the morbidity of MGM. Finally, an example is presented to illustrate the practicality of our results.
TL;DR: In this article, the authors derived existence results for a second-order singular multipoint boundary value problem at resonance using coincidence degree arguments and showed that the existence result holds for the case where the second order multipoint value problem is solvable at resonance.
Abstract: The aim of this paper is to derive existence results for a second-order singular multipoint boundary value problem at resonance using coincidence degree arguments.
TL;DR: In this paper, the existence and uniqueness of solutions for boundary value problem of nonlinear fractional differential equation is studied and the main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.
Abstract: In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: , , , where , , , and . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.
TL;DR: A family of boundary value methods with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach to generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration.
Abstract: A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.
TL;DR: In this article, a hybrid algorithm and regularization method were proposed to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements, where the evolution of the heat is given by a degenerate parabolic equation with singular potential.
Abstract: A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given.
TL;DR: In this article, a modified Taylor series approach was proposed to solve nonlinear boundary value problems (BVPs), where the order of the differential equation under consideration is determined by the Taylor series.
Abstract: Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A -step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the -step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.
TL;DR: In this paper, the authors propose nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption.
Abstract: This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle.
TL;DR: In this paper, the problem of existence and uniqueness of solutions of stochastic differential equations with jumps and concentration points is solved, and the theoretical results are illustrated by one example of one example.
Abstract: In this article the problem of existence and uniqueness of solutions of stochastic differential equations with jumps and concentration points are solved. The theoretical results are illustrated by one example.
TL;DR: In this article, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed, and the convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators.
Abstract: For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state to an neighbourhood of the target state at time under the assumption that the nonlinear function is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained.
TL;DR: In this paper, two reaction-diffusion equations connected by a one-directional coupling function are considered and conditions that the evolution of the driven system closely tracks the development of the driver system are derived.
Abstract: We consider two reaction-diffusion equations connected by one-directional coupling function and study the synchronization problem in the case where the coupling function affects the driven system in some specific regions. We derive conditions that ensure that the evolution of the driven system closely tracks the evolution of the driver system at least for a finite time. The framework built to achieve our results is based on the study of an abstract ordinary differential equation in a suitable Hilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.
TL;DR: In this article, the replicator dynamics for the asymmetric Hawk-Dove game were analyzed using a dynamical systems approach, and the equilibrium points and their stability were derived.
Abstract: We analyze, using a dynamical systems approach, the replicator dynamics for the asymmetric Hawk-Dove game in which there is a set of four pure strategies with arbitrary payoffs. We give a full account of the equilibrium points and their stability and derive the Nash equilibria. We also give a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters and . We also give details on the connections between the results found in this work and those of the standard two-strategy Hawk-Dove game. We conclude the paper with some examples of numerical simulations that further illustrate some global behaviours of the system.
TL;DR: In this paper, the authors considered the Cauchy problem for the Ostrovsky-Hunter equation, and proved the global in time existence and the large time asymptotic behavior of solutions.
Abstract: We consider the Cauchy problem for the Ostrovsky-Hunter equation , , , , where . Define . Suppose that is a pseudodifferential operator with a symbol such that , , and . For example, we can take . We prove the global in time existence and the large time asymptotic behavior of solutions.