Authors:Jagadish Singh; Joel John Taura Pages: 385 - 396 Abstract: Abstract In this paper the equations of motion of the circular restricted three body problem is modified to include radiation of the bigger primary, triaxiality of the smaller primary; and gravitational potential created by a belt. We have obtained that due to the perturbations, the locations of the triangular libration points and their linear stability are affected. The points move towards the bigger primary due to the resultant effect of the perturbations. Triangular libration points are stable for \(0<\mu <\mu _c\) and unstable for \(\mu _c \le \mu \le \frac{1}{2}\) , where \(\mu _c\) is the critical mass ratio affected by the perturbations. The radiation of the bigger primary and triaxiality of the smaller primary have destabilizing propensities, whereas the potential created by the belt has stabilizing propensity. This model could be applied in the study of the motion of a dust particle near radiating -triaxial binary system surrounded by a belt. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0243-0 Issue No:Vol. 25, No. 3 (2017)

Authors:Abdelkarim-Nidal Akdad; Brahim Essebbar; Khalil Ezzinbi Pages: 397 - 416 Abstract: Abstract In this work, we present a new composition theorem of \(\mu \) -pseudo almost automorphic functions in the sense of Stepanov satisfying some local Lipschitz conditions. Using this results, we establish an existence result of \(\mu \) -pseudo almost automorphic solutions for some nonautonomous neutral partial evolution equation with Stepanov \(\mu \) -pseudo almost automorphic nonlinearity. An example is shown to illustrate our results. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0246-x Issue No:Vol. 25, No. 3 (2017)

Authors:Aadil Lahrouz; Adel Settati Pages: 417 - 430 Abstract: Abstract A stochastic Gilpin–Ayala population model with diffusion between two patches is studied. A sufficient conditions for extinction and persistence are established. Furthermore, the existence of a stationary distribution is showed. The analytical results are illustrated by computer simulations. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0244-z Issue No:Vol. 25, No. 3 (2017)

Authors:Donghyun Kim Pages: 431 - 451 Abstract: Abstract We study the Cauchy problem for a system of cubic nonlinear Klein–Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order \(O(t^{-1/2})\) in \(L^\infty \) as t tends to infinity without the condition of a compact support on the Cauchy data which was assumed in the previous works. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0259-5 Issue No:Vol. 25, No. 3 (2017)

Authors:A. M. Elaiw; N. H. AlShamrani Pages: 453 - 479 Abstract: Abstract In this paper, we study the global properties of two general models for viral infection with humoral immune response. The incidence rate of infection, the removal rate of infected cells, the production and neutralize rates of viruses and the activation and removal rates of B cells are given by more general nonlinear functions. The second model generalizes the first one by taking into account the latently infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For both models, we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of all equilibria of the models. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0247-9 Issue No:Vol. 25, No. 3 (2017)

Authors:Daniel Olmos; Ignacio Barradas; David Baca-Carrasco Pages: 481 - 497 Abstract: Abstract A set of epidemiological models and their basic reproductive numbers are presented. For some models, the expression of the basic reproductive number can be very complex to determine. In this work, we present some techniques to find the expressions of these basic reproductive numbers starting from submodels of the original one. Some particular models with their respective \(R_0\) , which can be applied to the study of some important diseases, are also presented. PubDate: 2017-07-01 DOI: 10.1007/s12591-015-0257-7 Issue No:Vol. 25, No. 3 (2017)

Authors:Taekyun Kim; Dae San Kim; Lee-Chae Jang; Hyuck In Kwon Abstract: Abstract In this paper, we derive differential equations from the generating function of higher-order Frobenius–Euler numbers. In addition, we give some new and explicit identities for higher-order Frobenius–Euler numbers arising from those differential equations. PubDate: 2017-07-29 DOI: 10.1007/s12591-017-0380-8

Authors:Md. Maqbul; A. Raheem Abstract: Abstract The existence and uniqueness of a strong solution for a class of partial functional differential equations with Dirichlet boundary conditions is established by applying Rothe’s method. As an application, we included an example to illustrate the main result. PubDate: 2017-07-06 DOI: 10.1007/s12591-017-0379-1

Authors:J. Herbrych; A. G. Chazirakis; N. Christakis; J. J. P. Veerman Abstract: Abstract We consider large but finite systems of identical agents on the line with up to next nearest neighbor asymmetric coupling. Each agent is modelled by a linear second order differential equation, linearly coupled to up to four of its neighbors. The only restriction we impose is that the equations are decentralized. In this generality we give the conditions for stability of these systems. For stable systems, we find the response to a change of course by the leader. This response is at least linear in the size of the flock. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with at least next nearest neighbor interactions. Analytical predictions are tested in numerical simulations. PubDate: 2017-07-06 DOI: 10.1007/s12591-017-0377-3

Authors:A. M. Nagy Abstract: Abstract A non-standard finite difference (NSFD) methodology of Mickens is a popular method for the solution of differential equations. In this paper, we discusses how we can generalize NSFD schemes for solving variable-order fractional problems. The variable-order fractional derivatives are described in the Riemann–Liouville and Grünwald–Letinkov sense. Special attention is given to the Grünwald–Letinkov definition which is used to approximate the variable-order fractional derivatives. Some applications of the variable-order fractional in viscous-viscoelasticity oscillator model and chaotic financial system are included to demonstrate the validity and applicability of the proposed technique. PubDate: 2017-07-04 DOI: 10.1007/s12591-017-0378-2

Authors:R. Chandra Guru Sekar; K. Murugesan Abstract: Abstract In this article, we deal with non-linear system of higher order Volterra integro-differential equations and their numerical solutions using the Single Term Walsh Series (STWS) method. The connections between STWS coefficients of the unknown functions and its derivatives are derived. The non-linear system of Volterra integro-differential equations are converted into a system of non-linear algebraic equations using the Single Term Walsh Series coefficients. Solving these system of algebraic equations, we obtain the discrete numerical solutions of the non-linear Volterra integro-differential equations. Numerical examples are presented to show the efficiency and applicability of this STWS method for solving the non-linear system of higher order Volterra integro-differential equations. PubDate: 2017-06-30 DOI: 10.1007/s12591-017-0376-4

Authors:OPhir Nave Abstract: Abstract In this paper we present the concept of singularly perturbed vector field (SPVF) method, and its application to thermal explosion of diesel spray combustion. Given a system of governing equations, which consist of hidden Multi-scale variables, the SPVF method transfer and decompose such system to fast and slow singularly perturbed subsystems. The resulting subsystem enables us to understand better the complex system, and to simplify the calculations. Later powerful analytical, numerical and asymptotic methods [e.g method of integral (invariant) manifold, the homotopy analysis method etc.] can be applied to each subsystem. In this paper, we compare the results obtained by the methods of integral invariant manifold and SPVF as applied to the spray droplets combustion model. PubDate: 2017-06-29 DOI: 10.1007/s12591-017-0373-7

Authors:Jaume Llibre; Ammar Makhlouf Abstract: Abstract We study the zero-Hopf bifurcation of the third-order differential equations $$\begin{aligned} x^{\prime \prime \prime }+ (a_{1}x+a_{0})x^{\prime \prime }+ (b_{1}x+b_{0})x^{\prime }+x^{2} =0, \end{aligned}$$ where \(a_{0}\) , \(a_{1}\) , \(b_{0}\) and \(b_{1}\) are real parameters. The prime denotes derivative with respect to an independent variable t. We also provide an estimate of the zero-Hopf periodic solution and their kind of stability. The Hopf bifurcations of these differential systems were studied in [5], here we complete these studies adding their zero-Hopf bifurcations. PubDate: 2017-06-27 DOI: 10.1007/s12591-017-0375-5

Authors:Mudassar Imran; Muhammad Usman; Muhammad Dur-e-Ahmad; Adnan Khan Abstract: Abstract In this paper, a deterministic model is proposed to perform a thorough investigation of the transmission dynamics of Zika fever. Our model, in particular, takes into account the effects of horizontal as well as vertical disease transmission of both humans and vectors. The expression for basic reproductive number \(R_0\) is determined in terms of horizontal and vertical disease transmission rates. An in-depth stability analysis of the model is performed, and it is shown, that model is locally asymptotically stable when \(R_0 < 1\) . In this case, there is a possibility of backward bifurcation in the model. With the assumption that total population is constant, we prove that the disease free state is globally asymptotically stable when \(R_0 < 1\) . It is also shown that disease strongly uniformly persists when \(R_0> 1\) and there exists an endemic equilibrium which is unique if the total population is constant. The endemic state is locally asymptotically stable when \(R_0> 1\) . PubDate: 2017-06-23 DOI: 10.1007/s12591-017-0374-6

Authors:Alka Chadha; Swaroop Nandan Bora Abstract: This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results. PubDate: 2017-06-19 DOI: 10.1007/s12591-017-0371-9

Authors:Kuldip Singh Patel; Mani Mehra Abstract: Abstract In this paper, an unconditionally stable compact finite difference scheme for the solution of linear convection–diffusion equation is proposed. In the proposed scheme, second derivative approximations of the unknowns are eliminated with the unknowns itself and their first derivative approximations while retaining the fourth order accuracy and tri-diagonal nature of the scheme. Proposed compact finite difference scheme which is fourth order accurate in spatial variable and second or lower order accurate in temporal variable depending on the choice of weighted time average parameter is applied to Asian option partial differential equation. A diagonally dominant system of linear equation is obtained from the proposed scheme which can be efficiently solved. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed compact finite difference scheme. PubDate: 2017-06-15 DOI: 10.1007/s12591-017-0372-8

Authors:Maysam Maysami Sadr Abstract: Abstract Let \(\mathcal {M}\) be a manifold, \(\mathcal {V}\) be a vector field on \(\mathcal {M}\) , and \(\mathcal {B}\) be a Banach space. For any fixed function \(f:\mathcal {M}\rightarrow \mathcal {B}\) and any fixed complex number \(\lambda \) , we study Hyers–Ulam stability of the global differential equation \(\mathcal {V}y=\lambda y+f\) . PubDate: 2017-06-13 DOI: 10.1007/s12591-017-0369-3

Authors:Wensheng Yang; Xuepeng Li Abstract: Abstract A diffusive predator–prey model with ratio-dependent Holling type III functional response is considered in this work. Sufficient conditions for the global asymptotical stability of the constant positive steady-state solution are derived by constructing recurrent sequences and using an iterative method. It is shown that our result supplements one of the main results of Shi and Li’s paper (Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response. Appl Math Comput 250:71–77, 2015). PubDate: 2017-06-12 DOI: 10.1007/s12591-017-0370-x

Authors:Hasib Khan; Hossein Jafari; Dumitru Baleanu; Rahmat Ali Khan; Aziz Khan Abstract: Abstract The study of boundary value problems (BVPs) for fractional differential–integral equations (FDIEs) is extremely popular in the scientific community. Scientists are utilizing BVPs for FDIEs in day life problems by the help of different approaches. In this paper, we apply monotone iterative technique for the existence, uniqueness and the error estimations of solutions for a coupled system of BVPs for FDIEs of orders \(\omega ,\epsilon \in \left( 3,4 \right] \) . The coupled system is given by $$\begin{aligned} D^{\omega }u\left( t \right)= & {} -G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\varepsilon }v\left( t \right)= & {} -G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) ,\\ D^{\delta }u\left( 1 \right)= & {} 0=I^{3-\omega }u\left( 0 \right) =I^{4-\omega }u\left( 0 \right) , u(1)=\frac{{\Gamma }\left( {\omega -\delta } \right) }{{\Gamma }\left( \omega \right) }I^{\omega -\delta }\\&\quad G_1 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) ,\\ D^{\nu }v\left( 1 \right)= & {} 0=I^{3-\varepsilon }v\left( 0 \right) =I^{4-\nu }v\left( 0 \right) , v(1)=\frac{{\Gamma }\left( {\varepsilon -\nu } \right) }{{\Gamma }\left( \varepsilon \right) }I^{\varepsilon -\nu }\\&\quad G_2 \left( {t,I^{\omega }u\left( t \right) ,I^{\varepsilon }v\left( t \right) } \right) \left( {t=1} \right) , \end{aligned}$$ where \(t\in \left[ {0,1}\right] \) , \(\delta ,\nu \in \left[ {1,2} \right] .\) The functions \(G_1 ,G_2 :\left[ {0,1} \right] \times R\times R\rightarrow R,\) satisfy the Caratheodory conditions. The fractional derivatives \(D^{\omega },D^{\varepsilon },D^{\delta },D^{\nu }\) are in Riemann-Liouville sense and \(I^{\omega },I^{\varepsilon },I^{3-\omega },I^{4-\omega },I^{3-\varepsilon },I^{4-\varepsilon },I^{\omega -\delta },I^{\varepsilon -\nu }\) are fractional order integrals. The assumed technique is a better approach for the existence, uniqueness and error estimation. The applications of the results are examined by the help of examples. PubDate: 2017-06-08 DOI: 10.1007/s12591-017-0365-7

Authors:Sanaa L. Khalaf; Ayad R. Khudair Abstract: Abstract This paper adopts the inverse fractional differential operator method for obtaining the explicit particular solution to a linear sequential fractional differential equation, involving Jumarie’s modification of Riemann–Liouville derivative, with constant coefficient s. This method depends on the classical inverse differential operator method and it is independent of the integral transforms. Several examples are then given to demonstrate the validity of our main results. PubDate: 2017-05-29 DOI: 10.1007/s12591-017-0364-8