Authors:Madhukant Sharma; Shruti Dubey Pages: 499 - 517 Abstract: This work deals with the existence of solutions for a class of nonlinear nonlocal fractional functional differential equations of neutral type in Banach spaces. In particular, we prove the existence of solutions with the assumptions that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear operator \(-A(t)\) generates analytic semigroup for each \(t \ge 0\) . We also investigate global existence of solution and study the continuous dependence of solution on initial data. We conclude the article with an application to the developed results. PubDate: 2017-10-01 DOI: 10.1007/s12591-016-0290-1 Issue No:Vol. 25, No. 4 (2017)

Authors:Assia Guezane-Lakoud; Allaberen Ashyralyev Pages: 519 - 526 Abstract: In this paper, we discuss by means of a fixed point theorem, the existence of positive solutions of a system of nonlinear Caputo fractional differential equations with integral boundary conditions. An example is given to illustrate the main results. PubDate: 2017-10-01 DOI: 10.1007/s12591-015-0255-9 Issue No:Vol. 25, No. 4 (2017)

Authors:J. Ahn; K. L. Kuttler; M. Shillor Pages: 527 - 552 Abstract: This work models, analyses and simulates a coupled dynamic system consisting of a thermoviscoelastic rod and a linear viscoelastic beam. It is motivated by recent developments in MEMS systems, in particular the “V-shape” electro-thermal actuator that realizes large displacement and reliable contact in MEMS switches. The model consists of a system of three coupled partial differential equations for the beam’s and the rods’ displacements, and the rod’s temperature. Moreover, the rod may come in contact with a reactive foundation at one end, which is the main aspect of the actuating or switching property of the system. The thermal interaction at the contacting end of the rod is described by Barber’s heat exchange condition. The system is analyzed by setting it in an abstract form for which the existence of a weak solution is shown by using tools from the theory of variational inequalities and a fixed point theorem. A numerical algorithm for the system is constructed; its implementation yields computational depiction of the system’s behavior, with emphasis on the combined vibrations of the beam-rod system, dynamic contact force and thermal interaction. PubDate: 2017-10-01 DOI: 10.1007/s12591-016-0301-2 Issue No:Vol. 25, No. 4 (2017)

Authors:Jocirei D. Ferreira; Alejandra M. Pulgarin Galvis; V. Sree Hari Rao Abstract: In this paper we study the interactions among two species of predators and one single species of prey populations, under a more generalized functional response arising out of the competition. The models presented in this research for these species exhibit rich dynamics for varying values of the vital parameters involved. Conditions for the existence and stability of equilibria of the model equations are established. PubDate: 2017-11-11 DOI: 10.1007/s12591-017-0398-y

Authors:A. Boudaoui; E. Lakhel Abstract: In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result. PubDate: 2017-11-05 DOI: 10.1007/s12591-017-0401-7

Authors:Junjun Zhou; Xiao Bie; Yangyang Wang Abstract: In this paper, we consider the multiplicity of solutions for some damped nonlinear impulsive differential equations by using Morse theory in combination with the minimax arguments. Under some assumptions, we get some new results on the existence of multiple nontrivial solutions for the problems. Thus, we improve and extend recent results. PubDate: 2017-10-27 DOI: 10.1007/s12591-017-0400-8

Authors:P. Pramod Chakravarthy; Kamalesh Kumar Abstract: In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction–diffusion type. A fitted operator finite difference scheme based on Numerov’s method is constructed. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method. PubDate: 2017-10-27 DOI: 10.1007/s12591-017-0399-x

Authors:S. Shahmorad; S. Pashaei; M. S. Hashemi Abstract: Numerical solution of a Riemann–Liouville fractional integro-differential boundary value problem with a fractional nonlocal integral boundary condition is studied based on a numerical approach which preserve the geometric structure on the Lorentz Lie group. A fictitious time \(\tau \) is used to transform the dependent variable y(t) into a new one \(u(t,\tau ):=(1+\tau )^{\gamma }y(t)\) in an augmented space, where \(0<\gamma \le 1\) is a parameter, such that under a semi-discretization method and use of a Newton-Cotes quadrature rule the original equation is converted to a system of ODEs in the space \((t,\tau )\) and the obtained system is solved by the Group Preserving Scheme (GPS). Some illustrative examples are given to demonstrate the accuracy and implementation of the method. PubDate: 2017-10-14 DOI: 10.1007/s12591-017-0395-1

Authors:Savannah Bates; Hayley Hutson; Jorge Rebaza Abstract: The few mathematical models available in the literature to describe the dynamics of Zika virus are still in their initial stage of stability and bifurcation analysis, and they were in part developed as a response to the most recent outbreaks, including the one in Brazil in 2015, which has also given more hints to its association with Guillain–Barre Syndrome (GBS) and microcephaly. The interaction between and the effects of vector and human transmission are a central part of these models. This work aims at extending and generalizing current research on mathematical models of Zika virus dynamics by providing rigorous global stability analyses of the models. In particular, for disease-free equilibria, appropriate Lyapunov functions are constructed using a compartmental approach and a matrix-theoretic method, whereas for endemic equilibria, a relatively recent graph-theoretic method is used. Numerical evidence of the existence of a transcritical bifurcation is also discussed. PubDate: 2017-10-14 DOI: 10.1007/s12591-017-0396-0

Authors:Ghasem A. Afrouzi; Shahin Moradi; Giuseppe Caristi Abstract: In this paper, we study the existence of solutions for impulsive beam equations of Kirchhoff-type. By using critical point theory, we obtain some new criteria for guaranteeing that impulsive fourth-order differential equations of Kirchhoff-type have infinitely many solutions. Some recent results are extended and improved. An example is presented to demonstrate the application of our main results. PubDate: 2017-10-13 DOI: 10.1007/s12591-017-0397-z

Authors:Vivek Kumar; Balaji Srinivasan Abstract: In this article, we study a novel adaptive mesh strategy for singularly perturbed problems (SPPs) of the parabolic convection-diffusion type that exhibit regular boundary layers. Our central insight is that the introduction of an auxiliary inequality for an entropy-like variable also serves as a remarkably effective adaptation indicator. The primary novelty of this method is that, unlike extant methods used for layer adapted meshes (in which enough mesh points exist in the layer region for well resolved numerical solution), the current method requires no a priori knowledge of the location and width of the boundary layers. Further, the current method, [(which is an extension of the methodology from Kumar and Srinivasan (Appl Math Model 39:2081–2091, 2015)] is completely independent of the perturbation parameter and results in accurate solutions for a wide range of problems. We include some preliminary error estimates and also the results of several numerical experiments including the Black–Scholes equation. The results exhibit the promise of the proposed strategy to generate efficient adaptive meshes for time dependent convection-diffusion problems. PubDate: 2017-09-30 DOI: 10.1007/s12591-017-0394-2

Authors:Aditya Kaushik; Geetika Gupta; Manju Sharma; Vishal Gupta Abstract: A wavelet based rationalized method is presented for the numerical solution of differential, integral and integro-differential equations. Rationalized Haar functions are used to estimate the solution. Their fundamental properties are discussed. A rigorous convergence analysis is presented. The operational matrix of the product of two rationalized Haar functions is used to reduce the dynamical system to an algebraic system. A variety of model problems are taken into account so as to test the efficiency of the proposed method. The result so obtained are compared with the available exact solutions. In addition, proposed scheme is compared with some state of the art existing methods. It is found that the Haar wavelet operational matrix is the fastest. Moreover, the results obtained are mathematically simple and the desired accuracy of the solution is obtained using small number of grid points. The main advantages of the wavelet method are its simplicity, fast transformation, possibility of implementation of fast algorithms and low computational cost with high accuracy. PubDate: 2017-09-19 DOI: 10.1007/s12591-017-0393-3

Authors:R. K. Mohanty; Gunjan Khurana Abstract: In this paper, we propose a new three level implicit method based on half-step spline in tension method of order two in time and four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form \(w_{tt}=K(x,t,w)w_{xx} + {\varphi }(x,t,w,w_{x},w_{t})\) . We describe spline in tension approximations and its properties using two half-step grid points. The new method for one dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function w(x, t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method when applied to Telegraphic equation is shown to be unconditionally stable. Further, the stability condition for 1-D linear hyperbolic equation with variable coefficients is established. Our method is directly applicable to hyperbolic equations irrespective of the coordinate system which is the main advantage of our work. The proposed method for scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems and numerical computations are provided to demonstrate the effectualness of the method. PubDate: 2017-09-14 DOI: 10.1007/s12591-017-0391-5

Authors:Arti Mishra; Sunita Gakkhar Abstract: In this paper, a non-linear stage-structured model has been formulated to study the effects of primary/secondary dengue infection on children and adults. It is assumed that a proportion of susceptible children have been previously infected by a serotype asymptomatically. After recovery from asymptomatic infection, the immunity is developed for the particular serotype but they remain susceptible to heterologous serotypes. These children may get secondary infection when exposed to a different serotype. The model has two equilibrium states. Global/local stability of equilibrium states have been discussed. The stability of disease-free state changes at \(R_0=1\) . Applying the center manifold theory, the existence of forward bifurcation is possible. This concludes that the primary/secondary infection in children as well as in adults population could be eradicated for \(R_0<1\) . Numerical results are used to explore the global behavior of disease-free/endemic state for choice of arbitrary initial conditions. PubDate: 2017-09-13 DOI: 10.1007/s12591-017-0387-1

Authors:Hariom Sharma; Rajan Arora Abstract: In the present work we have taken one-dimensional unsteady flow of non-ideal gas with magnetic effect under the presence of thermal radiation. The system is hyperbolic in nature and solved by similarity method using Lie Group of Transformations under the assumption that the system is constantly conformally invariant under the transformations. The similarity solutions are investigated behind a cylindrical shock which is a consequence of a sudden explosion or produced by an expanding piston. The shock is assumed to be strong and propagating into the medium which is at rest, with uniform density. The total energy of the shock is assumed to be time dependent and obeying the power law. By means of similarity method our system of PDEs transformed into the system of ordinary differential equations (ODEs), which in general are nonlinear. The effects of thermal radiation on the the flow variables velocity, density, pressure and magnetic field are investigated behind the shock. PubDate: 2017-09-12 DOI: 10.1007/s12591-017-0392-4

Authors:Komal Bansal; Kapil K. Sharma Abstract: The motive of the present work is to develop a parameter robust numerical scheme for the class of problems involving singularly perturbed parabolic differential-difference equations with delay, which often arise in computational neuroscience. The numerical schemes developed prior to this work are restricted either to the case of small values of delay argument or linear convergence with restriction on the mesh generation. In practice, the delay argument can be of arbitrary size. Parameter \(\varepsilon \) may take small enough values e.g., viscosity coefficient in Navier–Stokes equation for fluids with high Reynolds number. It is required to construct a higher order parameter robust numerical scheme without any restriction on the mesh generation for singularly perturbed parabolic differential-difference equations with state dependent delay of arbitrary size. A new class of non-standard finite difference method based on interpolation, \(\theta \) -method and Micken’s techniques is constructed to approximate the solution of singularly perturbed parabolic differential-difference equations with arbitrary values of delay. It is shown that proposed numerical scheme is parameter uniform convergent. It is proved that this method is unconditionally stable and is convergent for \(\frac{1}{2} \le \theta \le 1,\) without having any restriction on the mesh. Some numerical experiments are provided to illustrate the performance of the method. PubDate: 2017-09-11 DOI: 10.1007/s12591-017-0390-6

Abstract: In this work we state sufficient conditions for convergence for nonhyperbolic fixed points of multidimensional discrete dynamical systems and analyze their speed of convergence. We introduce a new concept of slow convergence based on one dimensional discrete systems and we establish a general classification for slow discrete dynamical systems which is of paramount importance in numerical analysis and in the development of algorithms. A variety of two dimensional examples are presented to illustrate the diverse possibilities of convergence existing in multidimensional systems. PubDate: 2017-09-07 DOI: 10.1007/s12591-017-0388-0

Abstract: In this paper we consider a thermoelastic system of second sound with internal delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Cattaneo’s law is still strong enough to uniformly stabilize the system even in the presence of time delay. PubDate: 2017-09-07 DOI: 10.1007/s12591-017-0389-z

Authors:R. Agarwal; S. Hristova; D. O’Regan Abstract: The stability properties of Caputo fractional differential equations with impulses are studied. Both types of impulses, non-instantaneous impulses as well as instantaneous impulses are considered. The two approaches in the literature for the interpretation ofsolutions of impulsive Caputo fractional differential equations are presented and discussed. A generalization of Mittag–Leffler stability with respect to both types of impulses is given. PubDate: 2017-08-24 DOI: 10.1007/s12591-017-0384-4

Authors:Patrice Ndambomve; Khalil Ezzinbi Abstract: This work concerns the study of the controllability for some nonlinear partial functional integrodifferential equation with finite delay arising in the modelling of materials with memory in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator. An example of applications is given for illustration. PubDate: 2017-08-23 DOI: 10.1007/s12591-017-0386-2