Authors:R. K. Mohanty; Sachin Sharma; Swarn Singh Abstract: In this article, we proposed a new two-level implicit method of accuracy two in time and four in space based on spline in compression approximations using two half-step points and a central point on a uniform mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations subject to appropriate initial and natural boundary conditions prescribed. The proposed method is derived directly from the continuity condition of the first order derivative of the non-polynomial compression spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we have solved generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, coupled Burgers-equations and parabolic equations with singular coefficients. We show that the proposed method enables us to obtain high accurate solution for high Reynolds number. PubDate: 2018-05-31 DOI: 10.1007/s12591-018-0427-5

Authors:Anjali Jaiswal; D. Bahuguna Abstract: We introduce the concept of a mild solution of conformable fractional abstract initial value problem. We establish the existence and uniqueness theorem using the contraction principle. As a regularity result for a linear problem, we show that the mild solution is in fact a strong solution. We give an example to demonstrate the applicability of the established theoretical results. PubDate: 2018-05-28 DOI: 10.1007/s12591-018-0426-6

Authors:Abhishek Kumar; Nilam Abstract: The present study aims to control the infectious diseases and epidemics in the human population. Therefore, in the present article, we have proposed a delayed SIR epidemic model along with Holling type II incidence rate and treatment rate as Monod–Haldane type. Model stability has been established in the three regions of the basic reproduction number \( {\text{R}}_{0} \) i.e. \( {\text{R}}_{0} \) equals to one, greater than one and less than one. The model is locally asymptotically stable for disease-free equilibrium \( {\text{Q}} \) when the basic reproduction number \( {\text{R}}_{0} \) is less than one ( \( {\text{R}}_{0} < 1) \) and unstable when \( {\text{R}}_{0} > 1 \) for time lag \( \tau \ge 0 \) . We investigated the stability of the model for disease-free equilibrium at \( {\text{R}}_{0} \) equals to one using central manifold theory. Using center manifold theory, we proved that at \( {\text{R}}_{0} = 1 \) , disease-free equilibrium changes its stability from stable to unstable. We also investigated the stability for endemic equilibrium \( {\text{Q}}^{ *} \) for time lag \( \tau \ge 0 \) . Further, numerical simulations are presented to exemplify the analytical studies. PubDate: 2018-05-10 DOI: 10.1007/s12591-018-0424-8

Authors:A. K. Tripathy; S. S. Santra Abstract: In this work, necessary and sufficient conditions for the oscillation of a class of second order neutral impulsive systems are established. PubDate: 2018-05-09 DOI: 10.1007/s12591-018-0425-7

Authors:Belal Almuaalemi; Haibo Chen; Sofiane Khoutir Abstract: In this paper, we concern with the following nonhomogeneous fourth-order quasilinear equations of the form $$\begin{aligned} \Delta ^2u - \Delta u+V(x)u -\frac{k}{2} \Delta (u)^2u = f(x,u)+h(x) ,\quad x\in \mathbb {R}^{N}, \end{aligned}$$ where \(\Delta ^2 :=\Delta (\Delta )\) is the biharmonic operator, \(k \ge 0\) , \(N \le 6\) , \(V\in C(\mathbb {R}^{N},\mathbb {R})\) , \(f\in C(\mathbb {R}^{N}\times \mathbb {R},\mathbb {R})\) and \(h(x) \in L^2(\mathbb {R}^{N})\) . Under some relaxed assumptions on the nonlinear term f, a new result of multiple nontrivial solutions is obtained via the Ekeland’s variational principle and the mountain pass theorem. PubDate: 2018-05-04 DOI: 10.1007/s12591-018-0421-y

Authors:Niklas L. P. Lundström; Gunnar Söderbacka Abstract: We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest. PubDate: 2018-05-03 DOI: 10.1007/s12591-018-0422-x

Authors:Khalil Ezzinbi; Mohamed Ziat Abstract: In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$ without the assumption of equicontinuity on the resolvent operator and without the assumption of separability on the Banach space X. The nonlocal initial condition is assumed to be compact. Our main result is new and its proof is based on a measure of noncompactness developed in Kamenskii et al. (Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Walter De Gruyter, Berlin, 2001) together with the well-known Mönch fixed point Theorem. To illustrate our result, we provide an example in which the resolvent operator is not equicontinuous. PubDate: 2018-04-27 DOI: 10.1007/s12591-018-0423-9

Authors:Ali Barzanouni Abstract: Let \(f:X\rightarrow X\) be a continuous map on a compact metric space (X, d) with no isolated points. We introduce the concept of \(\epsilon \) -equicontinuous point in (X, f), indeed a point \(x\in X\) , is called \(\epsilon \) -equicontinuous point, \(x\in Eq_{\epsilon }(f)\) , if there exists \(\delta > 0\) such that \(diam(f^{n}(B(x, \delta )))\le \epsilon \) for every \(n\in {\mathbb {Z}}_+\) We give a system (X, f) such that for every \(\epsilon >0\) , \(Eq_{\epsilon }(f)\ne \emptyset \) but f has no equicontinuous point. Next, we study its basic properties and compare some property of \(\epsilon \) -equicontinuous points in (X, f) with those in two related dynamical systems, inverse limit space ( \(\lim _{\leftarrow }(X,f), \sigma _{f})\) and hyper space \((2^{X}, 2^{f})\) . It is known that a topologically transitive with equicontinuity points is uniformly rigid, we give an \(\epsilon \) version of that result. We also introduce the concept \(\epsilon \) -chain continuity points and study relation between \(\epsilon \) -chain continuity points and two concepts \(\epsilon \) -shadowable points and \(\epsilon \) -equicontinuous points. PubDate: 2018-04-26 DOI: 10.1007/s12591-018-0418-6

Authors:Rajnee Tripathi; Hradyesh Kumar Mishra Abstract: In this article, we established an application of homotopy perturbation method using Laplace transform (LT-HPM) to elaborate the analytical solution of heat conduction equation in the heterogeneous casting-mould system. The solution of the problem is provided with our supposition of an ideal contact between the cast and the mould. In the proposed method, we have chosen initial approximations of unknown constants which can be implemented by imposing the boundary and initial conditions. Examples have been discussed and confirmed the usefulness of this method. PubDate: 2018-04-21 DOI: 10.1007/s12591-018-0417-7

Authors:Priyanka Ghosh; Xianbing Cao; Joydeep Pal; Sonia Chowdhury; Shubhankar Saha; Sumit Nandi; Priti Kumar Roy Abstract: Methanol (MeOH) poisoning is a burning issue mostly for the third world country. Toxic methanol is the major compound in impure alcohol when consumed. It causes severe health hazards and sometimes causes death. Methanol when breakdowns into formate in the presence of alcohol dehydrogenase enzymes in human liver, it becomes toxic. This enzyme catalyzes the substrate (MeOH) to produce toxic metabolites (i.e. formate). In methanol toxicity, ethanol is suggested to inhibit the metabolism of methanol as antidote. This is the most common treatment for averting toxicity of methanol in clinics. Based on the chemical kinetics of the reaction, we formulate a mathematical model for the treatment of methanol toxicity with the effects of a constant competitive substrate input (ethanol). Our mathematical study is revealed the dosing policy for administering ethanol as antidote for morbid intoxicated patient. We also find the minimal time interval of the antidote dosing which stops the harmful reaction in treated patient. Numerical simulation of the nonlinear model has confirmed our analytical studies. PubDate: 2018-04-19 DOI: 10.1007/s12591-018-0420-z

Authors:Vu Trong Luong; Do Van Loi; Hoang Nam Abstract: This paper deals with a class of two-term time fractional differential equations with nonlocal initial conditions. We establish the existence of mild solutions with explicit decay rate of polynomial type. To illustrate the abstract results, an example is also given. PubDate: 2018-04-17 DOI: 10.1007/s12591-018-0419-5

Authors:Gopal Priyadarshi; B. V. Rathish Kumar Abstract: In this paper, we study the existence and uniqueness of the solution of Fredholm integral equation of the first kind with convolution type kernel. These results are based on band-limited scaling function which is generated by a class of band-limited wavelets. Since these band-limited functions are infinitely differentiable and possess rapid decay property, methods based on these functions would be highly accurate. Finally, convergence analysis has been discussed to validate the approximate solution. PubDate: 2018-04-07 DOI: 10.1007/s12591-018-0416-8

Authors:Francisco J. Solis; Ignacio Barradas; Daniel Juarez Abstract: In this work we introduce a family of operators called discrete advection–reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection–reaction–diffusion partial differential equations. They consists of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory. PubDate: 2018-04-02 DOI: 10.1007/s12591-018-0415-9

Authors:Rowthu Vijayakrishna; B. V. Rathish Kumar; Abdul Halim Abstract: A PDE based binary image segmentation model using a modified Cahn–Hilliard equation with weaker fidelity parameter ( \(\lambda \) ) and double well potential has been introduced. The threshold of separation \(\gamma \) is flexibly chosen between 0 and 1. Convexity splitting is used for time discretization and then Fourier-spectral method is used to solve the proposed modified Cahn–Hilliard equation. The proposed model is tested on bio-medical images. PubDate: 2018-03-15 DOI: 10.1007/s12591-018-0414-x

Authors:Talat Körpınar Abstract: In this work, we construct energy of the moving particle on tangent spherical images in diverse force fields on dynamical and electrodynamical devices by way of Newtonian mechanics. We work with both physical but even more major geometrical strategy meant for the calculation concerning the energy on the moving particle in these types of force fields. We demonstrate that energy on the moving charged particle or energy on a massive body that follows a trajectory of a moving particle can be characterized by intrinsic geometric highlights of tangent spherical images. We likewise associate the connection among energy in moving particle for the distinct category of force fields. PubDate: 2018-03-05 DOI: 10.1007/s12591-018-0413-y

Authors:Pavel Nesterov Abstract: We construct the asymptotics for solutions of two second-order integro-differential equations of Volterra type as independent variable tends to infinity. These equations are considered as integral perturbations of the harmonic oscillator. The specific feature of the considered integral perturbations is an oscillatory decreasing character of their kernels. To obtain the asymptotic formulas we use the special method proposed for the asymptotic integration of the linear dynamical systems with oscillatory decreasing coefficients. The method uses the ideas of the averaging theory and some known asymptotic theorems. PubDate: 2018-03-01 DOI: 10.1007/s12591-018-0412-z

Authors:Jayanta Borah; Swaroop Nandan Bora Abstract: We establish a set of sufficient conditions for the existence of mild solution of a class of fractional mixed integro differential equation with not instantaneous impulses. The results are obtained by establishing two theorems by using semigroup theory, Banach fixed point theorem and Krasnoselskii’s fixed point theorem. Two examples are presented to validate the results of the theorems. PubDate: 2018-02-12 DOI: 10.1007/s12591-018-0410-1

Authors:Sabbavarapu Nageswara Rao Abstract: In this paper, we establish the criteria for the existence and uniqueness of solutions of a two-point BVP for a system of nonlinear fractional differential equations on time scales. $$\begin{aligned} \begin{aligned} \Delta _{a^{\star }}^{\alpha _{1}-1}x(t)&=f_{1}(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb {T},\\ \Delta _{a^{\star }}^{\alpha _{2}-1}y(t)&=f_{2}(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb {T},\\ \end{aligned} \end{aligned}$$ subject to the boundary conditions $$\begin{aligned} \begin{aligned} x(a)=0,&\quad x^{\Delta }(b)=0,\quad x^{\Delta \Delta }(b)=0,\\ y(a)=0,&\quad y^{\Delta }(b)=0,\quad y^{\Delta \Delta }(b)=0. \end{aligned} \end{aligned}$$ where \(\mathbb {T}\) is any time scale (nonempty closed subsets of the reals), \(2<\alpha _{i}<3\) and \(f_{i}\in C_{rd}([a,b]\times \mathbb {R}\times \mathbb {R}, \mathbb {R})\) and \(\Delta _{a^{\star }}^{\alpha _{i}-1}\) denotes the delta fractional derivative on time scales \(\mathbb {T}\) of order \(\alpha _{i}-1\) for \(i=1, 2\) . By using the Banach contraction principle. Finally, an example is given to illustrate the main result. PubDate: 2018-02-09 DOI: 10.1007/s12591-018-0409-7

Authors:Shihua Zhang; Rui Xu Abstract: In this paper, an SIS epidemic model with age of vaccination is investigated. Asymptotic smoothness of the semi-flow is proved. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state is discussed. It is shown that if the basic reproduction number is greater than unity, the system is permanent. By constructing two Lyapunov functionals, it is proved that the endemic steady state is globally asymptotically stable if the basic reproduction number is greater than unity, and sufficient conditions are derived for the global asymptotic stability of the disease-free steady state. Numerical simulations are given to illustrate the asymptotic stabilities of the disease-free steady state and endemic state. PubDate: 2018-02-09 DOI: 10.1007/s12591-018-0408-8

Authors:G. Gopi Krishna; S. Sreenadh; A. N. S. Srinivas Abstract: The present investigation deals with the flow of a viscous fluid in an inclined deformable porous layer bounded by rigid plates. The lower and upper moving plates are maintained at constant different temperatures. A heat source of strength \(Q_0 \) is introduced in the porous layer. The coupled phenomenon of the fluid movement and solid deformation in the porous layer has been considered. An exact solution of governing equations has been obtained in closed form. The expressions for the fluid velocity, solid displacement and temperature distribution are obtained. The influence of pertinent parameters on flow quantities is discussed. In the inclined deformable porous medium, it is observed that the fluid velocity and temperature decreases with increasing viscous drag \(\delta \) . But the solid displacement increases with increasing viscous drag \(\delta \) . A table of comparison is made for flux in the present work and that of flux observed by Nield et al. (Transp Porous Media 56:351–367, 2004) for viscous flow in a horizontal undeformable porous medium. One of the important observations is that the volume flow rate is less for deformable porous media when compared with undeformable (rigid) porous media. The present result coincides with the findings of Nield et al. (2004). The results obtained for the present flow characteristics reveal many interesting behaviors that warrant further study of viscous fluid flow in an inclined deformable porous media. PubDate: 2018-02-08 DOI: 10.1007/s12591-018-0411-0