Abstract: Abstract Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion. PubDate: 2020-09-01

Abstract: Abstract The problem of analyzing the Itô stochastic differential system and its filtering has received attention. The classical approach to accomplish filtering for the Itô SDE is the Kushner equation. In contrast to the classical filtering approach, this paper presents filtering for the stochastic differential system affected by weakly coloured noise, i.e., \(\dot{x}_{t} = f\left( {x_{t} } \right) + g\left( {x_{t} } \right)\xi_{t} ,\) where the input process \(\xi_{t}\) is a weakly coloured noise process. As a special case, the process \(\xi_{t}\) can be regarded as the Ornstein–Uhlenbeck (OU) process, i.e., \(d\xi_{t} = - \alpha \xi_{t} dt + \beta dB_{t} ,\) where \(\alpha > 0.\) More precisely, the filtering model of this paper can be cast as \(\dot{x}_{t} = f\left( {x_{t} } \right) + g\left( {x_{t} } \right)\xi_{t} ,\) \(z_{t} = \int\limits_{{t_{0} }}^{t} {h(x_{\tau } )d\tau + \eta_{t} ,}\) where \(h(x_{t} )\) is the measurement non-linearity and \(\eta = \{ \eta_{t} ,t_{0} \le t < \infty \}\) is the Brownian motion process. The former expression describes the structure of a noisy dynamical system, and the latter is the observation equation. The novelties of this paper are two (1) the extension of the filtering theory for the Itô stochastic differential system to the filtering theory for the ‘weakly coloured noise-driven’ stochastic differential system (2) the theory of this paper is based on a pioneering contribution of Ruslan Stratonovich involving the perturbation-theoretic approach to noisy dynamical systems in combination with the notion of the ‘filtering density’ evolution. The stochastic evolution of condition moment is derived by utilizing the filtering density evolution equation. A scalar Duffing system driven by the OU process is employed to test the effectiveness of the filtering theory of the paper. Numerical simulations involving four different sets of initial conditions and system parameters are utilized to examine the efficacy of the filtering algorithm of this paper. PubDate: 2020-08-31

Abstract: Abstract The authors present some new sufficient conditions for the oscillation of second order quasilinear neutral delay differential equation $$\begin{aligned} (a(t)(z'(t))^{\beta })'+q(t)x^{\gamma }(\sigma (t))=0,\;t\ge t_0>0, \end{aligned}$$ where \(z(t)=x(t)+p(t)x(\tau (t))\) . Our oscillation results depend on only one condition and essentially improve, complement and simplify many related ones in the literature. Examples are provided to illustrate the value of the main results. PubDate: 2020-08-24

Abstract: Abstract In this paper, we investigate a differential game problem of multiple number of pursuers and a single evader with motions governed by a certain system of first-order differential equations. The problem is formulated in the Hilbert space \(\ell_2,\) with control functions of players subject to integral constraints. Avoidance of contact is guaranteed if the geometric position of the evader and that of any of the pursuers fails to coincide for all time t. On the other hand, pursuit is said to be completed if the geometric position of at least one of the pursuers coincides with that of the evader. We obtain sufficient conditions that guarantees avoidance of contact and construct evader’s strategy. Moreover, we prove completion of pursuit subject to some sufficient conditions. Finally, we demonstrate our results with some illustrative examples. PubDate: 2020-08-19

Abstract: Abstract Distributed delay is included in a simple predator–prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent “discount” factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the “discounting” term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions. PubDate: 2020-08-14

Abstract: Abstract In this paper, we study the dynamic of a multi-strain SEIR model with both saturated incidence and treatment functions. Two basic reproduction numbers are extracted from the epidemic model, noted \(R_{0,1}\) and \(R_{0,2}\) . Using the Lyapunov method, we investigate the global stability of the disease free equilibrium and prove that it is globally asymptotically stable when \(R_{0,1}\) and \(R_{0,2}\) are less than one. Moreover, we formulate the optimal control problem, solve it, and perform some numerical simulations, to support the analytical results and test how well the proposed model may be applied in practice. PubDate: 2020-08-13

Abstract: Abstract The objective of this paper is to investigate a mathematical model describing the interactions between hepatitis B virus with DNA-containing capsids, liver cells (hepatocytes) and the adaptive immune response. This adaptive immunity will be represented by cytotoxic T-lymphocytes and antibody immune responses. The positivity and boundedness of solutions for non negative initial data are proved which is consistent with the biological studies. The local stability of the equilibria is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibria is fulfilled by using appropriate Lyapunov functions. Finally, numerical simulations are performed to support our theoretical findings. PubDate: 2020-08-13

Abstract: Abstract Local perturbation analysis and oscillation of eigenvalues of the two parameter Sturm–Liouville system 1 $$\begin{aligned} -y_{1}^{\prime \prime } + q_{1} y_{1}\,\, \,= \,\,\, ( \lambda r_{11} + \mu r_{12} ) y_{1} \,\,\, \text{ on } \,\, [0,1] \end{aligned}$$ with the boundary conditions $$\begin{aligned} \frac{y_{1}' (0)}{y_{1} (0) } = \cot \alpha _{1} \,\, \text{ and } \,\,\, \frac{y_{1}' (1) }{y_{1} (1) } = \frac{a_{1} \lambda + b_{1} }{c_{1} \lambda + d_{1} } \end{aligned}$$ and 2 $$\begin{aligned} -y_{2}^{\prime \prime } + q_{2} y_{2} \,\, \,= \,\,\, ( \lambda r_{21} + \mu r_{22}) y_{2} \end{aligned}$$ with the boundary conditions $$\begin{aligned} \frac{y_{2}' (0) }{y_{2} (0)} = \cot \alpha _{2} \,\, \text{ and } \,\,\, \frac{y_{2}' (1) }{y_{2} (1)} = \frac{ a_{2} \mu + b_{2} }{c_{2} \mu + d_{2} }, \end{aligned}$$ are studied on the compact interval [0,1]. Here \(q_{i}\) and \(r_{ij}\) are continuous real valued functions on [0, 1], the angle \(\alpha _{i}\) is in \([0,\pi )\) and \(a_{i} , b_{i} , c_{i}, d_{i}\) are real numbers with \(\delta _{i} = a_{i} d_{i} - b_{i}c_{i} > 0\) and \(c_{i} \ne 0\) for \(i,j=1,2\) . Results on local analysis, algebraic multiplicity and oscillation of eigenvalues are given. Analysis on the presence of complex eigenvalues is conducted and an upper bound for the number of nonreal eigenvalues is obtained. PubDate: 2020-08-11

Abstract: Abstract We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in \(\mathrm L^p\) -spaces, involving first derivatives of the pressure. PubDate: 2020-08-08

Abstract: In this paper, we study boundary value problems for parameter-dependent elliptic differential-operator equations with variable coefficients in smooth domains. Uniform regularity properties and Fredholmness of this problem are obtained in vector-valued \( {\text{L}}_{p} \) -spaces. We prove that the corresponding differential operator is positive and is a generator of an analytic semigroup. Then, via maximal regularity properties of the linear problem, the existence and uniqueness of the solution to the nonlinear elliptic problem is obtained. As an application, we establish maximal regularity properties of the Cauchy problem for abstract parabolic equations, Wentzell–Robin-type mixed problems for parabolic equations, and anisotropic elliptic equations with small parameters. PubDate: 2020-08-01

Abstract: By using the measure theory on time scales, we extend the notion of weighted Stepanov-like pseudo almost periodicity to time scales and study some of its basic properties. To illustrate our abstract results, we study the existence and uniqueness of weighted pseudo almost periodic solutions to some classes of nonautonomous dynamic equations involving weighted Stepanov-like pseudo almost periodic forcing terms on time scales. PubDate: 2020-07-22

Abstract: Abstract In this study, the effect of the induced electromagnetic fields on the linear instability of a viscoelastic nanofluid liquid layer is discussed. Two physical cases are considered for the channel flow. The first one is considered for the free–free boundaries and the second for the rigid–rigid boundaries. The instability is investigated analytically due to the normal mode analysis and numerically according to the relation between the energy density function and time. The Routh–Hurwitz criteria are applied. The dispersion relation is obtained as a sixth-degree equation of the growth rate. Also, the stationary, as well as oscillatory states of the thermal Rayleigh number, are obtained. The numerical solution is compared with the exact solution for a special choice of the parameters and the results show higher match between the results. The main important result of the study concludes that the Brownian motion of the nanoparticles destabilizes the nanofluid layer. This means that the pure fluid layer is more stable than the nanofluid layer. Also, the results confirm that the flow between the rigid–rigid boundaries is more stable than the free–free boundaries. Finally, due to the Lorentz force, the induced magnetic field destabilizes the motion due to the temperature rise. PubDate: 2020-07-20

Abstract: Abstract We deal with a system of quasilinear fractional differential equations, subjected to the Cauchy–Nicoletti type boundary conditions. The task of explicit solution of such problems is difficult and not always solvable. Thus we suggest a suitable numerical–analytic technique that allows to construct an approximate solution of the studied fractional boundary value problem with high precision. PubDate: 2020-07-10

Abstract: Abstract In this paper, we study an optimal control problem for a generalized stochastic SIVR model as well as for the corresponding deterministic model. We consider two control strategies in the optimal control model, namely: the successful practice of non-pharmaceutical interventions and vaccination for susceptible strategies. The existence of optimal control in the deterministic case is proved and it is solved by using Pontryagins Maximum Principle. Moreover, the stochastic optimal control problem is discussed by using Dynamic programming approach and the results are obtained numerically through simulation using an approximation based on the solution of the deterministic model. Outputs of the simulations show that our control strategies play important role in the minimization of infectious population with minimum cost. PubDate: 2020-07-08

Abstract: Abstract Strichartz estimates for a time-decaying harmonic oscillator were proven with some assumptions of coefficients for the time-decaying harmonic potentials. The main results of this paper are to remove these assumptions and to enable us to deal with the more general coefficient functions. Moreover, we also prove similar estimates for time-decaying homogeneous magnetic fields. PubDate: 2020-07-06

Abstract: Abstract In this article we present the analysis of generalized S-mesh (denoted by \(S(\ell )\) ) based hybrid algorithm for a class of second-order singularly perturbed differential equations with discontinuous convection coefficient. The solution of the problem exhibits interior layer because of the discontinuity in convection coefficient. We have demonstrated the generation of \(S(\ell )\) mesh for a domain with interior layer and also estimated that the algorithm is perturbation parameter ( \(\epsilon \) ) uniformly convergent with error asymptotic to \(N^{-2}(\ln ^\ell (N))^2\) where \( \ell \ll N ~ \& ~ \ell \in {\mathbb {N}}\) . The numerical experiments of the algorithm on Shishkin mesh, B-type mesh and \(S(\ell )\) mesh are carried out for couple of examples, to demonstrate the efficiency of the proposed hybrid algorithm on \(S(\ell )\) mesh. It was observed that upwind algorithm on B-type mesh is more efficient than upwind algorithm on Shishkin or \(S(\ell )\) mesh and the proposed hybrid algorithm on \(S(\ell )\) mesh is of second order and observed to be efficient. PubDate: 2020-07-01

Abstract: Abstract The present article is devoted to develop an adaptive scheme for the numerical solution of fractional integro-differential equations with weakly singular kernel. The adaptive scheme is based on the product integration method of Huber. An error estimate is provided for the discretisation error occurring at each step to calculate the step size of next integration step. A parameter known as error tolerance is predefined to control local discretisation error. Computations verify that by controlling local discretisation error, the true global errors match fairly well to the error tolerance parameter. The first integration step size \(h_{{ initial}}\) is introduced in order to make a controlled evaluation of local discretisation error at the first integration step also, where the error estimate is not available. Finally, the computations and results of some numerical experiments validate the accuracy and applicability of the adaptive scheme. PubDate: 2020-07-01

Abstract: Abstract Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known co-efficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods. PubDate: 2020-07-01

Abstract: Abstract Spatially long-range interactions for linearly elastic media resulting in dispersion relations are modelled by an integro-differential equation of convolution type (IDE) that incorporate non-local effects. This type of IDE is nonstandard (hence, it is almost impossible to obtain exact solutions) and plays an important role in modeling various applied science and engineering problems. In this article, such an IDE describing a linear elastic wave phenomenon has been studied. First, a discrete equivalent of the model IDE in space is proposed and then a class of forward backward average one step \(\theta \) scheme for the semi-discrete time dependent numerical method has been developed. Further, stability and accuracy of the developed method has been analyzed rigorously. PubDate: 2020-07-01