Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. In this article a numerical method for solving the one‐dimensional hyperbolic telegraph equation is presented. The method is based upon Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with Galerkin method are then utilized to reduce the telegraph equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical simulation of reaction‐diffusion telegraph systems in unbounded domains

We present a boundary element method for computing numerical solutions of the reaction‐diffusion telegraph equation in unbounded domains. This technique does not need artificial boundary conditions at the computational domain and uses a new algorithm to compute the Fourier transform, the convolution theorem, and the fact that the exact solution of the telegraph equation can be written as an integral transform in terms of the fundamental solution. We use the logistic growth model to find how the population of an organism evolves according to its growth rate. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 326–335, 2015     

A numerical method for solving the hyperbolic telegraph equation

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Traveling wavefronts in temporally discrete reaction–diffusion equations with delay

This paper is concerned with the existence of traveling wavefronts of a temporally discrete reaction–diffusion equation with delay. By using monotone iteration and upper–lower solution technique, the existence of traveling wavefronts for the temporally discrete reaction–diffusion equation with delay is established. As an application, we consider an abstract diffusive equation, which includes a single species diffusive model as a particular case. Our result implies the temporally discrete model is a good approximation of corresponding continuous time model in sense of propagation.     

Reaction diffusion equation with spatio-temporal delay

We investigate reaction–diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction–diffusion equation with spatio-temporal delay. Applying this theory to Lotka–Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem’s steady-state solution.     

Analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods

In this note we show the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods. Taking inspiration from previous papers by Dattoli et al. [4], [5] and [6] about spectral properties of Laguerre derivative, we here generalize some of their results to fractional evolution equations. Besides that, we have two interesting generalized examples. One is about telegraph equation with time dependent coefficient. The other, that could be of some interest for realistic applications, is the fractional diffusion with a space-dependent diffusion coefficient.     

Optimal time delays in a class of reaction-diffusion equations

ABSTRACT

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.     

Traveling wavefront in a Hematopoiesis model with time delay

This paper is concerned with a reaction–diffusion equation with time delay, which describes the dynamics of the blood cell production. The existence of the traveling wavefront is given by using the upper–lower solution technique and the monotone iteration.     

Fitted Galerkin spectral method to solve delay partial differential equations

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction‐diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.     

Modulation equations and parabolic limits of reaction random-walk systems

Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

On Persistence and Stability of Some Spatially Inhomogeneous Systems

Local and global stability and persistence of some coupled map lattices (CMLs) and partial differential equations are studied. A logistic CML with noninteger time step and delay is introduced. The persistence results for reaction-diffusion equations are extendable to the telegraph reaction-diffusion equation for a sufficiently small delay parameter. The stability and persistence results are applied to ecology, physics, economics, and immunology.     

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.     

Stochastic delay foraging arena predator–prey system with Markov switching

Abstract

In this paper, we introduce white noise, telegraph noise and time delay to the two-dimensional foraging arena population system describing the prey and predator abundance. The aim is to find out how the interactions between white noise, telegraph noise and time delay affect the dynamics of the population system. Firstly, the existence of a global positive solution is verified. Then the long-time properties including the stochastically ultimate boundedness, extinction and some other asymptotic pathwise estimation of this population system are studied. Finally, the main results are illustrated by two examples.     

On the uniqueness of semi-wavefronts for non-local delayed reaction–diffusion equations

We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction–diffusion equation. This result is obtained by using a generalization of the Diekmann–Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction–diffusion equations with distributed time delay are also considered.     

Time-fractional telegraph equations and telegraph processes with brownian time

We study the fundamental solutions to time-fractional telegraph equations of order 2. We are able to obtain the Fourier transform of the solutions for any and to give a representation of their inverse, in terms of stable densities. For the special case =1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.This research has been partially supported by the NATO grant No. SA (PST.CLG.976361) 5437.     

Qualitative analysis of a coupled reaction-diffusion model in biology with time delays

Certain biochemical reaction can be modeled by a coupled system of time-delayed ordinary differential equations and linear parabolic partial differential equations. In a three-compartment model these equations are coupled through the boundary conditions. The aim of this paper is to give a qualitive analysis of this unusual coupled system. The analysis includes the existence and uniqueness of a global solution, explicit upper and lower bounds of the solution, and global stability of a steady-state solution. The global stability result is with respect to any nonnegative initial perturbation and is independent of the time delays in the process of reaction. Special attention is given to the Goodwin model for biochemical control of genes by a negative feedback mechanism with time delay and diffusion.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.