An approximate method for solving a class of singular perturbation problems

AbstractAn approximate method for the numerical solution of a class of singularly perturbed two point boundary value problems is presented. The given region is divided into inner and outer regions. The original second-order problem is replaced by an asymptotically equivalent first-order problem and solved as an initial value problem in the inner region. A terminal boundary condition is then obtained from the solution of the inner region problem. In turn, an outer region problem is obtained, by replacing the second-order differential equation by an approximate first-order differential equation with a small deviating argument, and solved efficiently by employing the trapezoidal formula coupled with a discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point of the inner region problem. Several numerical examples have been solved to demonstrate the applicability of the method

Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition

AbstractIn this paper, a mixed finite difference method is proposed to solve singularly perturbed differential difference equations with mixed shifts, solutions of which exhibit boundary layer behaviour at the left end of the interval using domain decomposition. A terminal boundary point is introduced into the domain, to decompose it into inner and outer regions. The original problem is reduced to an asymptotically equivalent singular perturbation problem and with the terminal point the singular perturbation problem is treated as inner region and outer region problems separately. The outer region and the modified inner region problems are solved by mixed finite difference method. The method is repeated for various choices of the terminal point. To validate the computational efficiency of the method model examples have been solved for different values of perturbation, delay and advanced parameters. Convergence of the proposed scheme has also been investigated

Special Second Order Non Symmetric Fitted Method for Singular Perturbation Problems

In this paper, we present a special second order non symmetric fitted difference method for solving singular perturbed two point boundary value problems having boundary layer at one end. We introduce a fitting factor in the special second order non symmetric finite difference scheme which takes care of the rapid changes occur that in the boundary layer. The value of this fitting factor is obtained from the theory of singular perturbations. The discrete invariant imbedding algorithm is used to solve the tridiagonal system obtained by the method. We discuss the existence and uniqueness of the discrete problem along with stability estimates and the convergence of the method. We present the maximum absolute errors in numerical results to illustrate the proposed method. Keywords: Singularly perturbed two-point boundary value problem, Boundary layer, Fitting factor, Maximum absolute erro

Heterogeneous Delays in Neural Networks

We investigate heterogeneous coupling delays in complex networks of excitable elements described by the FitzHugh-Nagumo model. The effects of discrete as well as of uni- and bimodal continuous distributions are studied with a focus on different topologies, i.e., regular, small-world, and random networks. In the case of two discrete delay times resonance effects play a major role: Depending on the ratio of the delay times, various characteristic spiking scenarios, such as coherent or asynchronous spiking, arise. For continuous delay distributions different dynamical patterns emerge depending on the width of the distribution. For small distribution widths, we find highly synchronized spiking, while for intermediate widths only spiking with low degree of synchrony persists, which is associated with traveling disruptions, partial amplitude death, or subnetwork synchronization, depending sensitively on the network topology. If the inhomogeneity of the coupling delays becomes too large, global amplitude death is induced

Isolation and evolutionary analysis of Australasian topotype of bluetongue virus serotype 4 from India

Bluetongue (BT) is a Culicoides-borne disease caused by several serotypes of bluetongue virus (BTV). Similar to other insect-borne viral diseases, distribution of BT is limited to distribution of Culicoides species competent to transmit BTV. In the tropics, vector activity is almost year long, and hence, the disease is endemic, with the circulation of several serotypes of BTV, whereas in temperate areas, seasonal incursions of a limited number of serotypes of BTV from neighbouring tropical areas are observed. Although BTV is endemic in all the three major tropical regions (parts of Africa, America and Asia) of the world, the distribution of serotypes is not alike. Apart from serological diversity, geography-based diversity of BTV genome has been observed, and this is the basis for proposal of topotypes. However, evolution of these topotypes is not well understood. In this study, we report the isolation and characterization of several BTV-4 isolates from India. These isolates are distinct from BTV-4 isolates from other geographical regions. Analysis of available BTV seg-2 sequences indicated that the Australasian BTV-4 diverged from African viruses around 3,500 years ago, whereas the American viruses diverged relatively recently (1,684 CE). Unlike Australasia and America, BTV-4 strains of the Mediterranean area evolved through several independent incursions. We speculate that independent evolution of BTV in different geographical areas over long periods of time might have led to the diversity observed in the current virus population

Mathematical model of plant-virus interactions mediated by RNA interference

Cross-protection, which refers to a process whereby artificially inoculating a plant with a mild strain provides protection against a more aggressive isolate of the virus, is known to be an effective tool of disease control in plants. In this paper we derive and analyse a new mathematical model of the interactions between two competing viruses with particular account for RNA interference. Our results show that co-infection of the host can either increase or decrease the potency of individual infections depending on the levels of cross-protection or cross-enhancement between different viruses. Analytical and numerical bifurcation analyses are employed to investigate the stability of all steady states of the model in order to identify parameter regions where the system exhibits synergistic or antagonistic behaviour between viral strains, as well as different types of host recovery. We show that not only viral attributes but also the propagating component of RNA-interference in plants can play an important role in determining the dynamics