Patricia M. Lumb

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t), defined for t@?[-1,k], (k@?N), that satisfies this equation almost everywhere on [0,k-1] and assumes specified values on the intervals [-1,0] and (k-1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.

Numerical Modelling of a Functional Differential Equation with Deviating Arguments Using a Collocation Method

This paper is concerned with the approximate solution of a functional differential equation of the form: x′(t) = α(t)x(t)+β(t)x(t−1)+γ(t)x(t+1). We search for a solution x, defined for t∈[−1,k],(k∈N), which takes given values onn the intervals [−1,0] and (k−1,k]. Continuing the work started in [10], we introduce and anlyse some new computational methods for the solution of this problem which are applicable both in the case of constant and variable coefficients. Numerical results are presented and compared with the results obtained by other methods.

Finite element solution of a linear mixed-type functional differential equation

This paper is devoted to the approximate solution of a linear first-order functional differential equation which involves delayed and advanced arguments. We seek a solution x, defined for t ∈ (0, k − 1],(k ∈ IN ), which takes given values on the intervals [ − 1, 0] and (k − 1, k]. Continuing the work started in previous articles on this subject, we introduce and analyse a computational algorithm based on the finite element method for the solution of this problem which is applicable both in the case of constant and variable coefficients. Numerical results are presented and compared with the results obtained by other methods.

Mathematical modelling of plant species interactions in a harsh climate

This paper is concerned with the mathematical modelling of complex interactions between plant species in a harsh environment such as in the arctic. The aim of the paper is to consider whether interactions between the species change in character as environments change. For example, if the effect of climate change is to make harsh climates more benign, will this imply changes in the way species interact and affect biodiversity? We consider the interaction of two species of grass. Our model is constructed based on the notion of a summer season when the plants grow, followed by a winter season when there is no growth but when the plants are subject to the effects of events such as winter storms. Our aim is to investigate changes when the summer season is lengthened, when the climate becomes more benign, when the susceptibility of plants to damage as a result of storms is increased, and when the intensity (and number) of winter storms is varied. The models we consider provide new insights into the known behaviour of plant species interactions in such situations and a basis for further modelling and prediction.

Analysis and Computational Approximation of a Forward–Backward Equation Arising in Nerve Conduction

This paper is concerned with the approximate solution of a nonlinear mixed-type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a solution defined on the whole real axis, which tends to given values at ±∞.The numerical algorithms, developed previously by the authors for linear problems, were upgraded to deal with the case of nonlinear problems on unbounded domains. Numerical results are presented and discussed.

Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh-Nagumo equation

We have analysed a stochastic functional equation, which contains both delayed and advanced arguments.We have created a new computational algorithm to approximate this equation, based on the Euler-Maruyama method.We have analysed noise induced changes in the dynamical behaviour of equations.We observe that a low level of noise is enough to produce a significantly different dynamical behaviour of the solutions.We further observe that the effect of noise is much stronger in the region where the solutions change faster. In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh-Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation.

Mathematical Modelling and Numerical Simulations in Nerve Conduction

In this paper we are concerned with the numerical solution of the discrete FitzHugh-Nagumo equation. This equation describes the propagation of impulses across a myelinated axon. We analyse the asymptotic behaviour of the solutions of the considered equation and numerical approximations are computed by a new algorithm, based on a finite difference scheme and on the Newton method. The efficiency of the method is discussed and its performance is illustrated by a set of numerical examples.

Numerical Approximation of a Nonlinear Boundary Value Problem for a Mixed Type Functional Differential Equation Arising in Nerve Conduction

This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) with deviating arguments arising from nerve conduction theory. The considered equation describes conduction in a myelinated nerve axon in which the myelin totally insulates the membrane. As a consequence, the potential change jumps from node to node. As described in [2], this process is modelled by a first order nonlinear functional‐differential equation with deviated arguments. We search for a solution of this equation defined in R, which tends to given values at ±∞. Following the approach introduced in [13] and [8], we propose and analyze some new computational methods for the solution of this problem. Numerical results are obtained and compared with the ones presented in [2].