Multiderivative methods for periodic initial value problems

A family of two-step multiderivative methods based on Pade approximants to the exponential function is developed. The methods are analysed and periodicity intervals in PECE mode are calculated. Two of the methods are tested on two problems from the literature and one predictor-corrector combination is tested on two further problems

A family of difference schemes for fourth order parabolic partial differential equations

A family of methods is developed for the numerical solution of fourth order parabolic partial differential equations in one- and two-space variables. The methods are seen to evolve from multiderivative methods for second order ordinary differential equations. The methods are tested on three model problems, with constant coefficients and variable coefficients, which have appeared in the literature

Stability regions for one-step multiderivative methods

Stability regions are plotted for certain members of a family of one-step multiderivative predictor-corrector methods developed by the authors in an earlier paper. The methods discussed are tested on a linear system where the matrix of coefficients has constant complex eigenvalues and on a stiff non-linear system arising in reactor kinetics

Global extrapolation procedures for special and general initial value problems

Two- and three- grid global extrapolation procedures are considered fohe special and general initial value problems of arbitrary order r tq. Extrapolation formulas are developed for consistent numerical methods of arbitrary order p . The global extrapolations of a number of existing numerical methods are considered and tested on three problems from the literature

Lo - stable methods for parabolic partial differential equations

In recent years much attention has been devoted in the literature to the development, analysis and implementation of extrapolation methods for the numerical solution of partial differential equations with mixed initial and boundary values specified, see, for example, Lawson and Morris [5], Lawson and Swayne [6] and Gourlay and Morris [3]. The essential theme of these papers was to develop Lo-stable methods for the solution of parabolic partial differential equations in which splitting methods, such as the Crank-Nicolson method, are less than satisfactory when a time discretization is used with time steps which are too large relative to the spatial discretization. In the present paper a family of new Lo-stable methods based on Padé approximants to the exponential function is developed, and. higher accuracy is achieved. The methods are tested on heat equations in one and two space dimensions in which discontinuities exist between the initial and boundary conditions

Backward difference replacements of the space derivative in first order hyperbolic equations

Two families of two-time level difference schemes are developed for the numerical solution of first order hyperbolic partial differential equations with one space variable. The space derivative is replaced by (i) a first order, (ii) a second order backward difference approximant and the resulting system of first order ordinary differential equations is solved using A0-stable and L0-stable methods. The methods are tested on a number of problems from the literature involving wave-form solutions, increasing solutions with discontinuities in function values or first derivatives across a characteristic, and exponentially decaying solutions

A two-dimensional mathematical model of percutaneous drug absorption

Background When a drug is applied on the skin surface, the concentration of the drug accumulated in the skin and the amount of the drug eliminated into the blood vessel depend on the value of a parameter, r. The values of r depend on the amount of diffusion and the normalized skin-capillary clearence. It is defined as the ratio of the steady-state drug concentration at the skin-capillary boundary to that at the skin-surface in one-dimensional models. The present paper studies the effect of the parameter values, when the region of contact of the skin with the drug, is a line segment on the skin surface. Methods Though a simple one-dimensional model is often useful to describe percutaneous drug absorption, it may be better represented by multi-dimensional models. A two-dimensional mathematical model is developed for percutaneous absorption of a drug, which may be used when the diffusion of the drug in the direction parallel to the skin surface must be examined, as well as in the direction into the skin, examined in one-dimensional models. This model consists of a linear second-order parabolic equation with appropriate initial conditions and boundary conditions. These boundary conditions are of Dirichlet type, Neumann type or Robin type. A finite-difference method which maintains second-order accuracy in space along the boundary, is developed to solve the parabolic equation. Extrapolation in time is applied to improve the accuracy in time. Solution of the parabolic equation gives the concentration of the drug in the skin at a given time. Results Simulation of the numerical methods described is carried out with various values of the parameter r. The illustrations are given in the form of figures. Conclusion Based on the values of r, conclusions are drawn about (1) the flow rate of the drug, (2) the flux and the cumulative amount of drug eliminated into the receptor cell, (3) the steady-state value of the flux, (4) the time to reach the steady-state value of the flux and (5) the optimal value of r, which gives the maximum absorption of the drug. The paper gives valuable information which can be obtained by this two-dimensional model, that cannot be obtained with one-dimensional models. Thus this model improves upon the much simpler one-dimensional models. Some future directions of the work based on this model and the one-dimensional non-linear models that exist in the literature, are also discussed

A model of dengue fever

BACKGROUND: Dengue is a disease which is now endemic in more than 100 countries of Africa, America, Asia and the Western Pacific. It is transmitted to the man by mosquitoes (Aedes) and exists in two forms: Dengue Fever and Dengue Haemorrhagic Fever. The disease can be contracted by one of the four different viruses. Moreover, immunity is acquired only to the serotype contracted and a contact with a second serotype becomes more dangerous. METHODS: The present paper deals with a succession of two epidemics caused by two different viruses. The dynamics of the disease is studied by a compartmental model involving ordinary differential equations for the human and the mosquito populations. RESULTS: Stability of the equilibrium points is given and a simulation is carried out with different values of the parameters. The epidemic dynamics is discussed and illustration is given by figures for different values of the parameters. CONCLUSION: The proposed model allows for better understanding of the disease dynamics. Environment and vaccination strategies are discussed especially in the case of the succession of two epidemics with two different viruses

Dynamics of a disabled population in Morocco

BACKGROUND: The disabled population constitutes a class of people needing special care and necessitating important economic and social effort. METHODS: In this paper, using specific parameter settings, partial differential equations are used to model the temporal change of the proportion of the disabled population in Morocco. RESULTS: Combining different forms and values of the parameters, a numerical method is proposed and three scenarios are considered. These forms and values are determined by data fitting and simulation. CONCLUSIONS: The experiments show clearly the dynamical evolution of the disabled population with time and age according to each scenario

A mathematical model for the burden of diabetes and its complications

BACKGROUND: The incidence and prevalence of diabetes are increasing all over the world. Complications of diabetes constitute a burden for the individuals and the whole society. METHODS: In the present paper, ordinary differential equations and numerical approximations are used to monitor the size of populations of diabetes with and without complications. RESULTS: Different scenarios are discussed according to a set of parameters and the dynamical evolution of the population from the stage of diabetes to the stage of diabetes with complications is clearly illustrated. CONCLUSIONS: The model shows how efficient and cost-effective strategies can be obtained by acting on diabetes incidence and/or controlling the evolution to the stage of complications