We interpret a boundary-value problem arising in a cell growth model as a singular Sturm–Liouville problem that involves a functional differential equation of the pantograph type. We show that the probability density function of the cell growth model corresponds to the first eigenvalue and that there is a family of rapidly decaying eigenfunctions.

We present an application of optimal control theory to a simple SIR disease model of avian influenza transmission dynamics in birds. Basic properties of the model, including the epidemic threshold, are obtained. Optimal control theory is adopted to minimize the density of infected birds subject to an appropriate system of ordinary differential equations. We conclude that an optimally controlled seasonal vaccination strategy saves more birds than when there is a low uniform vaccination rate as in resource-limited places.

An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

The Poiseuille flow of a generalized Maxwell fluid is discussed. The velocity field and shear stress corresponding to the flow in an infinite circular cylinder are obtained by means of the Laplace and Hankel transforms. The motion is caused by the infinite cylinder which is under the action of a longitudinal time-dependent shear stress. Both solutions are obtained in the form of infinite series. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases. Finally, the influence of the material and fractional parameters on the fluid motion is brought to light.

We present qualitative and quantitative comparisons of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of American put options paying zero dividends. We analyse the asymptotic behaviour of these methods close to expiration, and introduce a new numerical scheme for computing the early exercise boundary. Our local iterative numerical scheme is based on a solution to a nonlinear integral equation. We compare numerical results obtained by the new method to those of the projected successive over-relaxation method and the analytical approximation formula recently derived by Zhu [‘A new analytical approximation formula for the optimal exercise boundary of American put options’, Int. J. Theor. Appl. Finance9 (2006) 1141–1177].

We consider the optimal proportional reinsurance from an insurer’s point of view to maximize the expected utility and minimize the value at risk. Under the general premium principle, we prove the existence and uniqueness of the optimal strategies and Pareto optimal solution, and give the relationship between the optimal strategies. Furthermore, we study the optimization problem with the variance premium principle. When the total claim sizes are normally distributed, explicit expressions for the optimal strategies and Pareto optimal solution are obtained. Finally, some numerical examples are presented to show the impact of the major model parameters on the optimal results.

A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results.

We propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has and iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.