Nonlocal calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of nonlocal problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including a model of the cell cycle developed recently by Simms, Bean and Koeber. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents.

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes.

We propose a modified projected Polak–Ribière–Polyak (PRP) conjugate gradient method, where a modified conjugacy condition and a method which generates sufficient descent directions are incorporated into the construction of a suitable conjugacy parameter. It is shown that the proposed method is a modification of the PRP method and generates sufficient descent directions at each iteration. With an Armijo-type line search, the theory of global convergence is established under two weak assumptions. Numerical experiments are employed to test the efficiency of the algorithm in solving some benchmark test problems available in the literature. The numerical results obtained indicate that the algorithm outperforms an existing similar algorithm in requiring fewer function evaluations and fewer iterations to find optimal solutions with the same tolerance.

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

Steady water infiltration in homogeneous soils is governed by the Richards equation. This equation can be studied more conveniently by transforming to a type of Helmholtz equation. In this study, a dual-reciprocity boundary element method (DRBEM) is employed to solve the Helmholtz equation numerically. Using the solutions obtained, numerical values of the suction potential are then computed. The proposed method is tested on problems involving infiltration from different types of periodic channels in a homogeneous soil. Moreover, the method is also examined using infiltration from periodic trapezoidal channels in three different types of homogeneous soil.

A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.

This paper presents an integrated guidance and control (IGC) design method for an unmanned aerial vehicle with static stability which is described by a nonlinear six-degree-of-freedom (6-DOF) model. The model is linearized by using small disturbance linearization. The dynamic characteristics of pitching mode, rolling mode and Dutch rolling mode are obtained by analysing the linearized model. Furthermore, an IGC design procedure is also proposed in conjunction with a proportional–integral–derivative (PID) control method and fuzzy control method. A PID controller is applied in the control loop of the elevator and aileron, and the attitude angle and attitude angular velocity are used as compensation feedback, giving a simple and low-order control law. A fuzzy control method is applied to perform the cross-coupling control of rolling and yawing. Finally, the 6-DOF simulation shows the effectiveness of the developed method.