In Part I of this thesis we extend the Lanchester Ordinary Differential Equations and construct a new physically meaningful set of partial differential equations with the aim of more realistically representing soldier dynamics in order to enable a deeper understanding of the nature of conflict. Spatial force movement and troop interaction components are represented with both local and non-local terms, using techniques developed in biological aggregation modelling. A highly accurate flux limiter numerical method ensuring positivity and mass conservation is used, addressing the difficulties of inadequate methods used in previous research. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions which has not been previously achieved in continuous combat modelling. In Part II, we reproduce for the first time apparently complex cellular automaton behaviour with simple partial differential equations, providing an alternate mechanism through which to analyse this behaviour. Our PDE model easily explains behaviour observed in selected scenarios of the cellular automaton wargame ISAAC without resorting to anthropomorphisation of autonomous 'agents'. The insinuation that agents have a reasoning and planning ability is replaced with a deterministic numerical approximation which encapsulates basic motivational factors and demonstrates a variety of spatial behaviours approximating the mean behaviour of the ISAAC scenarios. All scenarios presented here highlight the dangers associated with attributing intelligent reasoning to behaviour shown, when this can be explained quite simply through the effects of the terms in our equations. A continuum of forces is able to behave in a manner similar to a collection of individual autonomous agents, and shows decentralised self-organisation and adaptation of tactics to suit a variety of combat situations. We illustrate the ability of our model to incorporate new tactics through the example of introducing a density tactic, and suggest areas for further research.

Traveling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing (decreasing) traveling wave solutions are established. Some faults in previous studies are corrected.

Ordinary differential equations of various types appear in the mathematical modelling in mechanics. Differential equations obtained are usually rather complicated nonlinear equations. However, using suitable approximations of nonlinearities, one can derive simple equations that are either well known or can be studied analytically. An example of such "approximative" equation is the so-called Duffing equation. Hence, the question on the existence of a periodic solution to the Duffing equation is closely related to the existence of periodic vibrations of the corresponding nonlinear oscillator.

Student Number : 9202983Y - PhD thesis - School of Computational and Applied Mathematics - Faculty of Science / A derivation is given of the constitutive equation for an incompressible transversely isotropic hyperelastic material in which the direction of the anisotropic director is unspecified. The field equations for a transversely isotropic incompressible hyperelastic material are obtained. Nonlinear radial oscillations in transversely isotropic incompressible cylindrical tubes are investigated. A second order nonlinear ordinary differential equation, expressed in terms of the strain-energy function, is derived. It has the same form as for radial oscillations in an isotropic tube. A generalised Mooney-Rivlin strainenergy function is used. Radial oscillations with a time dependent net applied surface pressure are first considered. For a radial transversely isotropic thin-walled tube the differential equation has a Lie point symmetry for a special form of the strain-energy function and a special time dependent applied surface pressure. The Lie point symmetry is used to transform the equation to an autonomous differential equation which is reduced to an Abel equation of the second kind. A similar analysis is done for radial oscillations in a tangential transversely isotropic tube but computer graphs show that the solution is unstable. Radial oscillations in a longitudinal transversely isotropic tube and an isotropic tube are the same. The Ermakov-Pinney equation is derived. Radial oscillations in thick-walled and thin-walled cylindrical tubes with the Heaviside step loading boundary condition are next investigated. For radial, tangential and longitudinal transversely isotropic tubes a first integral is derived and effective potentials are defined. Using the effective potentials, conditions for bounded oscillations and the end points of the oscillations are obtained. Upper and lower bounds on the period are derived. Anisotropy reduces the amplitude of the oscillation making the tube stiffer and reduces the period. Thirdly, free radial oscillations in a thin-walled cylindrical tube are investigated. Knowles(1960) has shown that for free radial oscillations in an isotropic tube, ab = 1 where a and b are the minimum and maximum values of the radial coordinate. It is shown that if the initial velocity v0 vanishes or if v0 6= 1 but second order terms in the anisotropy are neglected then for free radial oscillations, ab > 1 in a radial transversely isotropic tube and ab < 1 in a tangential transversely isotropic tube. Radial oscillations in transversely isotropic incompressible spherical shells are investigated. Only radial transversely isotropic shells are considered because it is found that the Cauchy stress tensor is not bounded everywhere in tangential and longitudinal transversely isotropic shells. For a thin-walled radial transversely isotropic spherical shell with generalised Mooney-Rivlin strain-energy function the differential equation for radial oscillations has no Lie point symmetries if the net applied surface pressure is time dependent. The inflation of a thin-walled radial transversely isotropic spherical shell of generalised Mooney-Rivlin material is considered. It is assumed that the inflation proceeds sufficiently slowly that the inertia term in the equation for radial oscillations can be neglected. The conditions for snap buckling to occur, in which the pressure decreases before steadily increasing again, are investigated. The maximum value of the parameter for snap buckling to occur is increased by the anisotropy.

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Due to their convenience, computers have become a standard in society and therefore, need the utmost care. It is convenient and useful to model the behavior of digital virus outbreaks that occur, globally or locally. Compartmental models will be used to analyze the mannerisms and behaviors of computer malware. This paper will focus on a computer worm, a type of malware, spread within a business network. A mathematical model is proposed consisting of four compartments labeled as Susceptible, Infectious, Treatment, and Antidotal. We shall show that allocating resources into treating infectious computers leads to a reduced peak of infections across the infection period, while pouring resources into treating susceptible computers decreases the total amount of infections throughout the infection period. This is assuming both methods are receiving resources without loss. This result reveals an interesting notion of balance between protecting computers and removing computers from infections, ultimately depending on the business executives' goals and/or preferences.

No / We present an explicit polynomial solution method for surface generation. In this case the surface in question is characterized by some boundary configuration whereby the resulting surface conforms to a fourth order linear elliptic Partial Differential Equation, the Euler–Lagrange equation of a quadratic functional defined by a norm. In particular, the paper deals with surfaces generated as explicit Bézier polynomial solutions for the chosen Partial Differential Equation. To present the explicit solution methodologies adopted here we divide the Partial Differential Equations into two groups namely the orthogonal and the non-orthogonal cases. In order to demonstrate our methodology we discuss a series of examples which utilize the explicit solutions to generate smooth surfaces that interpolate a given boundary configuration. We compare the speed of our explicit solution scheme with the solution arising from directly solving the associated linear system.

In this dissertation, we have studied diffusion models and their applications in risk theory and insurance. Let Xt be a d-dimensional diffusion process satisfying a system of Stochastic Differential Equations defined on an open set G Rd, and let Ut be a utility function of Xt with U0 = u0. Let T be the first time that Ut reaches a level u^*. We study the Laplace transform of the distribution of T, as well as the probability of ruin, psileft(u_{0}right)=Prleft{ T&lt;inftyright} , and other important probabilities. A class of exponential martingales is constructed to analyze the asymptotic properties of all probabilities. In addition, we prove that the expected discounted penalty function, a generalization of the probability of ultimate ruin, satisfies an elliptic partial differential equation, subject to some initial boundary conditions. Two examples from areas of actuarial work to which martingales have been applied are given to illustrate our methods and results: 1. Insurer's insolvency. 2. Terrorism risk. In particular, we study insurer's insolvency for the Cram'{e}r-Lundberg model with investments whose price follows a geometric Brownian motion. We prove the conjecture proposed by Constantinescu and Thommann. / Mathematics

no / In this paper, we extend the use of the partial differential equation (PDE) method to head visualization with computed tomography (CT) data and show how the two primary medical visualization means, surface reconstruction, and volume rendering can be integrated into one single framework through PDEs. Our scheme first performs head segmentation from CT slices using a variational approach, the output of which can be readily used for extraction of a small set of PDE boundary conditions. With the extracted boundary conditions, head surface reconstruction is then executed. Because only a few slices are used, our method can perform head surface reconstruction more efficiently in both computational time and storage cost than the widely used marching cubes algorithm. By elaborately introducing a third parameter w to the PDE method, a solid head can be created, based on which the head volume is subsequently rendered with 3D texture mapping. Instead of designing a transfer function, we associate the alpha value of texels of the 3D texture with the PDE parameter w through a linear transform. This association enables the production of a visually translucent head volume. The experimental results demonstrate the feasibility of the developed head visualization method.

No / Wounds arising from various conditions are painful, embarrassing and often requires treatment plans which are costly. A crucial task, during the treatment of wounds is the measurement of the size, area and volume of the wounds. This enables to provide appropriate objective means of measuring changes in the size or shape of wounds, in order to evaluate the efficiency of the available therapies in an appropriate fashion. Conventional techniques for measuring physical properties of a wound require making some form of physical contact with it. We present a method to model a wide variety of geometries of wound shapes. The shape modelling is based on formulating mathematical boundary-value problems relating to solutions of Partial Differential Equations (PDEs). In order to model a given geometric shape of the wound a series of boundary functions which correspond to the main features of the wound are selected. These boundary functions are then utilised to solve an elliptic PDE whose solution results in the geometry of the wound shape. Thus, here we show how low order elliptic PDEs, such as the Biharmonic equation subject to suitable boundary conditions can be used to model complex wound geometry. We also utilise the solution of the chosen PDE to automatically compute various physical properties of the wound such as the surface area, volume and mass. To demonstrate the methodology a series of examples are discussed demonstrating the capability of the method to produce good representative shapes of wounds.