1 
The Dirichlet problemWyman, Jeffries January 1960 (has links)
Thesis (M.A.)Boston University / The problem of finding the solution to a general eliptic type partial differential equation, when the boundary values are given, is generally referred to as the Dirichlet Problem. In this paper I consider the special eliptic equation of ∇2 J=0 which is Laplace's equation, and I limit myself to the case of two dimensions. Subject to these limitations I discuss five proofs for the existence of a solution to Laplace's equation for arbitrary regions where the boundary values are given. [TRUNCATED]

2 
An Approximate Solution to the Dirichlet ProblemRedwine, Edward William 08 1900 (has links)
In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

3 
Partial Differential Equations for Modelling Wound GeometryUgail, Hassan 20 March 2022 (has links)
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 boundaryvalue 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.

4 
Explicit polynomial solutions of fourth order linear elliptic Partial Differential Equations for boundary based smooth surface generationArnal, A., Monterde, J., Ugail, Hassan January 2011 (has links)
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 nonorthogonal 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.

5 
A PDEbased head visualization method with CT dataChen, C., Sheng, Y., Li, F., Zhang, G., Ugail, Hassan 30 November 2015 (has links)
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.

6 
Combat modelling with partial differential equationsKeane, Therese Alison, Mathematics & Statistics, Faculty of Science, UNSW January 2009 (has links)
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 nonlocal 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 selforganisation 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.

7 
Combat modelling with partial differential equationsKeane, Therese Alison, Mathematics & Statistics, Faculty of Science, UNSW January 2009 (has links)
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 nonlocal 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 selforganisation 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.

8 
On the holomorphic solution of nonlinear totally characteristic equations with several space variablesChen, Hua, Lua, Zhuangehu January 1998 (has links)
In this paper we study a class of nonlinear singular partial differential
equation in complex domain Csub(t) x C n sub(x). Under certain assumptions, we prove the existence and uniqueness of holomorphic solution near origin of Csub(t) x C n sub(x).

9 
On the holomorphic solution of nonlinear totally characteristic equationsChen, Hua, Hidetoshi, Tahara January 1998 (has links)
The paper deals with a nonlinear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by GérardTahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.

10 
On the method of lines for singularly perturbed partial differential equationsMbroh, Nana Adjoah January 2017 (has links)
Magister Scientiae  MSc / Many chemical and physical problems are mathematically described by partial
differential equations (PDEs). These PDEs are often highly nonlinear and
therefore have no closed form solutions. Thus, it is necessary to recourse to
numerical approaches to determine suitable approximations to the solution
of such equations. For solutions possessing sharp spatial transitions (such as
boundary or interior layers), standard numerical methods have shown limitations
as they fail to capture large gradients. The method of lines (MOL)
is one of the numerical methods used to solve PDEs. It proceeds by the
discretization of all but one dimension leading to systems of ordinary di erential
equations. In the case of timedependent PDEs, the MOL consists of
discretizing the spatial derivatives only leaving the time variable continuous.
The process results in a system to which a numerical method for initial value
problems can be applied. In this project we consider various types of singularly
perturbed timedependent PDEs. For each type, using the MOL, the
spatial dimensions will be discretized in many different ways following fitted
numerical approaches. Each discretisation will be analysed for stability and
convergence. Extensive experiments will be conducted to confirm the analyses.

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