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Upton, Deborah Susan. "Students' solution strategies to differential equations problems in mathematical and non-mathematical contexts." Thesis, Boston University, 2004. https://hdl.handle.net/2144/32845.
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Thesis (Ed.D.)--Boston University<br>PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.<br>The present study investigated undergraduate students' understanding of slope fields and equilibrium solutions as they solved problems in differential equations. The following questions were addressed: 1) Does performance on complex problems vary by context (mathematical, non-mathematical)? 2) When considering a complex problem in a mathematical and a non-mathematical context, are participants who answer the problem in one context correctly more likely to answer the corresponding problem in the other context correctly? 3) Does performance on simple problems predict performance on complex problems?
A written test, Differential Equations Concept Assessment (DECA), was designed and administered to 91 participants drawn from three introductory differential equations courses. Of those participants, 13 were interviewed. DECA consists of four complex problems, two in mathematical contexts and two in non-mathematical contexts, and six simple problems that assess aspects of slope fields and equilibrium solutions.
The data obtained from DECA and the interviews showed that participants performed significantly better on complex problems in non-mathematical contexts than on complex problems in mathematical contexts. There was a significant relationship found between performance on a problem in a mathematical context and performance on the isomorphic problem in the context of population growth, but a significant relationship was not found between a different pair of isomorphic problems, one in a mathematical context and the other in the context oflearning. However, for all the complex problems, participants illustrated a preference for algebraic rather than geometric methods, even when a geometric approach was a more efficient method of solution. Although performance on simple problems was not found to be a strong predictor of performance on complex problems, the simple problems proved to elicit difficulties participants had with aspects of slope fields and equilibrium solutions. For example, participants were found to overgeneralize the notion of equilibrium solution as being any straight line and as existing at all values where a differential equation equals zero. Participants were also found to identify slope fields as determining only equilibrium solutions.<br>2031-01-01
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Rout, Sweta. "The mathematical modelling and numerical solution of options pricing problems." Thesis, University of Greenwich, 2005. http://gala.gre.ac.uk/6285/.
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Accurate and efficient numerical solutions have been described for a selection of financial options pricing problems. The methods are based on finite difference discretisation coupled with optimal solvers of the resulting discrete systems. Regular Cartesian meshes have been combined with orthogonal co-ordinate transformations chosen for numerical accuracy rather than reduction of the differential operator to constant coefficient form. They allow detailed resolution in the regions of interest where accuracy is most desired, and grid coarsening where there is least interest. These transformations are shown to be effective in producing accurate solutions on modest computational grids. The spatial discretisation strategy is chosen to meet accuracy requirements as sell as to produce coefficient matrices with favourable sparsity and stability properties. In the case of single factor European options, a modified Crank-Nicolson, second order accurate finite difference scheme is presented, which uses adaptive upwind differences when the mesh Peclet conditions are violated. The resulting tridiagonal system of equations is solved using a direct solver. A careful study of grid refinement displays convergence towards the true solution and demonstrates a high level of accuracy can be obtained with this approach. Laplace inversion methods are also implemented as an alternative solution approach for the one-factor European option. Results are compared to those produced by the direct solver algorithm and are shown to be favourable. It is shown how Semi-Lagrange time-integration can solve the path-dependent Asian pricing problem, by integrating out the average price term and simplifying the finite difference equations into a parameterised Black-Scholes form. The implicit equations that result are unconditionally stable, second order accurate and can be solved using standard tridiagonal solvers. The Semi-Lagrange method is shown to be easily used in conjunction with co-ordinate transformations applied in both spatial directions. A variable time-stepping scheme is implemented in the algorithm. Early exercise is also easily incorporated, the resulting linear complementarity problem can be solved using a projection or penalty method (the penalty method is shown to be slightly more efficient). Second order accuracy has been confirmed for Asian options that must be held to maturity. A comparison with published results for continuous-average-rate put and call options, with and without early exercise, shows that the method achieves basis point accuracy and that Richardson extrapolation can also be applied.
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Penlington, Thomas Helm. "Exploring learners' mathematical understanding through an analysis of their solution strategies." Thesis, Rhodes University, 2005. http://hdl.handle.net/10962/d1007642.
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The purpose of this study is to investigate various solution strategies employed by Grade 7 learners and their teachers when solving a given set of mathematical tasks. This study is oriented in an interpretive paradigm and is characterised by qualitative methods. The research, set in nine schools in the Eastern Cape, was carried out with nine learners and their mathematics teachers and was designed around two phases. The research tools consisted of a set of 12 tasks that were modelled after the Third International Mathematics and Science Study (TIMSS), and a process of clinical interviews that interrogated the solution strategies that were used in solving the 12 tasks. Aspects of grounded theory were used in the analysis of the data. The study reveals that in most tasks, learners relied heavily on procedural understanding at the expense of conceptual understanding. It also emphasises that the solution strategies adopted by learners, particularly whole number operations, were consistent with those strategies used by their teachers. Both learners and teachers favoured using the traditional, standard algorithm strategies and appeared to have learned these algorithms in isolation from concepts, failing to relate them to understanding. Another important finding was that there was evidence to suggest that some learners and teachers did employ their own constructed solution strategies. They were able to make sense of the problems and to 'mathematize' effectively and reason mathematically. An interesting outcome of the study shows that participants were more proficient in solving word problems than mathematical computations. This is in contrast to existing research on word problems, where it is shown that teachers find them difficult to teach and learners find them difficult to understand. The findings of this study also highlight issues for mathematics teachers to consider when dealing with computations and word problems involving number sense and other problem solving type problems.
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Lawson, Jane. "Towards error control for the numerical solution of parabolic equations." Thesis, University of Leeds, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329947.
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Jaksic, Vojkan Simon Barry Simon Barry. "Solutions to some problems in mathematical physics /." Diss., Pasadena, Calif. : California Institute of Technology, 1992. http://resolver.caltech.edu/CaltechETD:etd-09122005-162352.
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Shyshkov, Andriy. "NUMERICAL SOLUTION OF ILL-POSED PROBLEMS." Kent State University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=kent1271700548.
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Murat, Ekrem Alper. "An allocation based modeling and solution framework for location problems with dense demand /." Thesis, McGill University, 2005. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=102685.
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In this thesis we present a unified framework for planar location-allocation problems with dense demand. Emergence of such information technologies as Geographical Information Systems (GIS) has enabled access to detailed demand information. This proliferation of demand data brings about serious computational challenges for traditional approaches which are based on discrete demand representation. Furthermore, traditional approaches model the problem in location variable space and decide on the allocation decisions optimally given the locations. This is equivalent to prioritizing location decisions. However, when allocation decisions are more decisive or choice of exact locations is a later stage decision, then we need to prioritize allocation decisions. Motivated by these trends and challenges, we herein adopt a modeling and solution approach in the allocation variable space.<br>Our approach has two fundamental characteristics: Demand representation in the form of continuous density functions and allocation decisions in the form of service regions. Accordingly, our framework is based on continuous optimization models and solution methods. On a plane, service regions (allocation decisions) assume different shapes depending on the metric chosen. Hence, this thesis presents separate approaches for two-dimensional Euclidean-metric and Manhattan-metric based distance measures. Further, we can classify the solution approaches of this thesis as constructive and improvement-based procedures. We show that constructive solution approach, namely the shooting algorithm, is an efficient procedure for solving both the single dimensional n-facility and planar 2-facility problems. While constructive solution approach is analogous for both metric cases, improvement approach differs due to the shapes of the service regions. In the Euclidean-metric case, a pair of service regions is separated by a straight line, however, in the Manhattan metric, separation takes place in the shape of three (at most) line segments. For planar 2-facility Euclidean-metric problems, we show that shape preserving transformations (rotation and translation) of a line allows us to design improvement-based solution approaches. Furthermore, we extend this shape preserving transformation concept to n-facility case via vertex-iteration based improvement approach and design first-order and second-order solution methods. In the case of planar 2-facility Manhattan-metric problems, we adopt translation as the shape-preserving transformation for each line segment and develop an improvement-based solution approach. For n-facility case, we provide a hybrid algorithm. Lastly, we provide results of a computational study and complexity results of our vertex-based algorithm.
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Trickett, S. A. "The numerical solution of elliptic problems." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375307.
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Ak, Aykagan. "Berth and quay crane scheduling problems, models and solution methods /." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/26652.
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Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2009.<br>Committee Chair: Erera, Alan L.; Committee Member: Ergun, Ozlem; Committee Member: Savelsbergh, Martin; Committee Member: Tetali, Prasad; Committee Member: White III, Chelsea C.. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Thareja, Rajiv R. "Efficient single-level solution of hierarchical problems in structural optimization." Diss., Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/71195.
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Engineering design is hierarchical in nature, and if no attempt is made to benefit from this hierarchical nature, design optimization can be very expensive. There are two alternatives to taking advantage of the hierarchical nature of structural design problems. Multi-level optimization techniques incorporate the hierarchy at the formulation stage, and result in the coordinated optimization of a hierarchy of subsystems. The use of multi-level optimization techniques often necessitates the use of equality constraints. These constraints can sometimes cause numerical difficulties during optimization. Single-level decomposition techniques take advantage of the hierarchical nature to reduce the optimization cost.
In this research the decomposition approach has been followed to reduce the computational effort in a single-level design space. A decoupling technique has been developed that retains the advantages of a partitioned system of smaller independent subsystems without an increase in the total number of design variables. A penalty function formulation using Newton's method for the solution of a sequence of unconstrained minimizations was employed. The optimization of the decoupled system is cheaper due to (i) cheaper evaluation of the hessian matrix by taking advantage of its sparsity, (ii) fewer global analyses for constraint derivative calculations, and (iii) utilizing the decoupled nature of the hessian matrix in the solution process. Further, the memory requirements of the decoupled system are much less than that of the original coupled system. These benefits increase substantially for design problems with larger and larger number of detailed design variables.
Orthotropic material properties as stiffness global variables have been shown to be effective as global variables for panels in a hierarchical wing design formulation.
The proposed decoupling technique was implemented to minimize the volume of a portal frame and a wing box. Computational savings of up to 50 percent have been obtained for medium sized problems. The savings increase as the size of the problem and the amount of decoupling is increased. The procedure is simple to implement. For truly large systems this decoupling technique provides the necessary reduction of computational effort to make the optimization process viable.<br>Ph. D.
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Jain, Sachin. "Multiresolution strategies for the numerical solution of optimal control problems." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/22656.
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Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2008.<br>Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Min.
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Heilporn, Géraldine. "Network pricing problems : complexity, polyhedral study and solution approaches." Thèse, Universite Libre de Bruxelles, 2008. http://hdl.handle.net/1866/6451.
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Gülpinar, Nalân. "Analysis of large scale linear programming problems with embedded network structures : detection and solution algorithms." Thesis, Brunel University, 1998. http://bura.brunel.ac.uk/handle/2438/7123.
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Linear programming (LP) models that contain a (substantial) network structure frequently arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection algorithms perform very well computationally and make positive contributions to the known body of results for the embedded network detection. For computational solution a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution made by the research reported in this thesis.
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Robertson, Mairi Laidlaw. "The adaptive numerical solution of phase change problems." Thesis, University of Strathclyde, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366834.
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Al-Kharafi, Abdulmohsen A. H. "Finite element solution of initial/boundary value problems." Thesis, Loughborough University, 1986. https://dspace.lboro.ac.uk/2134/32335.
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In the last few decades, the Finite Element Method (F.E.M.) has become one of the best techniques used to solve a vast variety of the world's initial/boundary value problems. When such a powerful method is facilitated by a 'user friendly' computer program that possess both pre- and post-processors together with an automatic mesh generation and refinement processor, it becomes indeed a powerful tool to solve a wide range of problems in Applied Mathematics and Engineering. This thesis is an attempt to show the potential of the method which is implemented by a general purpose program used to solve problems governed by partial differential equations (P.D.E.s).
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Adhikary, Junas. "Forest harvest scheduling problem, studying mathematical and constraint programming solution strategies." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/mq24076.pdf.
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Almoustafa, Samira. "Distance-constrained vehicle routing problem : exact and approximate solution (mathematical programming)." Thesis, Brunel University, 2013. http://bura.brunel.ac.uk/handle/2438/7640.
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The asymmetric distance-constrained vehicle routing problem (ADVRP) looks at finding vehicle tours to connect all customers with a depot, such that the total distance is minimised; each customer is visited once by one vehicle; every tour starts and ends at a depot; and the travelled distance by each vehicle is less than or equal to the given maximum value. We present three basic results in this thesis. In the first one, we present a general flow-based formulation to ADVRP. It is suitable for symmetric and asymmetric instances. It has been compared with the adapted Bus School Routing formulation and appears to solve the ADVRP faster. Comparisons are performed on random test instances with up to 200 customers. We reach a conclusion that our general formulation outperforms the adapted one. Moreover, it finds the optimal solution for small test instances quickly. For large instances, there is a high probability that an optimal solution can be found or at least improve upon the value of the best feasible solution found so far, compared to the other formulation which stops because of the time condition. This formulation is more general than Kara formulation since it does not require the distance matrix to satisfy the triangle inequality. The second result improves and modifies an old branch-and-bound method suggested by Laporte et al. in 1987. It is based on reformulating a distance-constrained vehicle routing problem into a travelling salesman problem and uses the assignment problem as a lower bounding procedure. In addition, its algorithm uses the best-first strategy and new branching rules. Since this method was fast but memory consuming, it would stop before optimality is proven. Therefore, we introduce randomness in choosing the node of the search tree in case we have more than one choice (usually we choose the smallest objective function). If an optimal solution is not found, then restart is required due to memory issues, so we restart our procedure. In that way, we get a multistart branch and bound method. Computational experiments show that we are able to exactly solve large test instances with up to 1000 customers. As far as we know, those instances are much larger than instances considered for other VRP models and exact solution approaches from recent literature. So, despite its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in literature. Moreover, this approach is general and may be used in solving other types of vehicle routing problems. In the third result, we use VNS as a heuristic to find the best feasible solution for groups of instances. We wanted to determine how far the difference is between the best feasible solution obtained by VNS and the value of optimal solution in order to use the output of VNS as an initial feasible solution (upper bound procedure) to improve our multistart method. Unfortunately, based on the search strategy (best first search), using a heuristic to find an initial feasible solution is not useful. The reason for this is because the branch and bound is able to find the first feasible solution quickly. In other words, in our method using a good initial feasible solution as an upper bound will not increase the speed of the search. However, this would be different for the depth first search. However, we found a big gap between VNS feasible solution and an optimal solution, so VNS can not be used alone unless for large test instances when other exact methods are not able to find any feasible solution because of memory or stopping conditions.
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Colville, Kevin William. "Friction models in the solution of nonstationary contact problems." Master's thesis, University of Cape Town, 1993. http://hdl.handle.net/11427/17334.
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Bibliography: pages 82-83.<br>In most implementations of the finite element method for the solution of contact problems the model of friction used is the classic Amontons-Coulomb. This dissertation is an attempt to rectify the current situation by considering four more advanced friction models, and coding them in FORTRAN for use with the finite element program ABAQUS. The new models are: a quasi-steady-state sliding model proposed by Zhang, Moslehy and Rice; a nonlinear pressure-dependent model proposed by Wriggers, vu Van and Stein; and a model that includes a film of lubricant proposed by Wilson, Hsu and Huang. The friction models are described in detail, including the algorithmic implementation. The contact problem is then formulated in the Total Lagrangian and Updated Lagrangian formulations for contact between an elastic-plastic (Mises plasticity) body and a rigid tool. The variational (weak) form of the formulation is given and this is then discretised by the finite element method. To test and compare the models three common metal forming processes are simulated: hemispherical punching of a disk, two-dimensional plane strain and three-dimensional cold rolling of a strip, and axisymmetric cup deep-drawing. The results are presented in the form of contour plots of the second invariant of stress (Mises), and the plastic yield and maximum stress. Also graphs for the thickness strain are given. These results are presented for each combination of friction model and process to allow easy comparison of frictional behaviour.
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Dykes, Laura R. "New Methods for Solution of Discrete Ill-posed Problems." Kent State University / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=kent1469796109.
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Nguyen, Fonzie T. "THE HAHN-BANACH THEOREM AND SOLUTION OF RELATED PROBLEMS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/115.
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Fry, Carol Jean. "Eye fixation patterns in the solution of mathematical word problems by young adults : relation to cognitive style and spatial ability /." The Ohio State University, 1987. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487584612164575.
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Hellman, Fredrik. "Towards the Solution of Large-Scale and Stochastic Traffic Network Design Problems." Thesis, Uppsala University, Department of Information Technology, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-130013.
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<p>This thesis investigates the second-best toll pricing and capacity expansion problems when stated as mathematical programs with equilibrium constraints (MPEC). Three main questions are rised: First, whether conventional descent methods give sufficiently good solutions, or whether global solution methods are to prefer. Second, how the performance of the considered solution methods scale with network size. Third, how a discretized stochastic mathematical program with equilibrium constraints (SMPEC) formulation of a stochastic network design problem can be practically solved. An attempt to answer these questions is done through a series ofnumerical experiments.</p><p>The traffic system is modeled using the Wardrop’s principle for user behavior, separable cost functions of BPR- and TU71-type. Also elastic demand is considered for some problem instances.</p><p>Two already developed method approaches are considered: implicit programming and a cutting constraint algorithm. For the implicit programming approach, several methods—both local and global—are applied and for the traffic assignment problem an implementation of the disaggregate simplicial decomposition (DSD) method is used. Regarding the first question concerning local and global methods, our results don’t give a clear answer.</p><p>The results from numerical experiments of both approaches on networks of different sizes shows that the implicit programming approach has potential to solve large-scale problems, while the cutting constraint algorithm scales worse with network size.</p><p>Also for the stochastic extension of the network design problem, the numerical experiments indicate that implicit programming is a good approach to the problem.</p><p>Further, a number of theorems providing sufficient conditions for strong regularity of the traffic assignment solution mapping for OD connectors and BPR cost functions are given.</p>
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Deivakkannu, Ganesan. "Data acquisition and data transfer methods for real-time power system optimisation problems solution." Thesis, Cape Peninsula University of Technology, 2014. http://hdl.handle.net/20.500.11838/1178.
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Thesis submitted in fulfilment of the requirements for the degree
Master of Technology: Electrical Engineering
in the Faculty of Engineering
at the Cape Peninsula University of Technology
2014<br>The electric power utilities play a vital role in the generation, transmission and distribution of the electrical power to the end users. The power utilities face two major issues, i.e. i) power grids are expected to operate close to the maximum capacity, and ii) there is a need for accurate and better monitoring and control of the power system
network using the modern technology and the available tools. These two issues are interconnected as better monitoring allows for better control of the power system. Development of the new standard-based power system technologies contributed to raising the ideas for building of a Smart grid. The challenges are that this process requires development of new control and operation architectures and methods for data acquisition, data transfer, and control computation. These methods require data for the full dynamic state of the power system in real-time, which leads to the introduction of the synchrophasor-based monitoring and control of the power system. The thesis describes the research work and investigations for integration of the existing new power system technologies to build fully automated systems for real-time solution of power system energy management problems, incorporating data measurement and acquisition, data transfer and distribution through a communication network, and data storage and retrieval in one whole system.
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Song, Yongcun. "An ADMM approach to the numerical solution of state constrained optimal control problems for systems modeled by linear parabolic equations." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/551.
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We address in this thesis the numerical solution of state constrained optimal control problems for systems modeled by linear parabolic equations. For the unconstrained or control-constrained optimal control problem, the first order optimality condition can be obtained in a general way and the associated Lagrange multiplier has low regularity, such as in the L²(Ω). However, for state-constrained optimal control problems, additional assumptions are required in general to guarantee the existence and regularity of Lagrange multipliers. The resulting optimality system leads to difficulties for both the numerical solution and the theoretical analysis. The approach discussed here combines the alternating direction of multipliers (ADMM) with a conjugate gradient (CG) algorithm, both operating in well-chosen Hilbert spaces. The ADMM approach allows the decoupling of the state constraints and the parabolic equation, in which we need solve an unconstrained parabolic optimal control problem and a projection onto the admissible set in each iteration. It has been shown that the CG method applied to the unconstrained optimal control problem modeled by linear parabolic equation is very efficient in the literature. To tackle the issue about the associated Lagrange multiplier, we prove the convergence of our proposed algorithm without assuming the existence and regularity of Lagrange multipliers. Furthermore, a worst case O(1/k) convergence rate in the ergodic sense is established. For numerical purposes, we employ the finite difference method combined with finite element method to implement the time-space discretization. After full discretization, the numerical results we obtain validate the methodology discussed in this thesis.
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Unsurangsie, Sumalee. "Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem." Thesis, North Texas State University, 1988. https://digital.library.unt.edu/ark:/67531/metadc331780/.
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In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.
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Ranjbar, Zohreh. "Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations." Doctoral thesis, Linköpings universitet, Beräkningsvetenskap, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-54300.
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Ill-posed mathematical problem occur in many interesting scientific and engineering applications. The solution of such a problem, if it exists, may not depend continuously on the observed data. For computing a stable approximate solution it is necessary to apply a regularization method. The purpose of this thesis is to investigate regularization approaches and develop numerical methods for solving certain ill-posed problems for parabolic partial differential equations. In thermal engineering applications one wants to determine the surface temperature of a body when the surface itself is inaccessible to measurements. This problem can be modelled by a sideways heat equation. The mathematical and numerical properties of the sideways heat equation with constant convection and diffusion coefficients is first studied. The problem is reformulated as a Volterra integral equation of the first kind with smooth kernel. The influence of the coefficients on the degree of ill-posedness are also studied. The rate of decay of the singular values of the Volterra integral operator determines the degree of ill-posedness. It is shown that the sign of the coefficient in the convection term influences the rate of decay of the singular values. Further a sideways heat equation in cylindrical geometry is studied. The equation is a mathematical model of the temperature changes inside a thermocouple, which is used to approximate the gas temperature in a combustion chamber. The heat transfer coefficient at the surface of thermocouple is also unknown. This coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using the convection boundary condition. In both steps the surface temperature and heat flux are approximated using Tikhonov regularization and the method of lines. Many existing methods for solving sideways parabolic equations are inadequate for solving multi-dimensional problems with variable coefficients. A new iterative regularization technique for solving a two-dimensional sideways parabolic equation with variable coefficients is proposed. A preconditioned Generalized Minimum Residuals Method (GMRS) is used to regularize the problem. The preconditioner is based on a semi-analytic solution formula for the corresponding problem with constant coefficients. Regularization is used in the preconditioner as well as truncating the GMRES algorithm. The computed examples indicate that the proposed PGMRES method is well suited for this problem. In this thesis also a numerical method is presented for the solution of a Cauchy problem for a parabolic equation in multi-dimensional space, where the domain is cylindrical in one spatial direction. The formal solution is written as a hyperbolic cosine function in terms of a parabolic unbounded operator. The ill-posedness is dealt with by truncating the large eigenvalues of the operator. The approximate solution is computed by projecting onto a smaller subspace generated by the Arnoldi algorithm applied on the inverse of the operator. A well-posed parabolic problem is solved in each iteration step. Further the hyperbolic cosine is evaluated explicitly only for a small triangular matrix. Numerical examples are given to illustrate the performance of the method.
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Kotik, Nikolai. "Solution to boundary-contact problems of elasticity in mathematical models of the printing-plate contact system for flexographic printing." Doctoral thesis, Karlstad : Faculty of Technology and Science, Mathematics, Karlstad University, 2007. http://www.diva-portal.org/kau/abstract.xsql?dbid=773.
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Abd, El Aziz Osama Mostafa. "Solution of time dependent problems using the Global Element Method." Thesis, University of Liverpool, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329416.
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Bängtsson, Erik. "Robust preconditioned iterative solution methods for large-scale nonsymmetric problems." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86353.
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We study robust, preconditioned, iterative solution methods for large-scale linear systems of equations, arising from different applications in geophysics and geotechnics. The first type of linear systems studied here, which are dense, arise from a boundary element type of discretization of crack propagation in brittle material. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems are nonsymmetric and indefinite and arise from finite element discretization of the partial differential equations describing the elastic part of glacial rebound processes. An equal order finite element discretization is analyzed and an optimal stabilization parameter is derived. The indefinite algebraic systems are of 2-by-2-block form, and therefore block preconditioners of block-factorized or block-triangular form are used when solving the indefinite algebraic system. There, the required Schur complement is approximated in various ways and the quality of these approximations is compared numerically. When the block preconditioners are constructed from incomplete factorizations of the diagonal blocks, the iterative scheme show a growth in iteration count with increasing problem size. This growth is stabilized by replacing the incomplete factors with an inner iterative scheme with a (nearly) optimal order multilevel preconditioner.
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Ivanova, Kseniya. "Mathematical model of multi-dimensional shear shallow water flows : problems and solutions." Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0642/document.
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Cette thèse porte sur la résolution numérique du modèle multi-dimensionnel d'écoulement cisaillé en eau peu profonde. Dans le cas d'un mouvement unidimensionnel, ces équations coïncident avec les équations de la dynamique de gaz pour un choix particulier de l'équation d'état. Dans le cas multi-dimensionnel, le système est complètement différent du modèle de la dynamique de gaz. Il s'agit d'un système EDP hyperbolique 2D non-conservatif qui rappelle un modèle de turbulence barotrope. Le modèle comporte trois types d'ondes correspondant à la propagation des ondes de surface, des ondes de cisaillement et à celle de la discontinuité de contact. Nous présentons dans le cas 2D un schéma numérique basé sur une nouvelle approche de ``splitting" pour les systèmes d'équations non-conservatives. Chaque sous-système ne contient qu'une seule famille d'ondes: ondes de surface ou ondes de cisaillement, et discontinuité de contact. La précision d'une telle approche est testée sur des solutions exactes 2D décrivant l'écoulement lorsque la vitesse est linéaire par rapport aux variables spatiales, ainsi que sur des solutions décrivant des trains de rouleaux 1D. Finalement, nous modélisons un ressaut hydraulique circulaire formé dans un écoulement convergent radial d'eau. Les résultats numériques obtenus sont clairement similaires à ceux obtenus expérimentalement: oscillations du ressaut et son rotation avec formation du point singulier. L'ensemble des validations proposées dans ce manuscrit démontre les aptitudes du modèle et de la méthode numérique pour la résolution des problèmes complexes d'écoulements cisaillés en eau peu profonde multidimensionnels<br>This thesis is devoted to the numerical modelling of multi-dimensional shear shallow water flows. In 1D case, the corresponding equations coincide with the equations describing non--isentropic gas flows with a special equation of state. However, in the multi-D case, the system differs significantly from the gas dynamics model. This is a 2D hyperbolic non-conservative system of equations which is reminiscent of a generic Reynolds averaged model of barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow (contact characteristics). First, we show the ability of the one-dimensional conservative shear shallow water model to predict the formation of roll-waves from unstable initial data. The stability of roll waves is also studied.Second, we present in 2D case a new numerical scheme based on a splitting approach for non-conservative systems of equations. Each split subsystem contains only one family of waves (either surface or shear waves) and contact characteristics. The accuracy of such an approach is tested on exact 2D solutions describing the flow where the velocity is linear with respect to the space variables, and on the solutions describing 1D roll waves. Finally, we model a circular hydraulic jump formed in a convergent radial flow of water. Obtained numerical results are qualitatively similar to those observed experimentally: oscillation of the hydraulic jump and its rotation with formation of a singular point. These validations demonstrate the capability of the model and numerical method to solve challenging multi--dimensional problems of shear shallow water flows
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Dharmakadar, Aida. "An algorithmic solution to the minimax resource allocation problem with multimodal functions." Thesis, This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-10062009-020310/.
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Palmgren, Myrna. "Optimal Truck Scheduling : Mathematical Modeling and Solution by the Column Generation Principle." Doctoral thesis, Linköping : Linköpings universitet, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-3590.
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Psihoyios, Georgios. "Advanced step-point methods for the solution of initial value problems." Thesis, Imperial College London, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.321767.
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Habbireeh, A. A. "The numerical solution of time dependant problems by finite element methods." Thesis, University of Liverpool, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.383495.
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Ulusoy, Suleyman. "The Mathematical Theory of Thin Film Evolution." Diss., Georgia Institute of Technology, 2007. http://hdl.handle.net/1853/16213.
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We try to explain the mathematical theory of thin liquid film evolution. We start with introducing physical processes in which thin film evolution plays an important role. Derivation of the classical thin film equation and existing mathematical theory in the literature are also introduced.
To explain the thin film evolution we derive a new family of degenerate parabolic equations. We prove results on existence, uniqueness, long time behavior, regularity and support properties of solutions for this equation.
At the end of the thesis we consider the classical thin film Cauchy problem on the whole real line for which we use asymptotic equipartition to show H^1(R) convergence of solutions to the unique self-similar solution.
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Birchall, Alan. "Mathematical solutions to problems in radiological protection involving air sampling and biokinetic modelling." Thesis, University of Bristol, 1998. http://clok.uclan.ac.uk/19011/.
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Intakes of radionuclides are estimated with the personal air sampler (PAS) and by biological monitoring techniques: in the case of plutonium, there are problems with both methods. The statistical variation in activity collected when sampling radioactive aerosols with low number concentrations was investigated. By treating man as an ideal sampler, an analytical expression was developed for the probability distribution of intake following a single measurement on a PAS. The dependence on aerosol size, specific activity and density was investigated. The methods were extended to apply to routine monitoring procedures for plutonium. Simple algebraic approximations were developed to give the probability of exceeding estimated intakes and doses by given factors. The conditions were defined under which PAS monitoring meets the ICRP definition of adequacy. It was shown that the PAS is barely adequate for monitoring plutonium at ALl levels in typical workplaceconditions. Two algorithms were developed, enabling non-recycling and recycling compartmental models to be solved. Their accuracy and speed were investigated, and methods of dealing with partitioning, continuous intake, and radioactive progeny were discussed. Analytical, rather than numerical, methods were used. These are faster, and thus ideally suited for implementation on microcomputers. The algorithms enable non-specialists to solve quickly and easily any first order compartmental model, including all the ICRP metabolic models. Nonrecycling models with up to 50 compartments can be solved in seconds: recycling models take a little longer. A biokinetic model for plutonium in man following systemic uptake was developed. The proposed ICRP lung model (1989) was represented by a first order compartmental model. These two models were combined, and the recycling algorithm was used to calculate urinary and faecal excretion of plutonium following acute or chronic intake by inhalation. The results indicate much lower urinary excretion than predicted by ICRP Publication 54.
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Adali, Ali. "Applications of the Quasi Solution Method to Interfacial and Other Nonlinear Problems." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1502990387553748.
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Locksley, Harold Walker. "Indirect methods for the numerical solution of ordinary linear boundary value problems." Thesis, University of Newcastle Upon Tyne, 1993. http://hdl.handle.net/10443/1958.
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This thesis is mainly concerned with indirect numerical solution methods for linear two point boundary value problems. We concentrate particularly on problems with separated boundary conditions which have a 'dichotomy' property. We investigate the inter-relationship of various methods including some which have first appeared since the work for this thesis began. We examine the stability of these methods and in particular we consider circumstances in which the methods discussed give rise to well conditioned decoupling transformations. Empirical comparisons of some of the methods are described using a set of test problems including a number of ill conditioned problams. 'stiff' and marginally In the past the main method of error estimation has been to repaat the whole calculation. Here an alternative error estimation technique is proposed and a related iterative improvement method is considered. Although results for this are not completely conclusive we think they justify the need for further research on the method as it shows promise of being a novel and reliable practical method of solving both well conditioned and ill conditioned problems.
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Lynch, M. A. M. "Accurate and efficient algorithms for the approximate solution of large hyperbolic problems." Thesis, Heriot-Watt University, 1992. http://hdl.handle.net/10399/1473.
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Moody, R. O. "The numerical solution of moving-boundary problems using moving-finite-element methods." Thesis, University of Reading, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.383463.
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Zang, Peng. "Scaling solutions to Markov Decision Problems." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/42906.
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The Markov Decision Problem (MDP) is a widely applied mathematical model useful for describing a wide array of real world decision problems ranging from navigation to scheduling to robotics. Existing methods for solving MDPs scale poorly when applied to large domains where there are many components and factors to consider.
In this dissertation, I study the use of non-tabular representations and human input as scaling techniques. I will show that the joint approach has desirable optimality and convergence guarantees, and demonstrates several orders of magnitude speedup over conventional tabular methods. Empirical studies of speedup were performed using several domains including a clone of the classic video game, Super Mario Bros. In the course of this work, I will address several issues including: how approximate representations can be used without losing convergence and optimality properties, how human input can be solicited to maximize speedup and user engagement, and how that input should be used so as to insulate against possible errors.
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Wikström, Gunilla. "Computation of Parameters in some Mathematical Models." Doctoral thesis, Umeå University, Computing Science, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-565.
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<p>In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning.</p><p>Computation of parameters often includes as a part solution of linear system of equations <i>Ax = b</i>. The corresponding pseudoinverse solution depends on the properties of the matrix <i>A</i> and vector <i>b</i>. The singular value decomposition of <i>A</i> can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution <i>x</i>. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of <i>A</i> into account</p>
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Bahcecioglu, Tunc. "Parallel Solution Of Soil-structure Interaction Problems On Pc Clusters." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12612954/index.pdf.
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Numerical assessment of soil structure interaction problems require heavy computational efforts because of the dynamic and iterative (nonlinear) nature of the problems. Furthermore,
modeling soil-structure interaction may require
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Chandrasiri, L. H. G. S. "The solution of steady-state free surface problems by the finite element method." Master's thesis, University of Cape Town, 1992. http://hdl.handle.net/11427/17333.
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Bibliography: pages 64-68.<br>This thesis is concerned with the development of a variational formulation for the problem of viscous incompressible free surface flows, and with the development and implementation of algorithms for the solution of this problem by finite elements. The study is restricted to two-dimensional steady problems. The approach differs from those in current use, in that it is based on a two-stage strategy suggested by theoretical (existence) studies of the problem. In the first stage the free surface is kept fixed and the resulting so-called auxiliary problem is solved. In the second stage the equation for the normal stress on the free surface is used to update the free surface. Both the auxiliary problem and the normal stress equation are formulated variationally; in the case of the latter problem the unknown variable is actually the slope of the free surface. Finite element approximations are used in both problems. Algorithms are developed for determining solutions at the two stages, and for the overall problem. The key example treated is the dieswell problem, for the plane and axisymmetric cases. Solutions obtained by the present method are presented, and compared with the solutions of others where available.
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Mudavanhu, Blessing. "A new renormalization method for the asymptotic solution of multiple scale singular perturbation problems /." Thesis, Connect to this title online; UW restricted, 2002. http://hdl.handle.net/1773/6794.
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Creswell, Steven Howard 1961. "Solution to a bay design and production sequencing problem." Thesis, The University of Arizona, 1989. http://hdl.handle.net/10150/277124.
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This thesis addresses the problem of setting up a surface mount placement machine for production. The objective is to minimize the number of machine changeovers made during a production run consisting of a number of circuit cards. The solution to the problem involves two separate decisions. The first decision considers determining how to combine feeders together in "bays" or groups of feeders, and how to assign the bays to the circuit cards. The second decision considers the circuit card production sequence. A mathematical programming formulation is given, however, its solution is very difficult for problems of a realistic size. Several heuristic approaches are suggested and used to solve actual and test problems. The heuristic for bay design uses clustering techniques used in Group Technology while the sequencing problem is solved using heuristics based on solution techniques for the Traveling Salesman problem.
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Hessing, Tiffany Marie. "Second Graders' Solution Strategies and Understanding of a Combination Problem." Diss., CLICK HERE for online access, 2006. http://contentdm.lib.byu.edu/ETD/image/etd1470.pdf.
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Stucky, Eric. "An Exposition of Kasteleyn's Solution of the Dimer Model." Scholarship @ Claremont, 2015. http://scholarship.claremont.edu/hmc_theses/89.
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In 1961, P. W. Kasteleyn provided a baffling-looking solution to an apparently simple tiling problem: how many ways are there to tile a rectangular region with dominos? We examine his proof, simplifying and clarifying it into this nearly self-contained work.
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Silva, Heloisa Helena Marino. "Iterated deferred correction schemes for the numerical solution of two-point boundary value problems." Thesis, Imperial College London, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275845.
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Whitbrook, Amanda Marie. "The efficient numerical solution of differential/algebraic boundary value problems arising in detonation modelling." Thesis, Nottingham Trent University, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.388868.
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