The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored. / Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos.

Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas. / The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed.

The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed. / Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas.

The objectives of this study are to investigate the dynamic behavior of an apple slab subjected to drying at constant external conditions and under changing in the drying temperatures and to determine the effects of temperature and time combinations at different steps during drying on the process dynamics parameters, time constant and process gain of the system. For this purpose, a semi-batch dryer system was simulated by using integral method of analysis. Initially, the dynamic behavior of the drying temperature was investigated by using first order system dynamic model. Process dynamic parameters, time constant and process gain of the system, for change in drying temperature were determined. Secondly, investigation of the drying kinetics of the apple slab was carried out under constant external conditions in a semi-batch dryer. A mathematical model for diffusion mechanism assumed in one dimensional transient analysis of moisture distribution was solved by using explicit finite difference method of analysis. Thirdly, investigation of the drying kinetics of the apple slab was carried out under change in drying temperature at different time steps during drying. Inverse response system model was used for the representation of the dynamic behavior of drying. Process dynamic parameters, time constant and process gain of the system were determined. Model predicted results for apple slab drying under constant external condition and under step change in the drying temperature were compared with the experimental data.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / The fast advancement in computer technologies in the recent years has made the use of simulation to estimate stock/equity performances and pricing possible; however, determining the optimal exercise time and prices of American options using Monte-Carlo simulation is still a computationally challenging task due to the involved computer memory and computational complexity requirements. At each time step, the investor must decide whether to exercise the option to get the immediate payoff, or hold on to the option until a later time. Traditionally, the stock options are simulated using Monte-Carlo methods and all stock prices along the path are stored, and then the optimal exercise time is determined starting at the final time period and continuing backward in time. Also, as the number of paths simulated increases, the number of simultaneous equations that need to be solved at each time step grow proportionally. Currently, two theoretical methods have emerged in determining the optimal exercise problem. The first method uses the concept of least-squares approach in linear regression to estimate the value of continuing to hold on to the option via a set of randomly generated future stock prices. Then, the value of continuing can be compared to the payoff at current time from exercising the option and a decision can be reached, which gives the investor a higher value. The second method uses the finite difference approach to establish an exercise boundary for the American option via an artificially generated mesh on both possible stock prices and decision times. Then, the stock price is simulated and the method checks to see if it is inside the exercise boundary. In this research, these two solution approaches are evaluated and compared using discrete event simulation. This allows complex methods to be simulated with minimal coding efforts. Finally, the results from each method are compared. Although a more conservative method cannot be determined, the least-squares method is faster, more concise, easier to implement, and requires less memory than the mesh method. The motivation for this research stems from interest in simulating and evaluating complicated solution methods to the optimal exercise problem, yet requiring little programming effort to produce accurate and efficient estimation results.

Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

Magnetic induction tomography (MIT) is a non-invasive, soft field imaging modality that has the potential to map the electrical conductivity (σ) distribution inside an object under investigation. In MIT, a number of exciter and receiver coils are distributed around the periphery of the object. A primary magnetic field is emitted by each exciter, and interacts with the object. This induces eddy currents in the object, which in turn create a secondary field. This latter is coupled to the receiver coils and voltages are induced. An image reconstruction algorithm is then used to infer the conductivity map of the object. In this thesis, the application of MIT for volumetric imaging of objects with low conductivity materials (< 5 Sm-1) and dimensions < 1 m is investigated. In particular, two low conductivity applications are approached: imaging cerebral stroke and imaging the saline water in multiphase flows. In low conductivity applications, the measured signals are small and the spatial sensitivity is critically compromised making the associated inverse problem severely non-linear and ill-posed.The main contribution from this study is to investigate three non-linear optimisation techniques for solving the MIT inverse problem. The first two methods, namely regularised Levenberg Marquardt method and trust region Powell's Dog Leg method, employ damping and trust region strategies respectively. The third method is a modification of the Gauss Newton method and utilises a damping regularisation technique. An optimisation in the convergence and stability of the inverse solution was observed with these methods compared to standard Gauss Newton method. For such non linear treatment, re-evaluation of the forward problem is also required. The forward problem is solved numerically using the impedance method and a weakly coupled field approximation is employed to reduce the computation time and memory requirements. For treating the ill-posedness, different regularisation methods are investigated. Results show that the subspace regularisation technique is suitable for absolute imaging of the stroke in a real head model with synthetic data. Tikhonov based smoothing and edge preserving regularisation methods also produced successful results from simulations of oil/water. However, in a practical setup, still large geometrical and positioning noise causes a major problem and only difference imaging was viable to achieve a reasonable reconstruction.

Wave phenomena appear in many fields of science such as acoustics, geophysics, and quantum mechanics. They can often be described by partial differential equations (PDEs). As PDEs typically are too difficult to solve by hand, the only option is to compute approximate solutions by implementing numerical methods on computers. Ideally, the numerical methods should produce accurate solutions at low computational cost. For wave propagation problems, high-order finite difference methods are known to be computationally cheap, but historically it has been difficult to construct stable methods. Thus, they have not been guaranteed to produce reasonable results. In this thesis we consider finite difference methods on summation-by-parts (SBP) form. To impose boundary and interface conditions we use the simultaneous approximation term (SAT) method. The SBP-SAT technique is designed such that the numerical solution mimics the energy estimates satisfied by the true solution. Hence, SBP-SAT schemes are energy-stable by construction and guaranteed to converge to the true solution of well-posed linear PDE. The SBP-SAT framework provides a means to derive high-order methods without jeopardizing stability. Thus, they overcome most of the drawbacks historically associated with finite difference methods. This thesis consists of three parts. The first part is devoted to improving existing SBP-SAT methods. In Papers I and II, we derive schemes with improved accuracy compared to standard schemes. In Paper III, we present an embedded boundary method that makes it easier to cope with complex geometries. The second part of the thesis shows how to apply the SBP-SAT method to wave propagation problems in acoustics (Paper IV) and quantum mechanics (Papers V and VI). The third part of the thesis, consisting of Paper VII, presents an efficient, fully explicit time-integration scheme well suited for locally refined meshes.

Este trabalho apresenta parte de um sistema de simulação integrado para escoamento de fluido incompressível bidimensional em malhas não estruturadas denominado UmFlow-2D. O sistema consiste de três módulos: um módulo modelador, um módulo simulador e um módulo visualizador. A parte do sistema apresentado neste trabalho é o módulo simulador. Este módulo, implementa as equações de Navier-Stokes. As equações governantes são discretizadas pelo método de diferenças finitas generalizadas e os termos convectivos pelo método semi-lagrangeano. Um método de projeção é empregado para desacoplar as componentes da velocidade e pressão. O gerenciamento da malha, não estruturada é feito pela estrutura de dados SHE. Os resultados numéricos obtidos pelo UmFlow-2D são comparados com soluções analíticas e soluções numéricas de outros trabalhos. / This work presents an integratc simulation system, called UmFlow-2D, wich aims a,t simulating two-dimensional íncompressible fluid flow using unstructed mesh. The system is divided three modules: modeling module, simulation module and visualization module. In this work we present the simulation module. The simulation module implements the Navier-Stokes equation. The governing equations are discretized by a generalized flnite dillerence method and the convective terms by semi-lagrangean method. A projection method is employed to uncouple the velocity componentes and pressure. The management at the unstructed mesh is ready using a data structure called SHE. The numérica! results are compared with analytical solutions and numerical simulations of other works.

Diploma thesis mainly deals with temperature management and calculation of temperature profile in oven by using SMD packages (PLCC, 1206) of different thermal capacitance on testing PCB. Above all shows theoretical consecution of temperature profile calculation in oven by using known mathematical method like the lumped capacitance method or finite difference method. Theoretical solution and measured values are compared. Diploma thesis also deals with fixation methods of thermocouples K type on assembly, comparison methods based on known and subexperiment, determines the deficiencies of methods. This thesis can perform as theoretical as well as experimental resource to prediction of temperature profiles of PCB´s with different assembly density.