An integral equation description for linear systems is developed and used as the basis for the development of various system identification, model reduction and order determination methods. The system integral equation is utilized in the problem of parameter identification in continuous linear single-input single-output, multi-input multi-output and linear in parameters nonlinear systems. The approach is developed in the time domain where the effect of non-zero initial conditions and additive disturbances occurs naturally. Parameter estimates are deduced using several weighted residual concepts which have previously been used to produce approximate solutions to differential equations.

The importance of the Stokes system stems from the fact that the Stokes system is the stationary linearised form of the Navier Stokes system [Te01, Chapter1]. This linearisation is allowed when neglecting the inertial terms at a low Reinolds numbers Re << 1. The Stokes system essentially models the behaviour of a non - turbulent viscous fluid. The mixed interior boundary value problem related to the compressible Stokes system is reduced to two different BDIES which are equivalent to the original boundary value problem. These boundary-domain integral equation systems (BDIES) can be expressed in terms of surface and volume parametrix-based potential type operators whose properties are also analysed in appropriate Sobolev spaces. The invertibility and Fredholm properties related to the matrix operators that de ne the BDIES are also presented. Furthermore, we also consider the mixed compressible Stokes system with variable viscosity in unbounded domains. An analysis of the similarities and differences with regards to the bounded domain case is presented. Furthermore, we outline the mapping properties of the surface and volume parametrix-based potentials in weighted Sobolev spaces. Equivalence and invertibility results still hold under certain decay conditions on the variable coeffi cient The last part of the thesis refers to the mixed boundary value problem for the stationary heat transfer partial di erential equation with variable coe cient. This BVP is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Chkadua, Mikhailov and Natroshvili in the paper [CMN09]. Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed in both bounded and unbounded domains.

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Simulation tools capable of analyzing electromagnetic (EM) field/wave interactions on complex penetrable scatterers have applications in various areas of engineering ranging from the design of integrated antennas to the subsurface imaging. EM simulation tools operating in the time domain can be formulated to directly solve the Maxwell equations or the integral equations obtained by enforcing fundamental field relations or boundary conditions. Time domain integral equation (TDIE) solvers offer several benefits over differential equation solvers: They require smaller number discretization elements/sampling points (both in space and time). Despite the advantages, TDIE solvers suffer from increased computational cost, stability issues of the time-marching algorithms, and limited applicability to complex scatterers. This thesis is focused on addressing the last two issues associated with time domain volume integral equation (TD-VIE) solvers, as the issue of increased computational cost has been addressed by recently developed acceleration methods. More specifically, four new closely-related, but different marching on-in-time (MOT) algorithms are formulated and implemented to solve the time domain electric and magnetic field volume integral equations (TD-EFVIE and TD-MFVIE). The first algorithm solves the TD-EFVIE to analyze EM wave interactions on high-contrast dielectric scatterers. The stability of this MOT scheme is ensured by using two-sided approximate prolate spherical wave (APSW) functions to discretize the time dependence of the unknown current density as well as an extrapolation scheme to restore the causality of matrix system resulting from this discretization. The second MOT scheme solves the TDMFVIE to analyze EM wave interactions on dielectric scatterers. The TD-MFVIE is cast in the form of an ordinary differential equation (ODE) and the unknown magnetic field is expanded using spatial basis functions. The time-dependent coefficients of this expansion are found by integrating the resulting ODE system using a linear multistep method. The third method is formulated and implemented to analyze EM wave interactions on scatterers with Kerr nonlinearity. The former scheme integrates in time a coupled of system of the TD-EFVIE and the nonlinear constitutive relation, which is cast in the form of an ODE system, for the expansion coefficients of the electric field and flux using a linear multistep method. The last method described in this thesis is developed to analyze EM wave interactions on ferrite scatterers.

Plasmonic structures are utilized in many applications ranging from bio-medicine to solar energy generation and transfer. Numerical schemes capable of solving equations of classical electrodynamics have been the method of choice for characterizing scattering properties of such structures. However, as dimensions of these plasmonic structures reduce to nanometer scale, quantum mechanical effects start to appear. These effects cannot be accurately modeled by available classical numerical methods. One of these quantum effects is the tunneling, which is observed when two structures are located within a sub-nanometer distance of each other. At these small distances electrons “jump" from one structure to another and introduce a path for electric current to flow. Classical equations of electrodynamics and the schemes used for solving them do not account for this additional current path. This limitation can be lifted by introducing an auxiliary tunnel with material properties obtained using quantum models and applying a classical solver to the structures connected by this auxiliary tunnel. Early work on this topic focused on quantum models that are generated using a simple one-dimensional wave function to find the tunneling probability and assume a simple Drude model for the permittivity of the tunnel. These tunnel models are then used together with a classical frequency domain solver. In this thesis, a time domain surface integral equation solver for quantum corrected analysis of transient plasmonic interactions is proposed. This solver has several advantages: (i) As opposed to frequency domain solvers, it provides results at a broad band of frequencies with a single simulation. (ii) As opposed to differential equation solvers, it only discretizes surfaces (reducing number of unknowns), enforces the radiation condition implicitly (increasing the accuracy), and allows for time step selection independent of spatial discretization (increasing efficiency). The quantum model of the tunnel is obtained using density functional theory (DFT) computations, which account for the atomic structure of materials. Accuracy and applicability of this (quantum corrected) time domain surface integral equation solver will be shown by numerical examples.

In this dissertation, a hybrid volume and surface integral equation is used to solve scattering problems. It is implemented with RWG basis on the surface and the edge basis in the volume. Numerical results shows the correctness of the hybrid VSIE in inhomogeneous medium. The MLFMM method is also implemented for the new VSIEs. Further more, a synthetic apature radar imaging method is used in a 2D microwave imaging for complex objects. With the mono-static and bi-static interpolation scheme, a 2D FFT is applied for the imaging with the data simulated with VSIE method. Then we apply a background cancelling scheme to improve the imaging quality for the targets in interest. Numerical results shows the feasibility of applying the background canceling into wider applications.

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<p>In this report we describe simulation of the surface current density on a PEC cylinder and the diffracted field for a line source above a finite PEC ground plane as a means to verify the Nero2D program. The results are compared with the exact solution and give acceptable errors. A terrain model for a communication link is studied in the report and we simulate the wave propagation for terrain with irregular shapes and different materials. The Nero2D program is based on the fast multipole method (FMM) to reduce computation time and memory. Gaussian sources are also studied to make the terrain model more realistic</p>

Information about the distribution of surface soil moisture can greatly benefit the management of agriculture and natural resource. However, direct measurement of soil moisture over larger areas can be impractical and expensive, which has led scientists to develop satellite based remote sensing techniques for soil moisture assessments. Retrieving soil moisture from radar satellite imagery often associated with the collection and use of ancillary field data on surface roughness. However, field data that is meant to characterize surface roughness is often unreliable, is expensive to collect and is nearly impossible to acquire for large scale applications. These issues represent barriers to the adoption and of radar data for mapping soil moisture over large areas.The research presented in the dissertation is aimed at the development of an operational soil moisture assessment system based solely on radar satellite data and a radar model, eliminating the field data requirements altogether. The research is directed towards a so-called equation-based solution of the problem as an alternative to the approach that requires the use of extensive field-data sets on surface roughness. This approach is based on the concept that if the number of equations are equal to the number of unknowns, then explicit solutions of all unknowns are possible. My research derived the necessary equations to solve for soil moisture and surface roughness. The derivation of the equations and how to use them to estimate soil moisture without using ancillary field data was demonstrated by my research. Validation results showed that the equation-based method that was developed is capable of providing more precise estimates of surface soil moisture than that of ancillary field-data supported method.

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