Nonlinear Fredholm-Volterra Integral Equation and its Numerical Solutions with Quadrature Methods

In this work, the existence and uniqueness of solution of the nonlinear Fredholm-Volterra integral equation (NF-VIE), with continuous kernels, are discussed and proved in the space L2(Ω)—C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a nonlinear system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced, using quadrature methods, to a nonlinear algebraic system (NAS). Then, the NAS can be solved, using two numerical methods. These methods are: Trapezoidal rule method and Simpson's rule method. Finally, some numerical examples are considered and the error estimate, in each case, is computed

Modal density of thin circular cylinders

Vibration modal response of thin cylindrical shell

The search for an optimal means of determining the minmax control parameter using sensitivity analysis

The use of computational methods for design and simulation of control systems allows for a cost-effective trial and error approach. In this work, we are concerned with the robust, real-time control of physical systems whose state space is infinite-dimensional. Such systems are known as Distributed Parameter Systems (DPS). A body whose state is heterogeneous is a distributed parameter. In particular, this work focuses on DPS systems that are governed by linear Partial Differential Equations, such as the heat equation. We specifically focus on the MinMax controller, which is regarded as being a very robust controller. The mathematical formulation of the MinMax controller involves a design parameter, &thetas;. This parameter provides a numerical measure of the robustness of the MinMax controller; hence it is very important. However, there exists no explicit formula for determining its value in advance of attempted control design. Currently, this parameter\u27s optimal value—optimal in the sense of robustness—is determined experimentally using an iterative process that seeks to maintain stability in the closed loop control system as well as an always positive definite result for [I − &thetas;2 PΠ] (i.e. [I − &thetas;2 PΠ \u3e 0) where 1 is the identity matrix, while P and Π are solutions to Algebraic Riccati Equations discussed in this dissertation. This process is obviously computationally expensive. The search for a more efficient means of determining &thetas;, including the possibility of the emergence of an explicit formula based on some mathematically rigorous criteria, is the driving force for this work. We use sensitivity analysis as a tool to mathematically investigate different criteria (such as the controller sensitivity, state sensitivity, Riccati equations\u27 sensitivity, etc.) to help achieve our goal of formulating a more efficient means of determining an optimal value for &thetas;. For each of the systems investigated, it was found that low &thetas; values (e.g. 0.05) are sufficient for adequate performance, robustness, and convergence of the MinMax controller

An Adaptive Method for Calculating Blow-Up Solutions

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as blow-up. The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting to other ad hoc methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure

Absolute Value Boundedness, Operator Decomposition, and Stochastic Media and Equations

The research accomplished during this period is reported. Published abstracts and technical reports are listed. Articles presented include: boundedness of absolute values of generalized Fourier coefficients, propagation in stochastic media, and stationary conditions for stochastic differential equations

Optical image Formation in terms of entropy transformations and intensity matrices.

Thesis (M.A.)--Boston UniversityThe entropy concepts in thermodynamics, statistical mechanics, information theory and optics are discussed along with their relationships. As a first step of the application of the entropy concept we evaluate the entropy loss in optical system (one- and two-dimensional cases) vs. defocussing from geometrical and physical optics viewpoints. After reviewing the image formation with Fourier analysis and bringing it to a general formulation, we consider the foundations of the optical image formulation in terms of matrices and apply it to several illustrative cases. We investigate the properties of the illumination matrix of optics proposed by Gabor and Gamo under the guidance of statistical mechanical density matrix. Comparison between the density matrix and the illumination matrix is considered. [TRUNCATED

The numerical solution of Fredholm integral equations of the first kind

A convergence theorem for Lee and Prenter\u27s filtered leastsquares method for solving the Fredholm first kind equationKf = g is corrected. Under suitable restrictions, the filtered least squares method is shown to be well-posed under compact perturbations in K and arbitrary perturbations in g;The M-solution of the Fredholm first kind equation Kf = g is the unique minimum norm element f (epsilon) M which minimizes (VBAR)(VBAR)Kf - g(VBAR)(VBAR), where M is a finite dimensional subspace of the given Hilbert space. Several convergence results are proved for the M-solution. A modified Gram-Schmidt method for calculating the M-solution of Kf = g is compared to a modification of the normal equations method which is used to calculate the M-solution of Kf = g. This modified Gram-Schmidt method is shown to be well-posed under compact perturbations in K and arbitrary perturbations in g;A convergence theorem for Marti\u27s method for solving the Fredholm first kind equation Kf = g is corrected. Marti\u27s method is shown to be well-posed under perturbations in g. It is shown that Marti\u27s method is not necessarily well-posed under compact perturbations in K. Also included are results relating Marti\u27s solution to the M-solution of Kf = g and to the least square solution of minimum norm of Kf = g;References;Lee, J. W; and Prenter, P. M. An Analysis of the Numerical Solution of Fredholm Integral Equations of the First Kind. Numerische Mathematik 30 (1978):1-23;Marti, J. T. An Algorithm for Computing Minimum Norm Solutions of Fredholm Integral Equations of the First Kind. SIAM Journal of Numerical Analysis 15 (December 1978):1071-1076

A numerical investigation of the Rayleigh-Ritz method for the solution of variational problems

PhD ThesisThe results of a numerical investigation of the Haylaigh·-H:l:tz method for the approxi.mate solution of two-point boundary value problems in ordinary differential equations are presented. Theoretical results are developed which indicate that the observed behaviour ie typical of the method in more general applications. In particular9 a number of choices of co-ord:i.natefunctions for certain second order equations are considerede A new algorit~~ for the efficient evaluation of an established sequence of functions related to the Legendre polynomials is desoribed, and the sequence is compared in use with a similar sequence related to the Chebyshev polynomials. Algebraic properties of the Rayleigh ••R1tz equations tor these and other co-ordinate systems are discussede The Chebyshev system is shown to lead to equations with oonvenient computational and theoretical properties, and the latter are used to characterize the asymptotic convergence of the approximations for linear equationse These results are subsequently extended to a certain type of non-linear equatione An orthonormalization approach to the solution ot the R~leigh- Ritz equations which has been suggested in the literature is compa.red in practice with more usual methods, and it is shown that the properties of the resulting approximations are not improvedo Since it is knoWli.that the method requires more work than established ones it cannot be recommendedo Quadrature approximations of elements of the ~leigh-Ritz matrices a.re investigated, and known results for a restricted class ot quadra·t';.re approximation are extended towards the more general case. In a final chapter extensions of the material of earlier chapters to partial differential equations are described, and new forms of the 'finite element' and 'extended Kantorovich' methods are proposed. A summary of the conclusions discerned from the investigation is given