This dissertation focuses on a novel approach for characterizing the mechanical behavior of an elastic body. In particular, we develop a mathematical tool for the estimation of residual stress field in an elastic body that has mechanical properties similar to that of the arterial wall, by making use of intravascular ultrasound (IVUS) imaging techniques. This study is a preliminary step towards understanding the progression of a cardiovascular disease called atherosclerosis using ultrasound technology. It is known that residual stresses play a significant role in determining the overall stress distribution in soft tissues. The main part of this work deals with developing a nonlinear inverse spectral technique that allows one to accurately compute the residual stresses in soft tissues. Unlike most conventional experimental, both in vivo and in vitro, and theoretical techniques to characterize residual stresses in soft tissues, the proposed method makes fundamental use of the finite strain non- linear response of the material to a quasi-static harmonic loading. The arterial wall is modeled as a nonlinear, isotropic, slightly compressible elastic body. A boundary value problem is formulated for the residually stressed arterial wall, the boundary of which is subjected to a constant blood pressure, and then an idealized model for the IVUS interrogation is constructed by superimposing small amplitude time harmonic infinitesimal vibrations on large deformations via an asymptotic construction of its solution. We then use a semi-inverse approach to study the model for a specific class of deformations. The analysis leads us to a system of second order differential equations with homogeneous boundary conditions of Sturm-Liouville type. By making use of the classical theory of inverse Sturm-Liouville problems, and root finding and optimization techniques, we then develop several inverse spectral algorithms to approximate the residual stress distribution in the arterial wall, given the first few eigenfrequencies of several induced blood pressures.

We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.

The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).