Abstract The dissertation considers the time fractional diffusion equation (TFDE) with the Dirichlet boundary condition in the sub-diffusion case, i.e. the order of the time derivative is α ∈ (0,1). In the thesis we have studied the solvability of TFDE by the method of layer potentials. We have shown that both the single layer potential and the double layer potential approaches lead to integral equations which are uniquely solvable. The dissertation consists of four articles and a summary section. The first article presents the solution for the time fractional diffusion equation in terms of the single layer potential. In the second and third article we have studied the boundary behaviour of the layer potentials for TFDE. The fourth paper considers the spline collocation method to solve the boundary integral equation related to TFDE. In the summary part we have proved that TFDE has a unique solution and the solution is given by the double layer potential when the lateral boundary of a bounded domain admits C1 regularity. Also, we have proved that the solution depends continuously on the datum in the sense that a nontangential maximal function of the solution is norm bounded from above by the datum in L2(ΣT). If the datum belongs to the space H1,α/2(ΣT), we have proved that the nontangential function of the gradient of the solution is norm bounded from above by the datum in H1,α/2(ΣT).

The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the base-pair level. One promising way to predict these properties as a function of sequence is to model DNA with a set of base-pair parameters that describe the local stacking of the different possible base-pair step combinations. In this dissertation research, we develop and study a computational model for predicting the diffusion coefficients of short, relatively rigid DNA fragments from the sequence and the base-pair parameters. We focus on diffusion coefficients because various experimental methods have been developed to measure them. Moreover, these coefficients can also be computed numerically from the Stokes equations based on the three-dimensional shape of the macromolecule. By comparing the predicted diffusion coefficients with experimental measurements, we can potentially obtain refined estimates of various base-pair parameters for DNA. Our proposed model consists of three sub-models. First, we consider the geometric model of DNA, which is sequence-dependent and controlled by a set of base-pair parameters. We introduce a set of new base-pair parameters, which are convenient for computation and lead to a precise geometric interpretation. Initial estimates for these parameters are adapted from crystallographic data. With these parameters, we can translate a DNA sequence into a curved tube of uniform radius with hemispherical end caps, which approximates the effective hydrated surface of the molecule. Second, we consider the solvent model, which captures the hydrodynamic properties of DNA based on its geometric shape. We show that the Stokes equations are the leading-order, time-averaged equations in the particle body frame assuming that the Reynolds number is small. We propose an efficient boundary element method with a priori error estimates for the solution of the exterior Stokes equations. Lastly, we consider the diffusion model, which relates our computed results from the solvent model to relevant measurements from various experimental methods. We study the diffusive dynamics of rigid particles of arbitrary shape which often involves arbitrary cross- and self-coupling between translational and rotational degrees of freedom. We use scaling and perturbation analysis to characterize the dynamics at time scales relevant to different classic experimental methods and identify the corresponding diffusion coefficients. In the end, we give rigorous proofs for the convergence of our numerical scheme and show numerical evidence to support the validity of our proposed models by making comparisons with experimental data. / text

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.