PhD Thesis Defence Presentations - Fragkogiannis Georgios

The ellipsoidal coordinate system is the most general rectangular curvilinear coordinate system, since it depicts the complete anisotropy of the three–dimensional space. However, the full spectral analysis of the Laplace differential operator leads to the Lamé functions, which are not all available analytically. Hence, the present thesis is involved with the derivation of new expressions and the introduction of a methodology of the computational calculation of the Lamé functions of any degree, accompanied by two applications. The first application is part of a series of studies on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable. The nutrient and the inhibitor concentration, as well as the pressure field are provided in analytical fashion via closed form series solutions in terms of ellipsoidal eigenfunctions, while the evolution equation of all the tumour’s ellipsoidal interfaces is solved numerically. The second application is related to the analytic computation of electric and magnetic fields near sharp edges and corners, which mathematically represent singularities. A new method related to the geometry and the analysis of the ellipsoidal coordinate system is introduced, in which the asymptote of a two–sided or one–sided hyperboloid of two sheets with elliptic cross–section is utilized, which is a general non–circular double cone. In particular, the Lamé functions are adopted and new hyperboloidal solid and surface harmonics are constructed, followed by the respective orthogonality properties. The above method is demonstrated to the solution of two boundary value problems in electrostatics. Both refer to a non–penetrable two–hyperboloid of elliptic cross–section and its double–cone limit, the first one being charged and the second one scattering off a plane wave, while closed form expressions are derived for the related fields. The last part of the thesis examines the ellipsoidal system through Jacobi’s ellipsoidal coordinates. Using the new coordinates, each point in space is uniquely identified and the geometric degeneracy of the ellipsoidal system to the spheroidal or spherical geometry becomes clear, while the process of results’ reduction is facilitated because any indeterminacies are eliminated.