Lectures on Functional Theory and Partial Differential Equations

Paper represents a summary of three lectures delivered in Anderson Hall on April 25, 26, 29, 196

Numerical analysis of a coupled pair of Cahn-Hilliard equations with non-smooth free energy

A mathematical analysis has been carried out for a coupled pair of Cahn-Hilliard equations with a double well potential function with infinite walled free energy, which appears in modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. Existence and uniqueness are proved for a weak formulation of the problem, which possesses a Lyapunov functional. Regularity results for the weak formulation are presented. Semi and fully discrete finite element approximations are proposed where existence and uniqueness of their solutions are proven. Their convergence to the solution of the continuous solutions are presented. Error bound between semi-discrete and continuous solutions, between semi-discrete and fully discrete solutions, and between fully discrete and continuous solutions are all investigated. A practical algorithm to solve the fully discrete finite element formulation at each time step is introduced and its convergence is shown. Finally, a linear stability analysis of the equations in one dimension space is presented and some numerical simulations in one and two dimension spaces are preformed

A model-trust region algorithm utilizing a quadratic interpolant

AbstractA new model-trust region algorithm for problems in unconstrained optimization and nonlinear equations utilizing a quadratic interpolant for step selection is presented and analyzed. This is offered as an alternative to the piecewise-linear interpolant employed in the widely used “double dogleg” step selection strategy. After the new step selection algorithm has been presented, we offer a summary, with proofs, of its desirable mathematical properties. Numerical results illustrating the efficacy of this new approach are presented

On the moments of central values of modular L-functions

The thesis studies the integer-power moments of the central values of families of modular L -functions. The two families under consideration in the thesis are those quadratic twists of a L -function associated with a cusp form and L -functions of a Hecke-basis of the space of cusp forms. Appropriate moment estimates are derived for each family. Applications of the derived estimates are given