Selected Topics in Partial Differential Equations.

Partial differential equations with random coefficients (random PDEs) is a very developed and popular field. The variety of applications, especially in biology, motivate us to consider the random PDEs on curved moving domains. We introduce and analyse the advection-diffusion equations with random coefficients on moving hypersurfaces.

Nonlinear singular partial differential equations arise naturally when studying models from such areas as Riemannian geometry, applied probability, mathematical physics and biology. The purpose of this thesis is to develop analytical methods to investigate a large class of nonlinear elliptic PDEs underlying models from physical and biological.

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems.

In this thesis I extend the theory of integrable partial difference equations (PAEs) and reductions of these systems under scaling symmetries. The main approach used is the direct linearization method which was developed previously and forms a powerful tool for dealing with both continuous and discrete equations. This approach is further developed and applied to several important classes of.

A mechanical algorithm for constructing an isomorphism image (a Lie algebra with the same structural constants) of the Lie algebra of symmetry of a differential equations without solving maximum symmetry of the equations is given based on differential form Wus.

This thesis is devoted to the comprehensive investigation of the application of Lie groups in the study of partial differential equations. The current thesis consists of four separate main chapters.

Mathematics MSc dissertations. The Department of Mathematics and Statistics was host until 2014 to the MSc course in the Mathematics of Scientific and Industrial Computation (previously known as Numerical Solution of Differential Equations) and the MSc course in Mathematical and Numerical Modelling of the Atmosphere and Oceans.

The material in this thesis is the result of a year's experience in the solution of problems on the Caltech Electric Analog Computer. Although much work has been done elsewhere, the solution of partial differential equations is a relatively new field for the Caltech Computer. It is natural that such an undertaking should initiate points of view and techniques that differ from those of other.

Many problems arise from various real life applications may lead to mathematical models which can be expressed as initial value problems (IVPs) and boundary value problems (BVPs) of first and second ordinary differential equations (ODEs).These problems might not have analytical solutions, thus numerical methods are needed inapproximating the solutions.

The work in this thesis considers rational solutions of nonlinear partial differential equations formed from polynomials. The main work will be on the Boussinesq equation and the Kadomtsev-Petviashvili-I (KP-I) equation, the nonlinear Schroedinger equation will also be included for completeness. Rational solutions of the Boussinesq equation model rogue wave behaviour.

Mathematics at IST Austria is an open, curious community, active in research that is inspired by both purely mathematical concepts and applications in physics, computer science, and other fields. The groups on campus have strengths in a variety of areas, including discrete and computational topology and geometry (.