Some studies within applied mathematics with focus on conditional symmetries of partial differential equations and bending waves in plates
Abstract (Summary)This is thesis in mathematics with special focus on applied mathematics and mathematical physics. There is an introduction (in Svedish), two papers and two addendums to the first paper. The second paper is published in an international journal. The purpose of the introduction is to sketch in an informal way some of the basic ideas and concepts concerning the use of symmetry methods for solving differential equations (DE). Roughly speaking, a symmetry of an object is a mapping (a transformation), which leaves the objekt unchanged, i.e. the object is mapped to itself. A symmetry of a DE is a group of transformations that maps the set of all solutions to the DE on itself. A special solution is mapped to another solution or to itself. We have omitted some mathematical details and no attemtemts are made to make the introduction completely rigorous. We discuss the connection between some ad hoc methods for solving special types of DE and symmetry groups, how symmetry groups can be used to lower the order of a DE, or to obtain similarity solutions to a PDE. A number of illustrative examples are concidered. There are also some historical notes. In paper 1 we have calculated conditional Q-symmetries of a class of nonlinear wave equations with a variable wave speed. This class o wave equations have many applications in the study of wave propagation in different areas, for example in the studies of one-dimensional gas flow, shallow water theory and electromagnetic transmission lines. By using the obtained conditional Q-symmetry generators to get similarity "ansatzes" and reductions to ordinary differential equations we have obtained several solutions which can not be obtained by the classical method of Sophus Lie. In an addendum we discuss with help of conditional symmetries connections between the linear heat equation and some nonlinear evolution equations.In particular, we show the relation between the well-known Cole-Hopf transformation, which transforms Burgers' equation to the linear heat equation and a corresponding conditional Q-symmetry to the linear heat equation. In paper 2 we used the general form of the standard Kirchoff plate equation for an anisotrophic plate to derive integral representation formulas for describing the propagation of bending waves in infinite plates.We started with the most general case with an anisotrophic plate and very modest restrictions on the outher force. After that we concidered various special cases and obtained both new and well-known formulas. For example in the simplest case with an isotrophic plate then the outher force is an unit impulse both in space and time we obtained the clasical Boussinesq solution. Our starting point was to use Fourier analysis. Some of our results contains a Fourier transform as factor in the integrand, which in some cases can be simplified. We also proved some cruical formulas, which are of great importance for this paper. Moreover we have compared theoretical results with experimental data and the agreement was very good.
School:Luleå tekniska universitet
Source Type:Master's Thesis
Date of Publication:01/01/2005