Lie group symmetries of delay and integro differential equations
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NO FULL TEXT AVAILABLE. Access is restricted indefinitely. ----- In this thesis we extend the classical algorithm for computing point symmetries of differential equations to delay- and integro differential equations of several variables. Similar efforts have been made in the past but never with a rigorous geometrical framework. We will close this gap. Our approach provides a tool to solve or reduce a large class of equations including recursive relations, delay differential equations, integro partial differential equations, discretised differential equations and initial value problems. We introduce a structure on a bundle, which allows us to define a coordinate system on the bundle suitable to represent the values of a function on different values in its domain. This structure is again a bundle, similar to the jet bundle used to represent differential equations as embedded submanifolds via derivative coordinates. The proof is a straight forward consequence of the classical results and is given in the first part of this thesis. This allows us to use all classical theorems regarding invariance of sets under (one-parameter) group actions. We show how to prolong group actions and their generators to the introduced bundle. In this setup it is possible to represent delay-, respectively functional equations as level sets and we are able to compute symmetries, i.e. actions leaving the space of solutions invariant. It turns out that already very simple examples of such equations have non-trivial symmetries, which can be computed in a straight forward manner. The symmetries allow us to solve equations via symmetry reduction or to construct non-trivial solutions from simple known solutions. We give various examples of this. We can apply a similar strategy to delay differential equations by adapting the notion of derivative coordinates to our set-up. The geometrical setting must only be altered slightly to achieve this. We compute various examples, where we solve delay differential equations in one and two unknowns via its group of symmetries. The extension to integral- and integro differential equations is done via replacing integrals with corresponding Riemann sums. This only requires expressing the values of an unknown function on a partition of the area of integration as coordinates on a bundle. The introduced coordinate system turns out to be capable of this. We combine this again with derivative coordinates and illustrate how to compute symmetries of such equations. We compute analytical solutions of integro partial differential equations. In all of the above we face the problem that the domain of definition cannot be chosen to be an arbitrary small (open) set in the domain of an unknown function, as in the case of differential equations. The non-local nature of the equations must be taken into account. We solve this problem by making a priori assumptions on the regularity of the transformed functions, respectively the group actions. Strictly speaking we are computing conditional symmetries. However, in most practical situations the restriction is unnecessary. This thesis is structured as follows. The first chapter consists of an introduction where we present an overview of the classical theory of symmetries of differential equations. We conclude the chapter with a summary of five methods developed by other authors to compute symmetries of integro differential equations. We point out some problems and flaws in those theories. The second chapter consists of the main body of work, i.e. our main results. We introduce the space Dπ which will be the foundation for all of our reasoning. The space is endowed with a bundle structure and we show how one-parameter group actions induce morphisms on Dπ. We then continue explaining how different types of equations - of increasing complexity - can be represented as level sets of real-valued functions and how the main idea, i.e. symmetries being transformations leaving the solution space invariant, can be expressed in terms of bundle morphisms. This climaxes in theorem 2.18, describing an algorithm for finding the symmetry groups of integro partial differential equations for functions of any number of variables and arbitrary domains of integration. In every section we compute numerous examples of symmetry groups of the various types of equations under consideration. In the third chapter we apply the same techniques to some major applications. They include transition and boundary-crossing probabilities of jump diffusions, initial value problems, discretised partial differential equations and the Vlasov-Maxwell equations. I would like to express my gratitude to Dr. Mark Craddock for his guidance over the past four years.
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