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In this thesis, we validate solutions of a fourth order boundary value problem arising from the study of microstructure in shape memory alloys using the computer-assisted ’Validated Continuation’ method described in Gameiro&Lessard (2010). The problem, as formulated in Healey&Miller (2007), considers a simplified two-dimensional model of an elastic solid undergoing forced anti-plane shear in the presence of interfacial energy. The resulting differential equation can be subdivided into a fourth order linear part and two nonlinearities of the form uxuxx and u 2 xuxx, where, for a particular loading parameter, the first nonlinearity vanishes which poses an important special case for the equation. Using the reciprocal of the capillarity coefficient as continuation parameter, several paths of equilibrium solutions bifurcating from the trivial solution are followed. Each numerical solution is then validated by a constructive analytical proof yielding a so-called validation radius. Existence of this radius allows to conclude that in the vicinity of the approximated numerical zero of the finite dimensional approximated equation there has to be a true solution of the partial differential equation. We achieve validation results for the special case for several numerical solutions at the beginning of different paths. However, due to computational limits, we can not pursue the paths to higher values of the continuation parameter, i.e. to solutions corresponding to a small capillarity coefficient and, thus, to small interfacial energy. We also analyze the difficulties of treating the general case with the same method and conclude with some ideas on different approaches for that problem. Declaration of Academic Integrity I hereby declare that I have created this work completely on my own and used no other sources or tools than the ones listed, and that I have marked any citations accordingly. Furthermore I declare that this thesis has never been published and / or submitted for any award to this or any other institution before. Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie Zitate kenntlich gemacht habe. Weiterhin versichere ich, dass die vorliegende Arbeit noch nie zuvor zur Erlangung einer Auszeichnung an diesem oder einem anderen Institut veröffentlicht und / oder eingereicht wurde. Friederike Schulze March 20, 2017

Die Methode der ‚Validierten Pfadverfolgung‘ angewandt auf ein vereinfachtes Modell eines zweidimensionalen Festkörpers