Hyperfinite methods for generalized smooth functions

The main aim of the proposed project is to develop hyperfinite methods for the solution of partial differential equations with generalized smooth functions. The nonlinear theory of generalized smooth function has recently emerged as a minimal extension of Colombeaus theory of generalized functions that allows for more general domains for generalized functions, resulting in the closure with respect to composition and a better behaviour on unbounded sets. By hyperfinite methods, we mean both the use of infinite integer Colombeau generalized numbers and the use of closed intervals with infinite boundary points. The former will be used to introduce a better notion of power series and hence a corresponding Cauchy-Kowalevski theorem. The latter will be used to define a Fourier transform applicable to any generalized smooth function (not only to those of tempered type). We also plan to study the method of characteristics, a hyperfinite Picard-Lindelöf theorem for partial differential equations, and to study operators defined by using hyperfinite methods. The project is situated within the larger scientific community interested in modelling singular phenomena

The main aim of the proposed project was to develop hyperfinite methods for the solution of partial differential equations with generalized smooth functions. The nonlinear theory of generalized smooth function has recently emerged as a minimal extension of Colombeau's theory of generalized functions that allows for more general domains for generalized functions, resulting in the closure with respect to composition and a better behaviour on unbounded sets. By hyperfinite methods, we mean both the use of infinite integer Colombeau generalized numbers and the use of closed intervals with infinite boundary points. The former has been used to introduce a better notion of power series and hence a corresponding Cauchy-Kowalevski theorem. The latter has been used to define a Fourier transform applicable to any generalized smooth function (not only to those of tempered type). We also plan to study the method of characteristics and a Picard-Lindelöf theorem for partial differential equations. The project is situated within the larger scientific community interested in modelling singular phenomena