The use of transforms in connection with difference-differential equations and related topics

Abstract

The theory of difference-differential equations is exanined with special reference to the problem of finding a solution by me-ns of transforms, and the discussion is confined mainly to the line r equation, only brief references being made to the non-linear tyre. The question of simple exponential solutions is considers, first. Following this the published material is dealt with chronologically, beginning with a paper by Schmidt in 1911, no proceeding to a discussion of the main contributions from men until in present day. between 1911 and 1922 s number of German mathematicians studied these equations in considerable detail, and some of these writers are shown to have used methods based on transforms. They are followed by Bochner and Titchmarsh, both of whom make aefinite use of the Fourier transform in their work on the subject, and finally some recent papers by Wright are considered, which clearly exhibit the power of the Laplace transform method. right avoids certain ssumptions made in the earlier papers, and his results ore seen to be by far the most important and far-reaching.The dissertation concludes with a reference to the way in which these equations have been treated in practical problems of various types.