Mathematical Sciences: Closed Form Solutions of DifferentialEquations
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This mathematical work will center on the algebraic                        structure of solutions of differential equations.  By this                      one means the analysis of how solutions of differential                         equations can be expressed in terms of solutions of certain                     special differential equations.  Particular instances of                        these problems include solving differential equations                           implicitly or explicitly in terms of elementary or                              liouvillian functions, or more generally, in terms of                           solutions of differential equations of lower order.  Related                    investigations into the calculation of associated Galois                        groups will also be carried out.                                                     The work divides into three parts.  In the first, work                     will be done on the algebraic structure of elementary and                       other special functions, with the aim of improving and                          extending existing algorithms for integration in finite                         terms.  The second thrust involves a continuation of efforts                    to develop a theory of liouvillian first integrals - first                      integrals expressible in terms of algebraic combinations of                     integrals and exponentials, and developing algorithms for                       finding such integrals.  By a first integral, one understands                   a function of several variables which vanishes on solutions                     of systems of differential equations.  A third goal of this                     project is to understand the algebraic structure of solutions                   of linear differential equations.  Here one is interested in                    using algebraic arguments, Lie theory and Galois theory to                      determine whether solutions of given differential equations                     can be expressed as combinations of solutions of equations of                   lower order.  Coupled with this are fundamental questions of                    providing algorithms for accomplishing the desired                              representations.                                                                     In addition to its applications to the general theory of                   differential equations, this research bears direct                              relationship with important work at the frontiers of                            computer science, particularly with regard to symbolic                          manipulation algorithms.
{'FirstName': 'Michael', 'LastName': 'Singer', 'PI_MID_INIT': 'F', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Michael F Singer', 'EmailAddress': 'singer@math.ncsu.edu', 'NSF_ID': '000121887', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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[{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '2865', 'Text': 'NUMERIC, SYMBOLIC & GEO COMPUT'}]
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