Mathematical Sciences: Differential and Integral Equations  for Physical Systems
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work on this project continues mathematical research on                    models of physical systems in which emphasis is placed on effects               which are non-local in time, in space or both.  Applications are                expected in the analysis of thermal and elastic properties of                   materials with memory, electromagnetic and elastic scattering and               population models.  The mathematical formulations involve                       differential equations, Volterra and Fredholm integral equations                and combinations of these.                                                           Three areas will receive attention.  The first concerns                    models which involve operators influenced by memory effects and                 are represented by integral operators with singular kernels.                    Efforts will be made to understand relations between problems and               solutions as the order of the singularity changes.  Both                        qualitative behavior as well as numerical approximation of                      solutions will be sought.                                                            Another line of investigation will consider interface                      problems in which parabolic-hyperbolic equations describe the                   same phenomenon on either side of a region's boundary.  In trying               to understand these problems a new concept of absorbing                         boundaries has evolved which shows great theoretical as well as                 numerical promise - but at this time there is no firm                           understanding as to why the boundaries work so well.                                 Finally, in work related to population diffusion, studies                  will continue on reaction-diffusion processes in which directed                 diffusion rather than random diffusion is the controlling                       hypothesis.  Models of this type have been used to analyze single               species populations diffusing to avoid crowding, extensions to                  age-dependent situations, models for epidemics and for predator-                prey situations.  Much of the theoretical work has lagged behind                the numerical because basic existence questions remain                          unanswered.  The search will continue.
{'FirstName': 'Richard', 'LastName': 'MacCamy', 'PI_MID_INIT': 'C', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Richard C MacCamy', 'EmailAddress': None, 'NSF_ID': '000428444', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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