Partial Differential Equations That Are Hard to Classify

Semi-linear n×n systems of the form A∂u/∂x+B∂u/∂y=f can generally be solved, at least locally, provided data are imposed on non-characteristic curves. There are at most n characteristic curves and they are determined by the coefficient matrices on the left-hand sides of the equations. We consider cases where such problems become degenerate as a result of ambiguity associated with the definition of characteristic curves. In such cases, the existence of solutions requires restrictions on the data and solutions might not be unique.     

The role of the Crank-Gupta model in the theory of free and moving boundary problems

A very brief review is given of the striking way in which the Crank-Gupta model has enhanced our understanding of the well-posedness of free and moving boundary problems.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

Alan Breach Tayler

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrialand Applied Mathematics, died on 28 January 1995, aged 63. Alan went up to Oxford in 1951 to read Mathematics at BrasenoseCollege. He obtained a first, and after a brief excursion tothe Bristol Aircraft Company, he returned to work for a DPhilwith George Temple. His thesis, completed in 1959 and entitled‘Problems in compressible flow’, contained a mixtureof analytic, approximate and numerical solutions which foreshadowedthe new practical applied mathematics that he embraced later.He became University Lecturer and Tutorial Fellow of St Catherine'sSociety in 1959. During the next twenty-five years, Alan Tayler brought a newethos to applied mathematics. This change came about throughhis recognition that the status quo in the 1960s, which compriseda delicate balance between theory and practice in the area ofapplied mechanics, was capable of far-reaching generalisation;indeed, he saw that such a development was essential since thefollowing decades were to be dominated by computers and an ever-increasingneed for mathematical modelling. In 1967, with Leslie Fox, heinitiated the mathematical Study Groups with Industry, whereinacademic and industrial researchers interact in week-long workshops.These were an immediate success: (1) with industry, who foundnew insights into their problems and new recruiting possibilities;(2) with students, whose enthusiasm to use their theoreticalknowledge soon led to the highly popular MSc in MathematicalModelling and Numerical Analysis; and (3) with faculty, bothpure and applied, who found an undreamed of source of fascinatingnew theoretical problems. For example, one intellectual consequencewas the use of industrial case studies to uncover the new fieldof ‘free boundary problems’, on which several thousandlearned articles have appeared since 1970.     

Edge diffraction of creeping rays

Explicit formulas are derived for waves modeled by the scalar two-dimensional Helmholtz equation for the field that is diffracted when surface creeping rays encounter an infinitely sharp edge. Both Neumann and Dirichlet boundary conditions are analyzed, and the diffracted field is found to be an order to magnitude smaller in the latter case.     

Continuum and discrete models of dislocation pile-ups. I. Pile-up at a lock

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip plane under the action of an external force in the direction of a locked dislocation in that plane is considered. As n→∞ there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away.