Leslie Fox - Mathematical Work

Mathematical Work

A detailed description of Fox's mathematical research can be found in obituaries and is summarised here. His early work with Southwell was concerned with the numerical solution of partial differential equations arising in engineering problems that, due to the complexity of their geometry, did not have analytical solutions. Southwell's group developed efficient and accurate relaxation methods, which could be implemented on desk calculators. Fox's contributions were particularly notable because he combined practical skills with theoretical advances in relaxation methods, which were to become important areas of research in numerical analysis. During the 1950 automatic electronic computers were replacing manual electro-mechanical devices. This led to different problems in the implementation of numerical algorithms, however the approach of approximating a partial differential equation by finite difference method and thus reducing the problem to a system of linear equations was the same. Careful analysis of the errors were a theme of many of Fox's early papers. His work at the Admiralty Computing Service and the National Physical Laboratory led to an interest in the computation of special functions, and his calculations were used in published tables. The techniques applied to the computation of special functions had much wider applicability including interpolation, stability of recurrence relations and asymptotic behaviour.

During the 1950s the group at the National Physics Laboratory worked on numerical linear algebra, and led to the publication of algorithms by Wilkinson and others. While not directly involved in development of numerical software he supported others in this endeavour. Fox worked on procedures for solving differential equations in which accuracy of the solution is estimated using asymptotic estimates. Fox's paper on this in 1947 led to the work of Victor Pereyra error-correcting algorithms for boundary-value problems and Stetter's results on defect correction and the resulting order of convergence.

Fox was also interested in the treatment of singularities in partial differential equations, the Stefan problem and other cases of free and moving boundaries. Many of these problems arose from his collaboration with mathematicians in industry through the Oxford Study Groups.

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