EPSRC Reference: 
GR/K24406/01 
Title: 
INTEGRAL EQUATIONS ON THE REAL LINE WITH APPLICATION TO DIFFRACTION GRATING AND RELATED PROBLEMS 
Principal Investigator: 
ChandlerWilde, Professor SN 
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Department: 
Mathematical Sciences 
Organisation: 
Brunel University London 
Scheme: 
Standard Research (PreFEC) 
Starts: 
01 January 1995 
Ends: 
31 December 1997 
Value (£): 
100,087

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Summary on Grant Application Form 
The programme will consider the solvability of second kind integral equations on the real line and establish results in the space of bounded continuous functions which enable existence of solution and continuous dependence of the solution on the inhomogenous terms to be obtained given uniqueness of solution. Based on these results and previous work for WeinerHopf integral equations on the halfline, numerical methods will be developed and their stability and convergence analysed. Using the Green's function for the Helmholtz equation in a halfplane with an impedance boundary condition, novel boundary integral equation formulations will be developed for various boundary value problems for the Helmholtz equation in a perturbed halfplane, including examples of practical interest in outdoor sound propogation and the design of diffraction gratings. These boundary integral equations can be written as integral equations on the real line of the type to be considered in the general theory. Once uniqueness of solution has been established for the boundary integral equations, which will be relatively straight forward in certain cases, and require a deep understanding of appropriate radiation conditions in the most general case, the general theory will be applicable, giving existence of solution of the boundary integral equation and boundary value problem and stability and convergence of numerical methods for the boundary integral equation.

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Organisation Website: 
http://www.brunel.ac.uk 