Mathematical Sciences: Integral and Integro-differential Equations in Applied Mathematics

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The project concerns the development of techniques for solving some new types of integral equations, and the application of these techniques to a variety of physical problems. Of particular interest are equations of Volterra type whose solution experiences a blow-up, and equations of Fredholm type whose solution exhibits boundary layer effects. The areas of application include (i) localized plastic straining effects in high-strength metals, (ii) the ignition of combustible solid materials, and (iii) the migration of atomic impurities in crystals. The project concerns the development and application of new techniques for solving certain scientific and engineering problems which can be formulated in mathematical terms as an integral or integro-differential equation. The results will be used to enhance the understanding of phenomena which exhibit explosive behavior, such as the ignition and burning of solid matter. Related applications concern the failure of high-strength metals which undergo viscoplastic strain due to extreme loading. Another area of potential use concerns the effects of impurities in crystalline materials.