Mathematical Sciences: Theory and Applications of Wave Front Propagation in Inhomogeneous Media

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This award supports mathematical research on nonlinear waves inhomogeneous media, focusing on properties of wave front phenomena arising in nonlinear partial differential equations. In particular reaction-diffusion equations and porous medium type equations will be treated. These equations come from premixed flame propagation problems in combustion theory and solute transport problems in underground water flow. Work will be done investigating general reaction-diffusion front propagation in the presence of periodic coefficients under the small diffusion fast reaction limit. Also to be studied is the reaction-diffusion front quenching problem in two dimensions from a numerical point of view and investigations into the existence of reaction-diffusion front propagation in quasiperiodic and random media. The concentrate front propagation for solute transport in homogeneous and inhomogeneous media will also be taken up. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.