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During the period covered by this report the investigators continued their research on the development and application of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. They continued their analysis of the stability of mesh moving schemes for one-dimensional parabolic problems. Their also developed at moving mesh scheme with local refinement for two-dimensional hyperbolic systems and are considering a similar scheme for parabolic problems. They are applying our methods to several interesting physical problems, such as, elastic-plastic solids, combustion, and a nonlinear Schrodinger equation which exhibits self-focusing. (Author)