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In the last ten years or so a considerable amount of work has been done to transform general relativlity into a mathematically rigorous disciple. With the work of Christodoulou and Klainerman on stability of Minkowski space-time, the work of Schoen and Yau on the positive energy theorem, the work of Christodoulou on the gravetational collapse, the work of Newman and others, on Yau's Lorentzian splitting conjecture, the work of Bartnick on maximal hypersurfaces in Lorentzian manifolds, general relativity has become a respectiable field of mathematical research.

Cauchy problem.

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Cauchy problem

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• Reference Manager
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In the last ten years or so a considerable amount of work has been done to transform general relativlity into a mathematically rigorous disciple. With the work of Christodoulou and Klainerman on stability of Minkowski space-time, the work of Schoen and Yau on the positive energy theorem, the work of Christodoulou on the gravetational collapse, the work of Newman and others, on Yau's Lorentzian splitting conjecture, the work of Bartnick on maximal hypersurfaces in Lorentzian manifolds, general relativity has become a respectiable field of mathematical research.

Cauchy problem.

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This thesis explores an accelerated alternating algorithm for solving the Cauchy problem associated with elliptic equations, utilizing Krylov subspaces. It addresses elliptic equations with variable coefficients and Helmholtz type equations, focusing on mixed boundary value problems, including Dirichlet and Robin conditions. The study reformulates the Cauchy problem as an operator equation, allowing the use of iterative methods such as the Conjugate Gradient Method and the Generalized Minimal Residual Method. The research demonstrates how these methods can be applied effectively, showing numerical success. The work is intended for mathematicians and researchers in science and engineering, offering insights into solving ill-posed inverse problems by improving stability through advanced algorithms.

Cauchy problem. --- Iterative methods (Mathematics)

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This dissertation by Chepkorir Jennifer focuses on methods for solving Cauchy problems related to elliptic and degenerate elliptic equations, which are typically ill-posed. The research addresses boundary value problems by dividing the boundary into parts with known Cauchy data and unknown Robin data. Various algorithms, such as the alternating algorithm and Krylov subspace methods, are analyzed for their effectiveness in solving these equations. The work includes numerical experiments that demonstrate the convergence and stability of the proposed methods, including applications to heat conduction problems and more general degenerate elliptic equations. The intended audience includes mathematicians and researchers in applied mathematics and engineering.

Cauchy problem. --- Boundary value problems.