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Analysis and Scientific Computing on Some Nonlocal PDEs Open Access Deposited

In this thesis we will perform analysis and computation on nonlocal PDEs. This dissertation consist of two parts. In the first part, we study the existence of solutions for the porous-media equation system, which is non-linear and nonlocal in both space and time: in T3 for 0 < s ≤ 1, and 0 < α ≤ 1. The term Dα t u denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian −∆)s represents the Levy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. The proof also involves passing to the time steplimit using an equivalent of Aubin-Lions compactness criterion for a discrete version of the Caputo time derivative.

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