Approximation Methods of Caputo Derivative and Applications to Non-local Porous Medium Equations
Open AccessThis thesis can be considered as two main parts. In the first part we study the fractional Caputo derivative and its discrete counterparts. Most importantly, we propose a strong compactness criterion for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. In the second part we investigate the existence of solutions for the porous-medium equation system with non-local diffusion effect in space, and non-local memory effect in time. The Caputo time derivative is used to model the memory effects involved. We prove global existence of non-negative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization.
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