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Nonlinear Evolution Equations: Analysis and Discretization Methods
The functional analytic description of time-dependent processes leads to (in general nonlinear) evolution equations of first or second order in time. Rather general assumptions on the structure of the underlying spatial differential operators such as monotonicity allow one to prove convergence of suitable numerical approximations and hereby also existence of solutions. Applications arise in fluid dynamics as well as in elasticity theory. An interesting aspect regarding equations of second order is their reduction to an equation of first order incorporating a simple integral operator in time. It will be part of future work to see whether more complicated nonlocality in time, such as memory or distributed delay, can be dealt with.
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