Discrete maximal regularity of time-stepping schemes for fractional evolution equations

In this work, we establish the maximal $\ell^p$-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order $\alpha\in(0,2)$, $\alpha\neq 1$, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis [48] and its discrete analogue due to Blunck [10]. These results generalize the corresponding results for parabolic problems

Well posedness for semidiscrete fractional Cauchy problems with finite delay

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay ∆ α u ( n ) = Tu ( n ) + β u ( n − τ ) + f ( n ) , n ∈ N , 0 < α ≤ 1 , β ∈ R , τ ∈ N 0 , (0.1) where is a bounded linear operator defined on a Banach space (typically a space of functions like ) and corresponds to the time discretization of the continuous Riemann–Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbol (( z − 1) α z 1 − α I − β z − τ − T ) − 1 , | z |= 1 , z ̸= 1 , whenever and satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations