A Numerical Method for an Integro-Differential Equation with Memory in Banach Spaces: Qualitative Properties

Runge–Kutta convolution quadrature methods for well-posed equations with memory

“…This representation reduces the error analysis to the basic results in [4]. On the other hand, for the implicit Euler method, the above representation implies that the numerical solution is an average of the exact one, showing that qualitative properties such as positivity or contractivity are inherited by the numerical solutions, thus extending well-known results in the context of IVPs [3,11,20] and fractional diffusion-wave equations [6]. As different from the approach in [16], we avoid the study of the approximation properties of the Runge-Kutta convolution quadrature (see [15] and references therein), an interesting issue on its own and which seems to be necessary for the analysis of the corresponding discretizations of (1) with source or nonlinear terms.…”

“…Therefore, we conclude that for abstract IVPs both implementations of the Runge-Kutta method coincide, a fact pointed out in [16] for parabolic IVPs. Representation (31) was also noticed for fractional diffusion-wave equations and the implicit Euler method in [6].…”

“…Finally, for the backward Euler method, it turns out that µ n,τ is a probability [3,6] so that the numerical solution inherits any convex restriction satisfied by all the points of the continuous orbit. In particular, positivity and contractivity are preserved by the numerical approximations.…”

On the numerical inversion of the Laplace transform of certain holomorphic mappings

“…2,2]. To this end, we fix Λ = 2, δ = π − π/β, divide It is interesting to notice that since f is not smooth enough in time, traditional time stepping methods for this problem exhibit order reduction (see [3][4][5]10]). …”

Convolution quadrature time discretization of fractional diffusion-wave equations

“…Numerical methods for the time discretization of (1.1) have been proposed by various authors [4,5,7,8,17,24,26,35,36,37]. A usual approach for the time discretization of (1.1) consists in treating separately the derivative and the integral term, which are approximated by a standard difference formula and by means of a suitable quadrature rule, respectively.…”