Efficient exponential time differencing methods with Padé approximations for the semilinear space-time-fractional Schrödinger equation

The semilinear space-time-fractional Schrödinger equation is solved numerically using one-step and two-step exponential time differencing methods in time, and a fractional centered difference scheme in space. The two-parametric Mittag–Leffler function arising in the time integral is computed with Padé approximations, which improves the efficiency of the scheme markedly. Numerical experiments for well-known models from literature are performed to show the effectiveness and efficiency of the proposed methods.

Interpolation of maps with values in semilinear space

Several known results about exact errors of piecewise linear interpolation of maps from $f\colon {\mathbb{R}}^n \rightarrow {\mathbb{R}^m}$ class are generalized to classes of set-valued mappings with given convex majorant of moduli of continuity.

Duality and quasi-normability for complexity spaces

<p>The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0,) ^ω. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual.</p> <p>We here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E,) is a biBanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (B*_E, _B* ) is biBanach, where B*_E = {f : E Σ^∞_n=0 2^-n( V ) } and _B* = Σ^∞_n=0 2^-n We deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and Smyth complete, not only in the case that this dual is a subspace of [0,)^ω but also in the general case that it is a subspace of F^ω where F is any biBanach normweightable space. We also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. Finally, we investigate completeness of the quasi-metric of uniform convergence and of the Hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively.</p>