Narrow Results

In the present paper, the smoothness of a Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS), as described by its Lipschitz exponent, is investigated. This is achieved by considering the simulation of a generally uneven surface using CHFIS. The influence of free variables and Lipschitz exponent on the smoothness of CHFIS is demonstrated by considering interpolation data generated from a sample surface.

This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order $\nu$ of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF.

In this paper, a new notion of super coalescence hidden-variable fractal interpolation function (SCHFIF) is introduced. The construction of SCHFIF involves choosing an IFS from a pool of several non-diagonal IFS at each level of iteration. Further, the integral of a SCHFIF is studied and shown to be a SCHFIF passing through a different set of interpolation data.

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$ and Fuchsian singularity at $z=0$, holomorphically depending on parameter $t$ within a polydisk in ${ℂ}^n$ centered at $t=0$. The eigenvalues of the leading matrix at $\infty$, which is diagonal, coalesce along a coalescence locus $\mathrm{\Delta }$ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at $z=0$, we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including $\mathrm{\Delta }$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of $\mathrm{\Delta }$, where it simplifies. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting $\mathrm{\Delta }$, and if suitable entries of the Stokes matrices vanish, then $\mathrm{\Delta }$ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.