Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y''(x)+Q(x)y'(x)+P(x)y(x)=R(x) with the Dirichlet''s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y'' can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology

Multivariate Affine Fractal Interpolation

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the p convergence of this type of interpolants for 1 = p < 8 extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuousfunctions defined on a multidimensional compact rectangle is studied

Generalized bivariate hermite fractal interpolation function

Abstract: Fractal interpolation provides an efficient way to describe a smooth or non-smooth structure associated with nature and scientific data. The aim of this paper is to introduce a bivariate Hermite fractal interpolation formula that generalizes the classical Hermite interpolation formula for two variables. It is shown here that the proposed Hermite fractal interpolation function and its derivatives of all orders are good approximations of original function even if the partial derivatives of original function are non-smooth in nature. © 2021, Pleiades Publishing, Ltd

Kantorovich-Bernstein a-fractal function in LP spaces

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein a-fractal operator in the Lebesgue space Lp(I), 1 = p = 8. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in Lp(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal Lp(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in Lp spaces is proven. Further, we derive the fractal analogues of some Lp(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein a-fractal function is developed

Fractal approximation of Jackson type for periodic phenomena

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed

Kantorovich-Bernstein _&#945;-Fractal function in Lp spaces

Fractal interpolation functions are xed points of contraction maps on suitable function spaces. In this paper, we introduce the&nbsp; Kantorovich-Bernstein -fractal operator in the Lebesgue space Lp(I); 1 ≤ p ≤ 1. The main aim of this article is to study the&nbsp; convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in Lp(I) spaces and Lipschitz spaces without aecting the non-linearity of the fractal functions. In the rst part of this paper, we introduce a new family of self-referential fractal Lp(I) functions from a given function in the same space. The existence of a Schauder basis consisting of selfreferential functions in Lp spaces is proven. Further, we derive the fractal analogues of some Lp(I) approximation results, for&nbsp; example, the fractal version of the classical Muntz-Jackson theorem. The one-sided approximation by the Bernstein -fractal function is developed. Mathematics Subject Classication (2010): 28A80, 41A25, 47A09, 47A05, 58C07. Key words: Fractal interpolation, -fractal operator, Bernstein-Kantorovich polynomial, function spaces, Schauder basis