Bivariate fractal interpolation functions on triangular domain for numerical integration and approximation

TL;DR

This paper introduces a novel method for constructing bivariate fractal interpolation functions over triangular domains using vertex coloring, and develops a double integration formula that aligns with fractal theory, supported by proofs and examples.

Contribution

It presents a new approach to partitioning triangles with a 3-color scheme and derives a double integration formula for fractal interpolation functions on these domains.

Findings

The double integral of the fractal interpolation function matches the fractal theory calculation.

A new partitioning method for triangles with a chromatic number of 3 is proposed.

The convergence of the integration method to the actual value is proven.

Abstract

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided. With the newly defined vertical scaling factor, it is clearly observed that the value of the double integral coincides with the integral value calculated using fractal theory. Further, a relation connecting the fractal interpolation function with the equation of the plane passing through the vertices of the triangle is established. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Sufficient examples are also provided to illustrate the method of construction and to verify the formula of double integration.