Fractal Dimension Through Numerical Integration of Fractal Interpolation Functions

Abstract

The initial objective of the paper is to propose an explicit relationship between the fractal dimension and fractal numerical integration of curves approximated through fractal interpolation from a discrete set of data points. Once the proposed relation is established, it is shown to be accurate by considering the data points of certain functions. The conventional box-counting dimension method has several drawbacks including the proper positioning of the boxes and determining the size of the boxes. The proposed relation becomes significant for such situations in providing the accurate determination of fractal dimension. Secondly, the paper aims to apply the derived relationship in the evaluation of two integral transforms of fractal interpolation functions, namely, the Fourier transform and the Laplace transform. The two integral transforms have been provided with alternate expressions, primarily, using the fractal dimension of fractal interpolation functions. Finally, considering these newly derived expressions and the proposed relation between fractal dimension and fractal numerical integration, this paper provides another method for the evaluation of the two integral transforms. This newly introduced method calculates the two integral transforms, via the fractal numerical integration of fractal interpolation functions.