Advanced Numerical and Semi-Analytical Methods for Differential Equations

18Preliminaries of Fractal Differential Equations

Fractal analysis is a subject of great interest in various science and engineering applications, especially in computer graphics, architecture [1], medicine [2], fractal antennas [3], etc. Differential equations over fractal domain are often referred to as fractal differential equations. In this context, a basic idea of fractals and notion of fractal differential equations have been incorporated.

18.1 Introduction to Fractal

Nature often exhibits irregularity, nonlinearity, and complexity. Even, the classical Euclidean geometrical shapes viz. lines, circles, quadrilaterals, spheres, etc. lack to abstractly define the complex structures of nature. For instance, the clouds, coastlines, rocks, trees, etc. may not be completely expressed in terms of the classical geometry. In this regard, a new class of geometry was developed by Benoit B. Mandelbrot, mainly referred to as fractal geometry (based on a new family of shapes named as fractals).

Benoit B. Mandelbrot [4] introduced the concept of fractals and fractal geometry in an imaginative way. A creative bench mark “The Fractal Geometry of Nature” [5] written by Mandelbrot serves as a standard book for elementary concepts of fractals. In a more conventional way, the definition of fractals given by Mandelbrot [5] is stated as,

A fractal is by definition a set for which the Hausdorff Besicovitch Dimension strictly exceeds the topological dimension.

—B. B. Mandelbrot

Here, Hausdorff ...