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 Besicovitch dimensional is referred to as the fractional dimension. In this regard, we illuminate glimpses of fractal dimension and its distinction from Euclid dimension. Generally, in the Euclidean space, the dimension is integer whereas in the case of fractals the fractal dimension may not be an integer [ 5 ,6]. For a delineated explanation of fractals, we encourage the readers to refer the interesting book [5] . The book brings a clear picture of fractals viz. Koch curves, snowflakes, Sierpinski gasket (SG) and its broad range of advanced ideas viz. fractal singularities of differential equations, fractal lattices, self‐mapping fractals, randomness, etc. Other standard books for a detailed study of fractals can be found in Refs. [ 6 –10] and further, fractal applications may be found in Refs. [ 2 , 3 11–13].

In the next section, we discuss few basic shapes of fractal geometry viz. triadic Koch curve, snowflake, and SG.

18.1.1 Triadic Koch Curve

Koch curve is one of the earliest and simplest fractal curve constructed using the Euclidean line segment of unit length as given in Table. 18.1.

Length of
Triadic Koch curve	Number of segments	Each segment	Koch curve
	1	1	1
	4
	42
	43
	44

The figures given in Table 18.1 have been reproduced using the MATLAB code given by D. Scherzinger in http://m2matlabdb.ma.tum.de/download.jsp?MC_ID=5&SC_ID=13&MP_ID=105. Similarly, a Koch snowflake (Figure 18.1) may be generated using an equilateral triangle instead of line segment considered in the Koch curve.

The snowflake given in Figure 18.1 has been reproduced using the MATLAB code given in https://codereview.stackexchange.com/questions/144700/plotting‐the‐koch‐snowflake.

18.1.2 Sierpinski Gasket

SG is named after Polish mathematician W. Sierpinski. The most commonly known SG given in Table 18.2 is formed using equilateral triangle by removal of middle inscribed triangle of th area. Sometimes, SG is also referred to as limit of pre‐gaskets G_m as

Number of iterations	Pre‐gaskets Gm	Sierpinski triangle
0	G0
1	G1
2	G2
4	G4
8	G8

Figures of G_m and SG have been reconstructed through MATLAB exchange file downloaded from https://in.mathworks.com/matlabcentral/fileexchange/50417‐sierpinski‐fractal‐triangle.

Further, the figures of G_m and SG have been incorporated into Table 18.2 .

Generally, the fractal analysis is not used as a direct approach for solving differential equations. But, the inverse approach of formulating differential equations from fractals is possible using construction of Laplacian on the SG and related fractals. In this regard, we have discussed briefly fractal differential equations over SG in the next section.

18.2 Fractal Differential Equations

A pioneer work on fractal differential equations on SG has been carried out by Dalrymple et al. [14]. In this regard, let us illustrate the work of Dalrymple et al. [14] for heat and wave equations in Sections 18.2.1 and 18.2.2, respectively.

18.2.1 Heat Equation

This section discusses the heat equation

(18.1)

subject to Dirichlet boundary conditions

(18.2)

and the initial condition

(18.3)

where Δ represents the symmetric Laplacian defined by Kigami [15] on a Sierpinski triangle. Here, u(x, t) is a function of x ∈ SG and 0 ≤ t ∈ ℜ.

The algorithms for computing the approximate solution of Eq. (18.1) may be found in greater detail in Ref. [14] . But, for the sake of completeness, the difference equation obtained using the procedure given in Ref. [14] are reproduced in Eq. (18.4) by restricting time t to integer multiples (in terms of k) for timescale h.

(18.4)

where E_m are the edges of pre‐gasket G_m.

18.2.2 Wave Equation

In this section, we consider wave equation

(18.5)

subject to boundary conditions u(p_j, t) = 0; j = 0, 1, 2 and initial conditions

(18.6)

where x ∈ SG and 0 ≤ t ∈ ℜ. Similar to the heat equation, the difference equation associated with Eq. (18.5) for computation of approximate solution is written as

18.7

using the procedure given in Ref. [14] .

It may be noted that the present chapter has been devoted only to illustrate the basic ideas of fractals and differential equations over fractals. As such, Sections 18.2.1 and 18.2.2 just give an overview of application of differential equations over fractals. The detailed analysis of Eqs. ( 18.1 ) and ( 18.5 ) may be found in Ref. [14] . Further, readers interested in having a thorough knowledge of fractal differential equations and fractal‐time dynamical systems are highly encouraged to refer Refs. [16,17].

References

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