EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 19-32
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Using fractal calculus to express electric potential and
electric field in terms of staircase and characteristic

functions

Nasibeh Delfan1, Amir Pishkoo2,∗, Mahdi Azhini1, Maslina Darus3

1 Department of Mathematics, Islamic Azad University, Science and Research branch, Tehran,
Iran.
2 Physics and Accelerators Research School, Nuclear Science and Technology Research
Institute, P.O. Box 14395-836, Tehran, Iran
3 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia

Abstract. The Dirac Delta function is usually used to express the discrete distribution of electric
charges in electrostatic problems. The integration of the product of the Dirac Delta function and
the Green functions can calculate the electric potential and the electric field. Using fractal calcu-
lus, characteristic function, χCn(x), as an alternative for dirac delta function is used to describe
Cantor set charge distribution which is typical example of a discrete set. In these cases we deal
with Fα-integration and Fα-derivative of the product of characteristic function and function of
staircase function, namely f(SαCn

(x)), which lead to calculation of electric potential and electric
field. Recently, a calculus based fractals, called Fα-calculus, has been developed which involve
Fα-integral and Fα-derivative, of orders α, 0 < α < 1, where α is dimension of F . In Fα-calculus
the staircase function and characteristic function have special roles. Finally, using COMSOL Mul-
tiphysics software we solve ordinary Laplace’s equation (not fractional) in the fractal region with
Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three
first iterations.

2020 Mathematics Subject Classifications: 26A33, 28A80, 31C20

Key Words and Phrases: Cantor set, fractal calculus, Fα-integral, Fα-derivative, potential
theory, discrete distribution

1. Introduction

Different types of charge distributions can be expressed by using Dirac delta functions.
Dirac delta function is in fact one kind of distribution that in one dimension, it is written
as δ(x − a) which mathematically is improper function having the following properties

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3609

Email addresses: apishkoo@gmail.com (A. Pishkoo), maslina@ukm.edu.my (M. Darus)

http://www.ejpam.com 19 c© 2020 EJPAM All rights reserved.



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 20

[16, 18]:
1- δ(x− a) = 0 for x 6= a.
2-

∫
δ(x− a)dx = 1 if the region of integration includes x = a, and otherwise 0.

In more than one dimension, we merely take products of delta functions in each dimension.

Example 1. 1- In spherical coordinates, a charge Q uniformly distributed over a spherical
shell of radius R0 is

ρ(r) =
Qδ(r −R0)

4πr2
.

Example 2. 2- In cylindrical coordinates, a ring of charge Q with radius a laying in the
xy plane with its center at the origin is described with

λ(ρ, z) =
Qδ(ρ− a)δ(z)

2πρ
.

Example 3. 3- The same ring of charge Q with radius a in spherical coordinates is
described by

λ(r, θ) =
Qδ(r − a)δ(θ − π

2 )

2πr2 sin θ
.

In ordinary calculus, we deal with discontinuity, lack of continuity, in some points or
intervals. There are also some situations where a derivative of a function fails to exist.
Discontinuity and non-differentiability are two common problems in ordinary calculus. On
the other hand, we observe fractals [15, 19] which are continuous or discontinuous, and
usually nowhere differentiable.

Fractals are often so irregular that defining smooth, differentiable structures on them
seem very difficult. In the past few years, the new calculus called Fα-calculus or fractal
calculus [10–12] have been introduced by Gangal, Parvate, and Satin. Unfortunately,
applying the methods of ordinary calculus on fractals are powerless. They study “Fokker-
Planck equation”, “Langevin equation” on fractal curves [21, 22].

Using Fα-calculus, Golmankhaneh and Fernandez define integrals and derivatives of
functions on Cantor tartan spaces with different dimensions among with their related
differential equations [1]. For example sub- and super- diffusion on Cantor sets which
are totally disconnected fractals are appeared when one needs to relax the continuum
requirement [4].

In [10, 11] a new calculus based on fractal subsets of the real line is formulated which
involves an integral of order α, 0 < α < 1, called Fα-integral and a derivative of order
α, 0 < α < 1, called Fα-derivative. This enables us to differentiate functions, like the
Cantor staircase, “changing” only on a fractal set. The Fα-derivative is local unlike the
classical fractional derivative. They generalize their work in Rn [12] so that this time a
new calculus on fractal curves, such as the von Koch curve, is formulated. A Riemann-like
integral along a fractal curve F , called Fα-integral, is defined where α is the dimension
of F . A derivative along the fractal curve called Fα-derivative, is also defined. Fractal
calculus has found many applications in physics and engineering [2, 3, 5–9, 13, 14, 17, 20].



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 21

Definition 1. [10] If F is an α-perfect set then the Fα-derivative of f at x is

DαF (f(x)) =

{
F − limy→x

f(y)−f(x)
SαF (y)−SαF (x) if x ∈ F

0 othervise,

}
(1)

if limit exist. The α-perfect sets are sets having properties necessary to define Fα-
derivative.

Like the first order derivative, the Fα-derivative is a limit of a quotient. But here
the limit is F -limit, and the denominator is the difference in the values of the staircase
function SαF at two points. Moreover, intuitively speaking, F is typically the set of change
of the function, and α is typically the γ-dimension of F .

Theorem 1. [10] Let F be such that F ∩ [a, b] is compact and SαF is finite on [a, b]. Let
f ∈ B(F ), and b > a. If f is F -continuous on F ∩ [a, b], then f is Fα-integrable on [a, b]
if supremum and infimum ∫ b

a
f(x) dαFx =

∫ b

a
f(x) dαFx. (2)

In that case the Fα-integral of f on [a, b], denoted by
∫ b
a f(x) dαFx is given by the common

value.

2. Main Results

The forms of functions in R and Fα-space are different. For instance we consider
g(x) = x2 and f(x) = (SαC(x))2 (see Fig.1). Using Eq. 1, their standard and Fα-derivatives
can be compared (Fig. 2). After defining Fα-derivative, Fα-integral is defined. In the
definition of Fα-integral, just the values of the function at points belonging to the set F
are considered. In this type of integral, instead of the length of subintervals (xi+1 − xi)
the difference between their values of the integral staircase function SαF (xi+1)−SαF (xi) are
inserted (Fig. 3).
Fα-integration of staircase and characteristic functions
Indefinite integral of characteristic function is defined as∫ x́

a
χF (x) dαFx = SαF (x́),

assume for the simplicity SαF (a) = 0.
Indefinite integral of staircase function is∫ x́

a
SαF (x) dαFx =

[SαF (x́)]2

2
.



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 22

Figure 1: Comparing graph of functions g(x) = x2 and f(x) = (SαC(x))2 in R and Fα-space, respectively on
the interval [0,1]

Figure 2: Comparing derivative of functions g(x) = x2 and f(x) = (SαC(x))2 in R and Fα-space, respectively
on the interval [0,1]

In other words, consecutive double integration of characteristic function give us∫ x́

a
dαF x́

∫ x́

a
χF (x) dαFx =

[SαF (x́)]2

2
.

N -times Fα-integration of characteristic function give the following formula

(N − times integration)

∫ x́

a
dαF x́...

∫ x́

a
χF (x) dαFx =

[SαF (x́)]N

N
.

Fα-integration for product of SαF and χF
Let F = C1 and α = ln 2

ln 3 = 0.63, namely Cantor set in the first iteration, we calculate the
following integral:



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 23

Figure 3: Comparing an integral of functions g(x) = x2 and f(x) = (SαC(x))2 in R and Fα-space, respectively
on the interval [0,1]

∫ 1

0
SαC1

(x)χC1(x) dαC1
x =

∫ 1
3

0
SαC1

(x) dαC1
x+ 0 +

∫ 1

2
3

SαC1
(x) dαC1

x

=

∫ 1

0
SαC1

(x) dαC1
x

=
[SαC1

(1)]2

2
−

[SαC1
(0)]2

2

=
1

2
− 0,

in which we have used SαC1
(1

3) = SαC1
(2

3).
Let F = C2 then the integral will be∫ 1

0
SαC2

(x)χC2(x) dαC2
x =

∫ 1
9

0
SαC2

(x) dαC2
x+ 0 +

∫ 1
3

2
9

SαC2
(x) dαC2

x+ 0

+

∫ 7
9

6
9

SαC2
(x) dαC2

x+ 0 +

∫ 9
9

8
9

SαC2
(x) dαC2

x

=

∫ 1

0
SαC2

(x) dαC2
x

=
[SαC2

(1)]2

2
−

[SαC2
(0)]2

2

=
1

2
− 0,

in which we have used SαC2
(1

9) = SαC2
(2

9), SαC2
(1

3) = SαC2
(6

9), and SαC2
(7

9) = SαC2
(8

9).

It can be expected that for the nth iteration we have∫ 1

0
SαCn(x)χCn(x) dαCnx =

1

2
. (3)



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 24

Electric charge distributed on Cantor set and electric potential Let the electric
charge Q is uniformly distributed over the Cantor set. At each stage of process of iteration:
zero iteration, first iteration, second iteration etc. the electric charge density increases with
the certain ratio, respectively (see Fig. 4).
Since the charge distribution is discrete, the characteristic function χC(x) can be used to

Figure 4: Distributed charge Q on Cantor set

describe it as charge density function. At first, assume that the unit charge (Qtotal = 1) is
uniformly distributed over the set C0 = [0, 1] while χC0(x) = 1 for all values of x ∈ [0, 1].
Now we obtain the constant k.

Q =

∫ b=1

a=0
χC0(x) dαC0

x = k[SαC0
(1)− SαC0

(0)]. (4)

So k = 1. If 0 < a < b < 1 then Q 6= Qtot and for it we have (see Fig. 5)

Q = SαC0
(b)− SαC0

(a). (5)

Example1 Let a = 0.2 and b = 0.7 then the charge Q in the interval [a, b] is Q =
0.7− 0.2 = 0.5 (coulomb).
Now suppose the charge is uniformly distributed on C1 = [0, 1

3 ] ∪ [2
3 , 1] (Fig. 6). while

χC1(x) = 1 for x ∈ C1 and otherwise χC1(x) = 0 (see Fig. 7).
Example2 Let a = 0.25 and b = 0.6 then the charge Q in the interval [a, b] is

Q = SαC1
(0.6)− SαC1

(0.25) = 0.5− 0.375 = 0.125 (coulomb).

Then, this time suppose the charge is uniformly distributed on C2 = [0, 1
9 ]∪ [2

9 ,
3
9 ]∪ [6

9 ,
7
9 ]∪

[8
9 , 1], while χC2(x) = 1 for x ∈ C2 and otherwise χC2(x) = 0 (see Fig. 8).

The amount of distributed charge from 0 to x can be obtained from the graph of staircase



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 25

Figure 5: Qtot from 0 to x at continuous state (zero iteration)

Figure 6: Qtot from 0 to x is changed as staircase function (at first iteration)

function at second iteration (Fig. 9).
Example3 Let a = 0.32 and b = 5

6 then the charge Q in the interval [a, b] is

Q = SαC2
(
5

6
)− SαC2

(0.32) = 0.75− 0.475 = 0.28 (coulomb).

Electric charge density for Cantor set charge distribution In this section the
linear charge density in the fractal space for the Cantor set is obtained. In the first
iteration, it is equal to

λC1 =
Q
2
3

χC1(x) =
3Q

2
χC1(x). (6)



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 26

Figure 7: Density function λ is changed as characteristic function (at first iteration)

Figure 8: Density function λ is changed as characteristic function (at second iteration)

For the second iteration we have

λC2 =
Q

(2
3)2

χC2(x) = (
3

2
)2QχC2(x). (7)

Finally at nth iteration we obtain (Fig. 10)

λCn =
Q

(2
3)n

χCn(x) = (
3

2
)nQχCn(x). (8)



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 27

Figure 9: Qtot from 0 to x is changed as staircase function (at second iteration)

Figure 10: To express density function in terms of characteristic function

When n→∞ charge density goes to infinity.
Given charge density function, in electrostatic problems we can calculate the electric po-
tential by the following integral (cgs system):

U(SαC2
(x)) =

∫ 1

0

9
4QχC2(x́)

SαC2
(x)− SαC2

(x́)
dαC2

x́,

=

∫ 1
9

0

9
4Q

SαC2
(x)− SαC2

(x́)
dαC2

x́+

∫ 3
9

2
9

9
4Q

SαC2
(x)− SαC2

(x́)
dαC2

x́+

∫ 7
9

6
9

9
4Q

SαC2
(x)− SαC2

(x́)
dαC2

x́

+

∫ 9
9

8
9

9
4Q

SαC2
(x)− SαC2

(x́)
dαC2

x́.

Regarding different values of staircase function

SαC2
(
1

9
) = SαC2

(
2

9
) = 0.25; SαC2

(
3

9
) = SαC2

(
6

9
) = 0.5



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 28

Figure 11: Instead of variables x and x́ we consider SαC2
(x) and SαC2

(x́)

SαC2
(
7

9
) = SαC2

(
8

9
) = 0.75; SαC2

(0) = 0; SαC2
(1) = 1

We have

U(SαC2
(x)) =

9

4
Q ln |

SαC2
(x)

SαC2
(x)− 1

| . (9)

Example3 Calculate the values of electric potential at xp = 1
6 , 0.4, 0.5, 0.6,

5
6 .

U(SαC2
(
1

6
)) =

9Q

4
(−1.098); U(SαC2

(0.4)) = 0;U(SαC2
(0.5)) = 0;

U(SαC2
(0.6)) = 0; U(SαC2

(
5

6
)) =

9Q

4
(1.098).

If instead of charge distribution at second iteration we have charge distribution at nth

iteration we deduce the following formula

U(SαCn(x)) = (
3

2
)nQ ln |

SαCn(x)

SαCn(x)− 1
| . (10)

Now by using the operator Dα
Cn

on electric potential, electric field can be deduced. Using
the following formula

Dα
Cn ln |f(SαCn(x))| =

Dα
Cn
f(SαCn(x))

f(SαCn(x))
, (11)

We have
Dα
Cn
SαCn(x)[SαCn(x)− 1]− SαCn(x)Dα

Cn
[SαCn(x)− 1]

(SαCn(x)− 1)2
, (12)



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 29

while Dα
Cn
SαCn(x) = χαCn(x), and E(SαCn(x)) = −Dα

Cn
[U(SαCn(x))]. Finally, we obtain

electric field

E(SαCn(x)) =

χαCn (x)

(SαCn (x)−1)2

SαCn (x)

SαCn (x)−1

=
χαCn(x)

SαCn(x)(SαCn(x)− 1)
. (13)

3. Electric potential for Koch snowflake boundary

In COMSOL Multiphysics one deal with two different environment: “model builder
desktop” and “application builder desktop”. The application builder desktop environment
show ones how to use the Form editor and the Method editor. Note that we can switch
between the model builder and application builder by clicking on their buttons. This
software uses Finite Element Method (FEM) to solve different types of problems numeri-
cally. For Koch snowflake boundary, Laplace equation interface is used to compute electric
potential. Choosing normal mesh, one may compute electric potential. Our results have
been summarized in Fig. 12, Fig. 13, Fig. 15, and Fig. 16 for the zero, first, second, and
third iteration, respectively (see also MP4 file) .

Figure 12: To create Koch snowflake, to plot mesh, and to compute electric field at zero iteration

4. Conclusions and Future works

In this paper, fractal calculus as a new mathematical language tool is used in physics
(electrostatics) to calculate and express the electric potential and electric field. If charge
distribution is of type discrete and fractal, then we can solve one problem with many dif-
ferent distributions that each of them is ith iteration, i = 0, 1, 2,etc. We solve the problem
with Cantor set fractal charge distribution while for other kind of discrete fractal distri-
butions this work can be continue in future. We have also studied the same problem but
with different boundaries which are ith iteration, i = 0, 1, 2,etc in COMSOL Multiphysics



A. Pishkoo et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 19-32 30

Figure 13: To create Koch snowflake, to plot mesh, and to compute electric field at first iteration

Figure 14: To create Koch snowflake, to plot mesh, and to compute electric field at second iteration

Figure 15: To create Koch snowflake, to plot mesh, and to compute electric field at third iteration



REFERENCES 31

software numerically by using finite element method (FEM).

Acknowledgments

The authors are grateful to the referees and the editor for valuable remarks which
contributed to the improvement of the paper. The fourth author was supported by FRGS
grant number stated: FRGS/1/2019/STG06/UKM/01/1.

5. Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of
this paper.

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