EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1738-1749
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

1

Solving conformable fractional differential equations2

with “EJS” software and visualization of sub-diffusion3

process4

Mahlagha Noshad1, Amir Pishkoo2,∗, Maslina Darus35

1 Department of Mathematics, Payame Noor University of Shiraz, Shiraz, Iran6

2 Physics and Accelerators Research School, Nuclear Science and Technology Research7

Institute, Tehran, Iran8

3 Department of Mathematical Sciences, Faculty of Science and Technology,9

Universiti Kebangsaan Malaysia, 43600, Bangi, Selangor, Malaysia10

11

Abstract. In this work, we numerically solve ordinary and conformable fractional differential
equations using Easy Java Simulations software. Their solutions, including homogeneous and non-
homogeneous parts, are compared in various time intervals. Using software’s visualization and
simulation features, we may better examine, compare, and evaluate solutions of analytical and
numerical fractional differential equations. A kind of oscillatory behavior is seen in long enough
times. In simulation of diffusion and sub-diffusion processes, two intriguing events have been
observed.

2020 Mathematics Subject Classifications: 26A33, 34A0812

Key Words and Phrases: Conformable fractional derivative, easy java simulations (EJS), frac-13

tional differential equation, sub-diffusion.14

15

1. Introduction16

Computer modeling and simulation are tightly intertwined. Modeling is the technique17

through which we build them. A model is a conceptual representation of a physical system18

and its features. A computer simulation is a model implementation that allows us to test19

the model in various scenarios to understand its behavior better. Easy Java/Javascript20

Simulations is a modeling application that enables non-computer scientists to develop sim-21

ulations in two programming languages. Easy Java/JavaScript Simulations (EJS) is a free,22

open-source application with over a thousand simulations accessible in the ComPADRE23

digital library [4, 13]. EJS automates operations like animation and solving ordinary24

differential equations numerically. Easy Java/Javascript Simulations has three modeling25

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4547

Email addresses: noshad.m107@gmail.com (M. Noshad),
apishkoo@gmail.com (A. Pishkoo), maslina@ukm.edu.my (M. Darus)

https://www.ejpam.com 1738 © 2022 EJPAM All rights reserved.



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1739

work panels. We utilize the application’s sequence of work panels to develop the model26

and its graphical user interface [16].27

28

As a result of advancements in computer hardware and software, industry, business,29

and research have successfully undergone technology-driven changes. All three industries30

have long made use of computer simulations. Programs were first created to make tasks31

simpler or quicker. Simulations work well for processes that take a very long period or32

happen very fast. Processes that are challenging, risky, or expensive are also excellent33

candidates. Computer simulations can help students comprehend science’s hidden mental34

realms through animation, resulting in a more abstract comprehension of scientific ideas.35

Students can manipulate and visualize quantitative data to form a qualitative mental36

image[23].37

38

Involving students in creating physical models to describe, explain, and predict events39

has been proven to minimize deficiencies in traditional teaching. Although the modeling40

technique may be used without a computer, employing one allows students to research41

difficult and time-consuming issues, illustrate their results, and communicate their discov-42

eries. Computer modeling, theory, and experiment may provide insights and information43

that cannot be achieved with a single method alone[12].44

45

By providing a central Website with computer modeling tools, simulations, educational46

resources like lesson plans, and a computational physics textbook that explains the algo-47

rithms used in the academic simulations, the Open Source Physics (OSP) project, located48

at www.compadre.org/osp/, aims to improve computational physics education (1). Our49

teaching materials are built around simple single-concept simulations with source codes50

that may be read, changed, recompiled, and disseminated. These interactive simulations51

will teach students to critically analyze and examine the premises and results of simula-52

tions [14].53

54

The Open Source Physics (OSP) project at www.compadre.org/osp/ aims to improve55

computational physics education by providing a central Web site with computer model-56

ing tools, simulations, curricular resources like lesson plans, and a computational physics57

textbook that explains the algorithms of the scientific simulations. Our tools are based58

on short, single-idea simulations with source code that can be looked at, changed, recom-59

piled, and shared freely. These simulations are used to teach important computer skills.60

All levels of students will benefit from these interactive simulations because they will learn61

to question and evaluate the simulation’s assumptions and results [14].62

63

Every technical task necessitates the use of the appropriate instrument. Easy Java64

Simulations is a Java authoring tool that was created exclusively for the construction of65

interactive simulations. Though it’s vital to distinguish between the finished output and66

the tool used to create it, theoretically, any existing programming language could be used67

to create the simulations we’ll be making with Ejs. This tool stems from the specific exper-68



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1740

tise accumulated over several years of experience in the creation of computer simulations69

and will thus be very useful to simplify our task, both technically and conceptually [16].70

71

The proper instrument is necessary for every technical task. Easy Java Simulations72

is an authoring tool created exclusively for developing Java-based interactive simulations.73

Although it’s important to distinguish between the final product and the tool used to make74

it, the simulations we’ll build with Ejs could theoretically also be constructed using any75

modern computer programming language. Consequently, this tool comes from the specific76

expertise developed over many years of experience in creating computer simulations. It77

will be of great assistance to simplify our task from both the technical and conceptual78

points of view [3].79

80

Using conformable fractional derivative, in this paper, we will solve some fractional81

differential equations numerically through Easy Java Simulations (EJSs) software.82

2. Fractional calculus and Fractional differential equations83

Calculus came before fractional derivatives. L’Hospital pondered the meaning of84

dnf
dxn if n = 1/2 in 1695. Since then, other academics have worked to clarify what a85

fractional derivative is. Different forms of fractional derivatives exist (for example, see86

[26, 27, 29, 31, 32]). Famous mathematicians including Riemann, Liouville, Letnikov,87

Sonin, Weyl, Riesz, Erdelyi, Kober, and others proposed fractional derivatives. In the88

natural sciences, processes and systems with spatial and temporal non-locality are often89

explained using fractional derivatives of non-integer orders (the non-locality in time is usu-90

ally called memory). In general, integro-differential operators are a subclass of fractional91

derivatives of non-integer orders.92

93

In recent years, fractional differential equations (FDEs) in the sense of Riemann-94

Liouville, Caputo, and Grunwald-Letnikov have played an essential role in modeling vari-95

ous real-world issues. FDEs have been used to describe real-world events in many fields,96

such as diffusion and dynamics in biology, fluid mechanics, fluid flow, signal processing,97

and others [15, 20, 22, 25, 30, 33]. An alternate formulation of the diffusion equation is98

presented to enhance anomalous diffusion modeling by employing a new derivative with99

fractional order known as the conformable derivative. The analytical solutions to the100

conformable derivative model are provided in terms of the Gauss kernel and the error101

function. The conformable diffusion model’s power law of the mean square displacement102

is examined using the time-dependent Gauss kernel. The Levenberg-Marquardt approach103

is used to calculate the parameters of the conformable derivative model from the experi-104

mental data of chloride ion transport in reinforced concrete. Fitting the data shows that105

the conformable derivative model and the experimental data match up better than the106

usual diffusion equation [36].107

108

Mathematical modeling is one of the viable ways of solving real-world issues. Modeling109



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1741

may be done in a variety of ways, including statistical approaches that can predict several110

occurrences. Health is now one of the most important fields of study in the world [17].111

Over the last three years, the world has been threatened by a new fatal disease known as112

COVID-19, caused mainly by the Coronavirus. This virus was discovered for the first time113

in Wuhan, China, and rapidly spread to the rest of the world. Numerous researchers have114

better developed mathematical models to comprehend the Coronavirus’s dynamics and115

intricacy. For the last three years, the globe has been threatened by a new lethal illness116

known as COVID-19, which is mostly caused by the Coronavirus. This virus was first117

detected in Wuhan, China, and quickly spread to the rest of the globe. Many researchers118

have produced mathematical models to better understand the dynamics and complexity of119

the Coronavirus [18, 19]. Using chaotic attractors, the conformable and fractal derivatives120

appear to better describe anomalous diffusion and even the flow of water inside a fracture121

system than the classical derivative [2].122

123

We provide a quick overview of the “local formulations” in this subsection. Given that124

they are, at most, a multiplicative factor of the derivative of order one, these formulations125

shouldn’t bear the moniker ”fractional,” according to a recent study [32]. Despite this,126

since 2010, similar plans, which first surfaced in the late 1990s of the 20th century, have127

become quite common. Chen introduced the local operator in 2006 in order to describe128

the turbulence [11] and anomalous diffusion [10] phenomena.129

130

In recent decades, fractional differential equations (FDEs) have become an important131

tool in mathematical modeling and studying the dynamics of many natural processes,132

such as viscoelastic materials, chaotic systems, optimal control problems, and financial133

markets. To solve these equations, several numerical and analytical methods are used.134

Among them, Chatibi et al. mention the continuous and discrete symmetry methods and135

efficient techniques to furnish various solutions for FDEs. The essential concept behind136

these approaches is the construction of transformations that preserve the form of the137

examined FDE [7]. The Lie group approach is used in [8] to derive the Lie symmetry138

algebra accepted by the time fractional Black-Scholes equation. The built symmetry139

generators are researched in order to build a family of precise solutions and conservation140

laws for the analyzed problem. Simultaneously, the family of solutions is expanded by using141

the invariant subspace approach. In [5], Chatibi et al. established the essential optimality142

requirements for variations problems of the Euler-Lagrange type, where the variational143

functional is dependent on the Atangana-Baleanu derivative. Examples are provided to144

show the outcomes that were produced. They provide a correct prolongation formula for145

conformable derivatives of the traditional prolongation formula of point transformations.146

The construction of a symmetry group accepted by conformable ordinary and partial147

differential equations uses this approach, which is shown. They also provide an exact148

solution to the conformable heat equation using Lie symmetry analysis [6]. The study149

[9] builds discrete symmetries for a variety of ordinary, partial, and fractional differential150

equations using the Hydon technique to find discrete symmetries for a differential equation.151

It is shown how to use those discrete symmetries to create new solutions out of existing152



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1742

ones.153

Fractional Taylor power series were recently introduced, and a nice theory was estab-154

lished. However, no work has been done on fractional Fourier series, while some work155

has been done on fractional Fourier transform. Conformable fractional Fourier series will156

be introduced in this study. For instance, we solve some fractional partial differential157

equations using fractional Fourier series [24, 28].158

The conformable fractional derivative, which is based on the derivative’s fundamental159

limit formulation in [21], is a novel fractional derivative that has just been presented.160

Following that, [1] creates fractional versions of chain laws, exponential functions, Gron-161

wall’s inequality, integration by parts, and Taylor power series expansions. In [34] the162

conformable fractional differential transform technique is described, and it is shown how163

it may be used to conformable fractional differential equations.164

A strong and efficient technique for modeling nonlinear systems is fractional calcu-165

lus. In order to explain the physical universe, Zhao developed a new class of fractional166

derivatives called general conformable fractional derivative (GCFD). From Khalil’s notion167

of conformable fractional derivative (CFD), the GCFD is generalized. We draw attention168

to the fact that the word ”t1” used in the definition of CFD is only a type of ”fractional169

conformable function” and is not necessary. We also provide physical and geometrical170

interpretations of GCFD, indicating possible engineering and physics applications. Since171

it is simple to show that CFD is a particular instance of GCFD, Zhao first discusses172

its physical and geometrical meanings [35]. Recently, the term conformable fractional173

derivative —a novel, straightforward, well-behaved formulation of the fractional derivative174

— was presented by the authors Khalil et al. [21].175

Definition 1. Given a function f : [0,∞] → R. Then, the “conformable fractional176

derivative” of f of order α is defined by177

f (α)(t) = lim
ε→0

f(t+ εt1−α)− f(t)

ε
(1)

for all t > 0, α ∈ (0, 1). If f is α-differentiable in some (0, a), a > 0, and limt→0+ f (α)(t)178

exists then define179

f (α)(0) = limt→0+ f (α)(t). It can be easily shown that f (α)(t) satisfies all the properties180

in the following theorem.181

Theorem 1. Let α ∈ (0, 1] and g, h be α-differentiable at a point t > 0 then.182

(i) f (α)(ag + bh) = af (α)g + bf (α)h, for all a, b ∈ R.183

(ii) f (α)(tp) = ptp−α for all p ∈ R.184

(iii) f (α)(λ) = 0, for all constant functions f(t) = λ.185

(iv) f (α)(gh) = gf (α)(h) + hf (α)(g).186



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1743

(v) f (α)( gh) =
gf (α)(h)−hf (α)(g)

h2 .187

(vi) If, in addition, f is differentiable, then f (α)(g(t)) = t1−α dg
dt (t).188

The conformable fractional derivative of some elementary functions is as follows.189

• f (α)(tp) = ptp−α for all p ∈ R.190

• f (α)(1) = 0.191

• f (α)(ecx) = cx1−αecx, c ∈ R.192

• f (α)(sin bx) = bx1−α cos bx, b ∈ R.193

• f (α)(cos bx) = −bx1−α sin bx, b ∈ R.194

• f (α)( 1α t
α)=1.195

One should notice that a function could be α-differentiable at a point but not differentiable.196

Example 1. Let g(t) = 2
√
t197

f ( 1
2
)(g)(0) = limt→0+ f ( 1

2
)(g)(t) = 1, where f ( 1

2
)(g)(t) = 1 for t > 0. However, f (1)(g)(0)198

does not exist.199

Using conformable fractional derivative, we will now solve some fractional differential200

equations numerically through Easy Java Simulations (EJSs) software.201

3. Solving conformable fractional differential equations by using EJS202

Easy Java simulation software can be used to visualize mathematical physics problems.
To utilize this program to solve the conformable fractional differential equation, we first
use the definition of the conformable fractional derivative.

To begin, we draw both the analytical and numerical solutions of the equation at the
same time to see if they coincide (Fig. 1, and Fig. 2 which is in fact the data table ). We
first solve the conformable fractional differential equations according to the definition.

y(
1
2
) + y = t2 + 2t

3
2 , y(0+) = 1. (2)

Using f (α)(t) = t1−α df
dt , and EJS software we solve this equation numerically through203

following steps (see Fig. 1), The index “h” refers to the solution of the homogeneous204

equation:205

• y
( 1
2
)

1h
+ y1h = 0, y

( 1
2
)

3 + y3 = t2 + 2t
3
2 .206

• y
(1)
2h

+ y2h = 0, y
(1)
4 + y4 = t2 + 2t

3
2 .207

Comparing the solutions of an ordinary differential equation with its fractional counterpart208

shows that from one point onwards, the fractional solution exceeds the ordinary solution209

(see both Fig.1).210



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1744

Figure 1: Two homogeneous solutions and two general solutions of ordinary and fractional differential equations.

4. Conformable sub-diffusion equation in EJS211

The standard diffusion equation established by Fick is written as

∂C(x, t)

∂t
= D

∂2C(x, t)

∂x2
, (3)

where C(x, t) is the concentration function, and units are mass percentage(%), kg/m3, or
mol/L. D stands for diffusivity or diffusion coefficient (m2/s). This model is ineffective in
characterizing the more intricate time-dependent diffusion processes, such as sub-diffusion.
Zhou et al. [36] suggest an effective model illustrating the time-dependent sub-diffusion to
get beyond this restriction. The conformable derivative model for sub-diffusion is derived
by considering one-dimensional diffusion.

∂αC(x, t)

∂tα
= Dα

∂2C(x, t)

∂x2
, (4)

where 0 < α ≤ 1, and Dα is the generalized diffusion coefficinet (m2/sα).
We may attempt to obtain an analytical solution to the given conformable diffusion

equation using the Gaussian kernel if we assume that the diffusion process happens in infi-
nite media and corresponds with the initial condition C(x, 0) = δ(x), where δ(x) indicates
the Dirac delta function. Applying the Fourier transform to both sides of the fractional
diffusion equation, Eq. (4), gives

∂αĈ(ξ, t)

∂tα
= Dα(2πiξ)

2Ĉ(ξ, t), (5)

or

f (α)(Ĉ(ξ, t)) = −4π2Dαξ
2Ĉ(ξ, t), (6)



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1745

Figure 2: From view elements in EJS software, we add “Data table” element to “tree element”

where the initial condition is Ĉ(ξ, 0) = 1. Using EJS, in this paper, we solve the following
equation numerically (see Fig. 3).

d

dt
Ĉ(ξ, t) + 4π2Dαξ

2tα−1Ĉ(ξ, t) = 0. (7)

It is possible to write the general solution of the ordinary differential equation, Eq. (7),
as

Ĉ(ξ, t) = c0 exp
−4Dαπ

2ξ2

α
tα. (8)

Changing the general solution to include the initial condition Ĉ(ξ, 0) = 1 on Eq. (8)
produces

Ĉ(ξ, t) = exp
−4Dαπ

2ξ2

α
tα. (9)

The analytical solution of the conformable diffusion equation, Eq. (5), is shown to be a
time-dependent Gaussian distribution resulting from the inverse Fourier transform. Ob-
viously, the conventional Gauss kernel model, which is the normal diffusion with α = 1,
and has an analytical solution, is an extension of the conformable derivative Gauss kernel
model that is being described. The analytical solution of the conformable diffusion equa-
tion, Eq. (9), is shown to be a time-dependent Gaussian distribution resulting from the
inverse Fourier transform.

C(x, t) =

√
α

4πDαtα
exp(− α

4Dαtα
x2). (10)



M. Noshad, A. Pishkoo, M. Darus / Eur. J. Pure Appl. Math, 15 (4) (2022), 1738-1749 1746

Figure 3: (a) Solving conformable differential equations Eq. (1.6) numerically (b) for different α = 1 (red), 0.9

(blue), 0.8 (green), 0.7 (yellow), 0.6 (black), and 0.5 (pink)-Cf ≡ Ĉ(ξ, t).

Figure 4: (a) Comparing diffusion process (blue curve for α = 1) with (b) sub-diffusion process (red curve for
α = 0.5).

Using a Gaussian of similar narrowness, the computation begins at t = 0.0001s to prevent212

the delta function’s singularity at time t = 0. It appears as a blue line before the com-213

mencement (see Fig. 4). After clicking the Run key, the maximum amplitude decreases214

(observe the shifting scale). Still, the width of the distribution rises proportionately.215

5. Conclusion216

The method used in this article differs from the other methods. At the same time as217

clicking the execute button to solve the differential equation numerically, the solution is218

also displayed as an animation diagram. In addition, it also provides the relevant data table219

in real time. With Easy Java simulations (EJSs) software, you may verify the analytical220

solution derived from another method with the numerical results of EJS software. This221

program has both educational and research purposes. Different algorithms for solving222

differential equations and adjusting the step size are available inside this application.223



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Acknowledgements224

The authors are sincerely grateful to the editor and anonymous referees for careful225

reading of the original manuscript and useful comments.226

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