=Paper=
{{Paper
|id=Vol-2879/paper27
|storemode=property
|title=The learning process simulation based on differential equations of fractional orders
|pdfUrl=https://ceur-ws.org/Vol-2879/paper27.pdf
|volume=Vol-2879
|authors=Oleksii P. Chornyi,Larysa V. Herasymenko,Victor V. Busher
}}
==The learning process simulation based on differential equations of fractional orders==
https://ceur-ws.org/Vol-2879/paper27.pdf
The learning process simulation based on differential
equations of fractional orders
Oleksii P. Chornyi1 , Larysa V. Herasymenko1 and Victor V. Busher2
1
Kremenchuk Mykhailo Ostrohradskyi National University, 20 Pershotravneva Str., Kremenchuk, 39600, Ukraine
2
National University "Odessa Maritime Academy", 8 Didrikhson, Odessa, 65029, Ukraine
Abstract
This article is an integrated study conducted to develop a learning model which would make it possible
to identify the students’ changes of knowledge, abilities and skills acquisition over time as well as the
formation of special features of their individual background. Authors have justified the application of
the cybernetic model based on fractional equations for the description and evaluation of the student’s
learning process. Learning is dealt as a transformation of young people’s knowledge, abilities and skills
into a complex background, which envisages its implementation in the future professional activity. The
advantage of the suggested model is better approximation characteristics which allow the consideration
of a wide range of factors affecting the learning process including the youth’s neurodynamic and psycho-
logical nature. The research has employed both mathematical modeling methods and psychodiagnostic
techniques (surveys, questionnaires). As a result of the findings, students who assimilate the content of
teaching information and form personal experience in different ways have compiled different groups;
the learning curve constructed on the basis of the heterogeneous differential equation of second order
with integer powers has been compared with the set of models with equations of fractional order of
aperiodic and fractional power components. The prospect of the issue to explore is the improvement of
the suggested model considering special characteristics of cognitive processes aimed at the formation of
an individual path of the student’s learning.
Keywords
learning process, learning simulation, cybernetic model, differential equations of fractional order, teaching
It will lead to a paradox, from which one day many useful consequences will be drawn.
– Leibniz on fractional derivatives in his letter to l’Hospital, September 30, 1695
1. Introduction
Changes in higher education in recent years focus on providing the system of high quality
training of stress resistant and creative specialists who acquire a complex of competencies and
social skills and are able to respond quickly to modern social and economic challenges. The
research will make it possible to track online the changes over time of knowledge, abilities and
CTE 2020: 8th Workshop on Cloud Technologies in Education, December 18, 2020, Kryvyi Rih, Ukraine
" alekseii.chornyi@gmail.com (O. P. Chornyi); gerasimenko24@gmail.com (L. V. Herasymenko);
victor.v.bousher@gmail.com (V. V. Busher)
~ http://www.kdu.edu.ua/PUBL/chorniyap.php (O. P. Chornyi)
0000-0001-8270-3284 (O. P. Chornyi); 0000-0003-3725-8681 (L. V. Herasymenko); 0000-0002-3268-7519
(V. V. Busher)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
473
skills acquisition as well as to reveal and suggest the best way to high quality education of
future professionals.
Modern information systems provide a means of development of models which allow us to
monitor the learning process, detect upturns and downturns of the learning activeness; all this
eventually affects the final result – knowledge, abilities and skills acquisition [1]. The purpose
of the research is to generate and describe a cybernetic model of learning which is based on
differential equations of fractional order.
Dmitriy A. Novikov has described the facilities of the learning model. Investigating the
iterative learning, which is the simplest type, the researcher has proved that the quantification
of the learning process can be demonstrated through curves and graphs if external impacts are
permanent. The result is meant to be a level of learning which might be measured by time,
speed and informational criteria and accuracy of the assignments [2].
The search for means of modelling is presented in works by Vsevolod V. Vasilyev and Lilija
A. Symak [3], Yong Zhou, Clara Ionescu and J. A. Tenreiro Machado [4], Devendra Kumar and
Dumitru Baleanu [5]. These researchers made a comparative analysis of fractional calculus and
classical mathematical analysis and revealed the possibility to use fractional calculus in different
scientific spheres as a means for description and modelling of the systems which change over
time. The advantage of this approach lies in the flexible transformation from one type of the
equation to another, change of the task’s physical parameters and the type of initial and final
conditions. However, the researchers think that the usage of these equations might be restricted
by the complication of the models having two independent variables.
There is an idea grounded in current studies that the obtained equations could be correlated
with the models of a blood system [6] and distribution of neural patterns’ signals in the nervous
tissues of biological objects [7, 8]. Thus, this research has used differential equations of fractional
order which consider the learning subjects’ nature in the most efficient way and allow us to
build an appropriate model that would disclose the change of knowledge and skills formation
over time being the basis of the gained experience.
2. Synthesis of the learning process simulation based on
differential equations of fractional orders
Having analysed the survey of 130 students of the Institute of Electromechanics, Energy Saving,
and Automatic Control Systems of Kremenchuk Mykhailo Ostrohradskyi National University it
has become evident that a large number of boys and girls are incapable of their own learning
strategy development, they struggle to master teaching information and decide on the best
learning method, etc. Tutorials as a means of individualising of learning do not contribute to
the task solution. For instance, it has been found out that 30% of students are hesitant about
attending tutorials as they think they are a waste of time, 52.3% are confident they can cope
with the problem on their own, 17.7% have troubles comprehending the subject’s content and
fail to formulate questions.
A similar situation is with the results of students’ independent work. Despite the individual
approach to the development of tasks for independent work (entry testing, potential mathe-
matical expertise evaluation, general and specific skills assessment) it should be acknowledged
474
that their accomplishment is merely pro forma for obtaining a mark and getting the ratings up,
which do not contribute to the main goal achievement – readiness for working for an enterprise,
comprehension of operating processes and making managerial decisions. In order to actualise
the need, virtual laboratory complexes (VLC) have been evolved.
They allow us to use a variety of virtual devices, measurement systems and software and hard-
ware complexes created with the help of diverse software tools of powerful modern computer
equipment. These systems are flexible and adaptable to a number of tasks, for example, they
facilitate the formation of engineering competencies trough planning skills training and con-
ducting engineering experiments with further analysis of their results. Furthermore, VLC can
be exploited as simulators that, due to their mathematical model, assist in studying properties
of electrical objects, as software tools to simulate and study the system modes (usual conditions,
pre-emergency, emergency), and as hardware and software tools for computer-assisted research
[1]. Moreover, VLC serve to record studying results and construct curves of learning.
Using the method of mathematical modelling we attempted to develop a model which would
clearly describe the pilot testing results and make it possible to reveal both specific features of
teaching information acquisition and formation of students’ individual background. Since the
obtained data form a curve similar to an exponent, the authors have chosen a cybernetic model
for the approximation. It is worth noting that starting from H. Ebbinghaus onwards many
researchers point at the exponential nature of memorisation and forgetting [4]. The exploration
of recent scientific papers [3, 4, 8, 9] places on record an appreciable quantity of suggested
cybernetic models with the second-order equation which would describe any process [2].
The suggested model of the rate of information flow assimilation expressed by a second-order
heterogeneous differential equation is an example of cybernetic approach:
𝑑2 𝑆 𝑑𝑆
𝑚 2
+𝑟 + (𝛼 − 𝑐) 𝑆 = 𝐻, (1)
𝑑𝑡 𝑑𝑡
where 𝑆 is the flow of digestible information as a function of time, 𝑡; 𝑟 – coefficient of resistance
to the learning process; 𝛼; 𝑐 – coefficients of forgetting and inference; 𝐻 – the flow of initial
information as a function of time, 𝑡; 𝑚 – inertia value.
In [10] the average values are indicated: coefficient of resistance to the process, 𝑟 = 0.5;
coefficient of forgetting 𝛼 = 0.3; inference coefficient 𝑐 = 0.25; inertia value 𝑚 = 0.65.
To calculate the coefficients of differential equations of the information assimilation (1) there
has been conducted a pilot testing among the third-year students of Kremenchuk Mykhailo
Ostrohradskyi National University during studying the discipline “Theory of Electric Drive”.
One hundred and thirty eight students who participated in the experiment took a test containing
20 multiple-choice questions with 5-6 options after each lecture. In total they had 14 tests. The
average values of students’ academic performance according to testing are presented in figure 1.
To process the data, a shift along ordinate axis has been performed so that the “transition
process” begins at zero initial conditions (Δ0 = 0.341). The mean-square error of the ap-
proximation is 𝑆𝑡𝑑.𝑒𝑟𝑟. = 0.0306597, the steady-state value is 0.434615. The final level of
knowledge corresponds to the value 𝑄 = 𝑘 + Δ0 = 0.776.
In modern science the number of applications of fractional calculus in various fields of science
and technology that use mathematical methods and computer simulation tools is increasing
rapidly.
475
Figure 1: Experimental data and their approximation by a second-order differential equation (1).
For more than three centuries specialists in theoretical mathematics have had no doubts
about existence of fractional derivatives. At the same time experts dealing with practical issues
have been looking for physical meaning of fractional derivatives.
Nevertheless, the following example clearly demonstrates the understanding of the introduc-
tion of fractional orders in degrees of equations. Svante Westerlund has proposed a generalisa-
tion of Newton’s second law and demonstrated that Hooke’s law in elasticity theory (𝐹 = 𝑘𝑥),
Newtonian fluid model (𝐹 = 𝑘𝑥′ ) and Newton’s second law (𝐹 = 𝑘𝑥′′ ) can be regarded as the
isolated cases of more general relation of the type: 𝐹 = 𝑘𝑥(𝜇) , where the order of derivative
𝜇 can be any real number. Of course, this generalisation cannot be called a conclusion; most
likely it is an interpolation between models of processes which are described by whole-order
derivatives.
From this point of view let us also consider the learning process the results of which have
been demonstrated above. Analysis of the curve construction (figure 1) on the basis of model
(1) does not allow us to assert unequivocally that the best solution has been found. Form the
mathematical viewpoint, the algorithm based on the minimum mean-square errors method has
provided the best possible approximation to the experimental data. Taking into account that
the learning process is connected with the information assimilation, short-term and long-term
memory, forgetting and inference, the nature of the model, the learning curve may also differ
from the classical ones [9]. Starting from the first classical works William Love Bryan and Noble
Harter [11] and Edward C. Tolman [12, 13] which studied the learning curve onwards, it is
noted that in the beginning the learning process goes quite fast and then starts to slow down
[4, 9]. The resulting dependence resembles a curve corresponding to an aperiodic first-order
unit or to series connection of two first-order units. Deviations from the experimental curve
were attributed to dissimilarity of the students’ physiological parameters, mutual influence in
the group, and individual abilities to remember and restore information which are stipulated by
the nervous system features. More importantly, scholars in the field of pedagogy did not have
mathematical methods to use fractional degree equations. The curve obtained from equation
476
Figure 2: Experimental data and their approximation by model (2): a) initial data and approximating
function changing over time; b) iterative process of finding a solution.
(1) will be approximately the same for all groups of students, which makes differentiation
by psychophysiological properties impossible. However, the works [3, 4, 9, 14] demonstrate
the possibility to use fractional calculus in various fields of science and technology and even
to describe both the processes of information assimilation and mastery and the process of
decision-making in a group. Scientists prove conclusively that the models and simulations of
the processes changing over time, which the processes of information accumulation are, go
beyond the framework of equations with derivatives of integer order.
Let us explore the models based on the fractional degree differential equations according to
the learning results.
To process the data at a constant step of one day a piecewise linear interpolation between
the known modules has been performed. Consider several models whose structure is similar.
Approximation by a fractional aperiodic unit:
𝑘
𝐻 (𝑝) = . (2)
𝑎0 𝑝𝜇 + 1
(1/𝜇)
Parameter 𝑎0 = 17.672 corresponds to the physical time constant being measured in
days. The graphs on the figure 2 show the initial data and the approximating function changing
over time; the graphs on the right display the changes in the mean square error and the order
of fractional component in the iterative search for a solution. It confirms the convergence of
this process. Also, the attention should be paid to the best agreement between the calculated
and experimental graphs in the middle and final parts (𝑡 > 20).
We also perform the approximation by the fractional aperiodic unit of order 1 + 𝜇 (figure 3)
that can be expressed in two forms:
𝑘 𝑘
𝐻 (𝑝) = = . (3)
𝑎1 𝑝𝜇+1 + 𝑎0 𝑝𝜇 + 1 𝑇0𝜇 𝑝𝜇 (𝑇1 𝑝 + 1) + 1
Physically, this transfer function is the inertial and fractional-integrating units with negative
(1/𝜇)
feedback. Parameter 𝑎0 = 𝑇0 = 21.78 corresponds to the physical time constant. If
𝑎1 = 𝑇1 𝑇0 than 𝑇1 = 0.8723 days.
477
Figure 3: Experimental data and their approximation by model (3): a) approximating function changing
over time; b) iterative process of finding a solution.
Figure 4: Experimental data and their approximation by model (4): a) approximating function changing
over time; b) iterative process of finding a solution.
Approximation by inertial and fractional aperiodic units (figure 4):
𝑘
𝐻 (𝑝) = . (4)
(𝑇 𝑝 + 1) (𝑎0 𝑝𝜇 + 1)
(1/𝜇)
Parameter 𝑎0 = 𝑇0 = 21.09 corresponds to the time constant which together with the
order of fractional aperiodic component corresponds to the physical time constant and is as
close as possible to the level gained previously. Moreover, it is clearly seen that 𝑇0 is close to
the break point between the initial and final stages.
Finally we perform the approximation by the unit (figure 5):
𝑘
𝐻 (𝑝) = . (5)
(𝑇 𝑝 + 1) (𝑎1 𝑝𝜇+1 + 𝑎0 𝑝𝜇 + 1)
Although this function is the most complex, its similitude to the almost the same solutions
at repeated starts, as in the previous cases, is worth noting. Here the inertial components
478
Figure 5: Experimental data and their approximation by model (5): a) initial data and approximating
function changing over time; b) iterative process of finding a solution.
(1/𝜇)
𝑇 = 1.865...1.971 and 𝑎0 = 𝑇0 = 28.14...29.46 correspond to the physical time constant
which is close to the values obtained in the previous case. Therewith, the additional time
constant is calculated: 𝑇1 = 0.3826...0.4027.
Summarized data of the models are given in table 1.
The obtained results are quite significant. Firstly, the accuracy of the approximation compar-
ing with the classical model (1) has increased almost by 10%. Secondly, the fractional order in
the transfer functions of models (2) – (4) is close to 0.5 which coincides with the description of
diffusion processes, turbulent and laminar flows, penetration of nonviscous liquids into porous
media, which also allows us to illustrate a concept by an analogy with the signal transmission
from axons to dendrites [2], [15].
3. Analisys of the personal data of students
To study the learning of individual students we adopt model (4) which is based on inertial and
fractional aperiodic components because it contains only two time constants and one exponent,
so, taking into account modern ideas about the learning process, it will be possible to give
physical meaning to these parameters.
In further studies the comparison of models’ parameters with information transfer processes
in the brain will allow the usage of more complex models, for example (5).
Analysing the obtained data, we can specify the elements which are characteristic of the
identified groups of students.
I. Students who quickly reached the maximum level of assimilation of information. The
speed of the content mastery and the achievement of high and medium levels are characteristic
of students of a strong type of nervous system which determines the speed of all cognitive
processes and flexibility. They work hard for a long period of time, respond flexibly to questions
of varying complexity, are able to use the gained knowledge in new situations, can easily update
the necessary information, correlate it with the new; compare, draw analogies, align with their
own experience, etc. Furthermore, their resistance allows them to confront negative phenomena
479
Table 1
Model summary
(1/𝜇)
Model 𝑇0 = 𝑎0 , 𝑇, 𝑇1 , days Graphics
days
𝐻 (𝑝) = 𝑎0 𝑝𝑘𝜇 +1 17.67
𝑘
𝐻 (𝑝) = 𝑎1 𝑝𝜇+1 +𝑎 𝜇
0 𝑝 +1
= 𝑇 𝜇 𝑝𝜇 (𝑇𝑘1 𝑝+1)+1 21.78 0.87
0
𝑘
𝐻 (𝑝) = (𝑇 𝑝+1)(𝑎 𝜇
0 𝑝 +1)
21.09 2.73
𝑘
𝐻 (𝑝) = (𝑇 𝑝+1)(𝑎1 𝑝𝜇+1 +𝑎0 𝑝𝜇 +1) 28.80 𝑇 = 1.92,
𝑇1 = 0.39
which arise during the learning activities (influence of classmates and the teacher, adverse
conditions of the organization of training, etc.).
Boys and girls fearlessly take progress tests and work efficiently under conditions of limited
time. High level of motivation, awareness of the value of the future profession and the necessity
of gradual development as well as intellectual and special aptitudes allow them to adsorb
teaching information quickly and effectively. Typical diagram for this group is presented on
figure 6 and numerical values of their parameters are 𝑘 = 0.3642, 𝑇 = 4.2012 , 𝑎0 = 3.086 ,
𝑇0 = 5.597 , 𝜇 = 0.6543 , Δ0 = 0.400.
II. Students who quickly reached the intermediate level and then slowly improved it. The
second group includes students with an inert type of nervous system; however, they are
motivated to study and conscious about the value and necessity of their own development
and mastery of the future profession. The same scheme of learning is true for boys and girls
480
Figure 6: Data of Student No 8.
Figure 7: Data of Student No 6.
with a strong nervous system yet lacking motivation, possessing weak intellectual and special
(professional) aptitudes (figure 7). Their parameters are 𝑘 = 0.9735, 𝑇 = 0.8759, 𝑎0 = 8.523,
𝑇0 = 116.6, 𝜇 = 0.4503, Δ0 = 0.250.
III. Students who slowly master the teaching material are of weak and inert types of nervous
system. They are fatigued by continuous, hard and responsible work. Even if the classroom
climate is favourable and the teacher is reserved and sensible, unexpected questions and false
answers significantly affect the final result of the content acquisition. In addition, progress
tests and work under conditions of limited time inhibit the progress significantly. Although the
students are well motivated and conscious about such values as development, freedom, etc., the
slowness of the cognitive processes exerts an impact.
Boys and girls with an inert type of nervous system with parametes 𝑘 = 0.3855, 𝑇 = 5.937,
𝑎0 = 12.263, 𝑇0 = 21.24, 𝜇 = 0.8202, Δ0 = 0.350 and typical diagram on figure 8 can more
481
Figure 8: Data of Student No 12.
easily withstand the difficult conditions of persistent and responsible work; nevertheless, a lack
of preliminary training and motivation, low level of awareness of values and imperfection of
special aptitudes determine the special features of the learning process which are characteristic
of the students of Group III.
Thus, the neurodynamic and psychological characteristics of a personality, which impact the
effectiveness of learning, can be taken into account by the cybernetic model using equations of
fractional order.
4. Conclusions
The developed cybernetic model of learning based on differential equations of second order with
fractional degrees allows us to describe the process of learning more accurately, quickly responds
to the changes in students’ acquisition of the information, increases the efficiency of the content
assimilation, enables us to guide the formation of an individual path of the student’s development
through the improvement and optimization of the class schedule, teaching methods and the
system of control measures. The proposed simulation of the learning process facilitates in flexible
and timely adjustments to the teaching process; the decay in knowledge and skills acquisition
can be altered by a methodically correct presentation with due regard to the neurodynamic and
psychological traits of students. So, to work with students of the second group it is necessary to
apply methods of motivational and value based stimulation emphasizing the practical value of
the information received, outlining the ways of its practical application, analyzing examples from
real enterprises, investigating the causes of possible pre-emergency and emergency situations.
The diversification of control methods, questioning aimed at stress reduction and creation of a
cooperative and emotionally favorable climate contribute to the effectiveness of learning. Work
with students of Group III requires additional diagnostic assessment of the level of training,
formation of general and special skills, motivational in order to decide on the methods of
learning activation.
482
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