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THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES
УДК 519.6:517.51
DOI: 10.25559/SITITO.16.202001.33-40
On Conservative Averaging Method in Spline Applications
H. Kalisa*, I. Kangrob
a University of Latvia, Riga, Latvia 19 Raina bulv, Riga LV-1586, Latvia * harijs.kalis@lu.lv
b Rezekne Academy of Technologies, Rezekne, Latvia 115 Atbrivosanas aleja, LV-4601, Rezekne, Latvia
We consider the conservative averaging method for solving the 3-D boundary-value problem of second order in multilayer domain. Looking back to the history of mathematics, integral parabolic splines relates to conservative averaging method (CAM) introduced by A. Kneser in 1914. In 1980's, A. Buikis had developed CAM method for partial differential equations with discontinuous coefficients, when he was modelling processes in environments with a layered structure. The special hyperbolic and exponential type splines, with middle integral values of piecewise smooth function interpolation, are considered. Using these type splines, the problems of mathematical physics in 3-D with piecewise coefficients are reduced to 2-D problems with respect to one coordinate. This procedure also allows reducing the 2-D problems to 1-D problems and the solution of the approximated problems can be obtained analytically. In the case of constant piecewise coefficients, we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem. Similarly, the approximation of the 3-D nonstationary problem is obtained with CAM. The numerical solution is compared with the analytical solution.
Keywords: special splines, averaging method, 3D problem, analytical solution.
For citation: Kalis H., Kangro I. On Conservative Averaging Method in Spline Applications. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2020; 16(1):33-40. DOI: https://doi.org/10.25559/SITITO.16.202001.33-40
Abstract
Контент доступен под лицензией Creative Commons Attribution 4.0 License. The content is available under Creative Commons Attribution 4.0 License.
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0 консервативном методе усреднения в сплайновых приложениях
Х. Калис1*, И. Кангро2
1 Латвийский университет, г. Рига, Латвия 1586, Латвия, г. Рига, Бульвар Райниса, д. 19 * harijs.kalis@lu.lv
2 Резекненская академия технологий, г. Резекне, Латвия 4601, Латвия, г. Резекне, аллея Освобождения, д. 115
Аннотация
В статье рассматривается консервативный метод усреднения для решения трехмерной краевой задачи второго порядка в многослойной области. Оглядываясь назад на историю математики, мы видим, что интегральные параболические сплайны относятся к консервативному методу усреднения (CAM), введенному А. Кнезером в 1914 году. В 1980-х годах А. Буйкис разработал метод CAM для уравнений в частных производных с разрывными коэффициентами, когда он моделировал процессы в средах со слоистой структурой. Рассматриваются специальные сплайны гиперболического и экспоненциального типов со средними интегральными значениями интерполяции кусочно-гладкой функции. Используя сплайны такого типа, задачи математической физики в трехмерном пространстве с кусочными коэффициентами сводятся к двумерным задачам относительно одной координаты. Эта процедура также позволяет свести двумерные задачи к одномерным задачам, и решение аппроксимированных задач может быть получено аналитически. В случае постоянных кусочных коэффициентов мы получаем точную дискретную аппроксимацию стационарной 1-й краевой задачи. Аналогично аппроксимация трехмерной нестационарной задачи получается с помощью CAM. Численное решение сравнивается с аналитическим решением.
Ключевые слова: специальные сплайны, метод усреднения, трехмерная задача, аналитическое решение.
Для цитирования: Калис, Х. О консервативном методе усреднения в сплайновых приложениях / Х. Калис, И. Кангро. - DOI 10.25559/SITITO.16.202001.33-40 // Современные информационные технологии и ИТ-образование. - 2020. - Т. 16, № 1. - С. 33-40.
Современные информационные технологии и ИТ-образование
Том 16, № 1. 2020 ISSN 2411-1473 sitito.cs.msu.ru
H. Kalis,
I. Kangro
THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES
Introduction
The paper is devoted to the memory of Latvian mathematician E. Grinbergs (1911-1982), pointed out his outstanding achievements in applied mathematics, namely: radio filters, hulls of tankers, graphs theory and integral circuits, especially his work in the construction of the tanker fleet of the USSR [1]. According to his calculations of tanker hulls, in the period 1963-1970, 23 veru large tankers were built in Leningrad. At that time the research was secret, and it was not reported in the open sources. A summary of the calculation methodology (without any reference to the application objects) contains the article [2]. Today this interpolation methodology relates to the mathematical methods of splins.
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multidimensional. A spline function is a piecewise polynomial function. The places where the pieces meet are named knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the knots. Integral parabolic splines (IPS) for the first time were defined by A.Buikis in [3] and developed in his doctoral dissertation [4]. Forty years ago, it was important to construct mathematical models for intensification of the crude oil or gas output. For the situation with layered media, integral splines were introduced to consider the energy or mass conservation in new simplified (less dimensional) formulation of the problem.
Looking back to the history of mathematics, integral parabolic splines relates to conservative averaging method (CAM) introduced in 1914 by A.Kneser [5]. In 1980's, A. Buikis had developed CAM method for partial differential equations with discontinuous coefficients, when he was modelling processes in environments with a layered structure. The conservative averaging method was developed as approximate analytical and numerical method for solving partial differential equations (PDE) with piecewise continuous coefficients. To apply this method for the layered media, a special type of quadratic spline was developed, namely: the integral averaged values interpolating quadratic spline.
Later, the concept of integral spline was generalized by defining other types of splines besides polynomials, namely: hyperbolic and exponential type splines [6] considered below in this paper.
Integral quadric spline
Given a continuous, smooth piece function U(x), x Є [a, b]. It is assumed that the first derivatives at the inner points ti of U(x) have a final jump: K-i U'(ti-0) = Ki U'(ti+0), i=1,...N, where Ki are given positive coefficients. The continuity executes the equations: U(ti-0) = U(ti+0), i=1,...N. The mean integral values for U(x) of Ui by subsegment [ti, ti+i] are given
Ui=—f 1+1 U(x)dx, Ht = £;+]_ - tt, i=0...N, t0=a, tN+i=b.
Щ Ч
It is necessary to approximate the function U(x) using the previous conditions and the following general boundary conditions (BC) at the interval endpoints x=a and x=b:
K0U'(a)- a(U(a)-F0)=0,
KNU’(b)+ β(υ^ι)=0,
where α, β are positive coefficients, Fi, F2 - given constants. Such BCs are typical one for the ordinary and partial differential
equations (the third kind or so called Robin BCs; the second kind or so called Neumann BCs (if α=β =0); the first kind, or Dirichlet BCs (if α=β =ro) and periodical BCs in case: U(a)=U(b), U'(a)=U'(b)). The proof is given that this interpolation problem can be solved by a second order interpolation polynomial spline, in other words, by the integral quadric spline ofthe form:
S(x) = ui + mi (x-tj + ei Gi -11) ,
where tl=(ti+ti+1)/2, Gi=Hi/Ki>0.
From the previous conditions the 2(1+ N) unknown coefficients mi, ei can be determined. The mean integral values of ui can be also determined by averaging the differential equations.
Hyperbolic and exponential type splines for 1-D stationary problem in N layered domains
In [6], we consider averaging methods for solving the 3-D boundary value problem in domain containing peat blocks. We consider the metal concentration in the peat block. A specific feature of these problems is that it is necessary to solve the 3-D boundary value problems for elliptic type partial differential equation of the second order (the diffusion equation) with piecewise diffusion coefficients in every direction. The special hyperbolic and exponential type splines, with interpolation of middle integral values by piecewise smooth function, are defined. This procedure allows us to transform the 3-D problem to 2-D and 1-D problems and the solution of the approximated problem is obtained analytically [7]. The numerical solution is compared below with the analytical solution.
Firstly, we consider the 1-D diffusion and diffusion-convections boundary value problems in N layered domains.
Example 1. We considered the following 1-D diffusion problem in [0, L] for N layers:
1) differential equations:
э2 2 ----
D^~i ui (z) — aiui (z) + Pi = 0, zЄ [zi_1, zt], i=1, N, z0 =0, zn = L,
2) third kind BCs for z=0:
ВіЭцэі(0) - a (ui(0)-U0)=0, and for z=L: Dn 9u"(L) +β (un(L) - ul)=0,
3) continuous conditions:
Ui(zi)=Ui+i(zi), DÎ-ψ^ = Di+iaUitl(Zi), i=i.N—1,
9z dz
where Di are constant diffusion coefficients, α, β - mass transfer coefficients, ai, Fi, U0, ul - fixed constant values.
For this problem we can obtain the analytical solution in the following form:
Ui(z)=mi sinh(ai*(z-zi*)) +ei cosh(ai*(z-zi*))+Fi/ai2,
zi*=(zi-i+zi)/2, a*= ·η= .
V
From BCs and the continuous conditions, we obtain N linear algebraic equations with tridiagonal matrix for determination of the unknown coefficients ei and then mi, using integral hyperbolic type splines:
Ui(z)= Uiz +mi
0.5 Hfsinh(a— (z-Zf *))
A=
sinh(0.5aiiHi)
0.5ajiHj
sinh(0. 5 afiHf}
+ Siz
cosh(afi (z-Zf t))-Af 8 sinh2(0.25
(1)
where Uiz=—fZl ut(z)dz are the mean integral values of u(z),
Hi=zi- zi-i, aii= at* = -?=.
V^t
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Similarly, from BCs and the continuous conditions we can determine the unknown coefficients etz and mtz depending on the mean integral values utz. By using the mean integral values of differential equation for determination of utz, we obtain linear algebraic equation in following form:
0.5 Dtetzaticoth(0.25 atiHt)/Ht-at2utz+Ft=0.
In limit case when the parameters ati for the hyperbolic spline (1) tends to zero, we have the integral parabolic spline, obtained from A. Buikis [3], [4]. We proved that the hyperbolic type splines give exact solution for the previous boundary-value problem with mt=0.5mtzHt/stnh(0.5 atiHt),
et=0.125etz/stnh2 (0.25 atiHt),
From BCs at z=0 and z=L and the continuous conditions:
П ut(zt)=rt+i ut+i(zt),
we obtain 2N linear algebraic equations for determination of the unknown coefficients et and Ct.
By using the integral exponential type splines, we get: ut(z)= utz +mtz(z - z*) + etz(exp (ai1(z - z*)) - qt), (4)
where qi = sinh-*2^1 , mtz=-Ft/rt.
j 2
Similarly, from BCs at z=0 and z=L and the continuous conditions, we determine the unknown coefficients etz and utz. We can see that using the mean integral values of differential equation we obtain the following identity:
2etzati stnh(0.5 atiHt)/Ht - ati (mtz Ht +etz 2etz stnh(0.5 atiHt) ) /Ht+ Ft /Dt =0.
We can see, that this exponential type splines gives exact solution with mt=mtz, et=etz and Ct =utz - etz qt.
Example 2. We consider following equations for 1-D diffusion -convection problem in [0,L] for N layers with continuous conditions and other BCs:
Dtr^ui(z) + ri J-Zui(z) - а2Щ(z) + Ft =0, ze [zl-1, Z;], t=1, N, ze =0, zn=L,
where ri is a constant convection velocity in every layer. For this problem we can obtain the analytical solution in the following form:
ut(z)=mt exp (at- (z-zt*)) + exp (a+ (z-zt*))+Ft/a2, zt*=(zt-i+zt)/2,
From BCs at z=0 and z=L and the continuous conditions:
rt ut(zt)=rt+i ut+i(zt), Dtau‘(Zl) = Di+idUl+1l'Zl) , t=i,..,N-1,
dz dz
we obtain 2N linear algebraic equations to determine the unknown coefficients et and mt.
Using integral exponential type splines, we get solution: ut(z)= utz +mtz (exp (at-(z - z*)) - q—) + etz ( exp (α^+(ζ-z*)) - Чі+і (2)
where qt- = —2—sinha'~Hl, qt+ = —2—sinha'+Hl .
Similarly, from BCs at z=0 and z=L and the continuous conditions, we determine the unknown coefficients etz and mtz depending on the mean integral values utz. By using the mean integral values of differential equation for determination of utz, we obtain a linear algebraic equation in the following form:
2mtzstnh(0.5 at-Ht)(Dtat-+rt) + 2etzstnh(0.5 a+Ht)(Dtat+ +rt) - Ht a,2utz+Ht Ft =0.
Thus, we can see that the exponential type splines give the exact solution with
mt=mtz, et=etz and Ft/a,2=utz - etz qt+ - mtz qt-.
Example 3. We considered following equations for simple (at=0) 1-D diffusion-convection problem in [0, L] for N layers with continuous conditions and BCs:
D^ui(z) + П J-Zui(z) + Fi = 0, ze [z-ι, zi],
t=1, FI, ze =0, zn=L, (3)
where rt Ф 0 is a constant convection velocity in every layer. For
this problem we can obtain the analytical solution in following
form:
ut(z)=mt (z-zt*) +et exp (ati (z-zt*))+Ct, ati=-rt/Dt, mt=-Ft/rt, zt*=(zt-i+zt)/2.
A particular case. For the given parameters: N=L=2, Hi=H2=i, ri=i, П2=2, Fi=i, F2=10, α=100, β=1, ue=0, ul=1, Di=0.0i, D2=0.0i we have discontinuous solutions: ui(1)=12.60, u2(1)=6.30, uiz=11.75, u2z=3.99 (Fig.1.)
If ri= П2=2 and Di=0.01, D2=0.02 then we have continuous solutions and discontinuous derivatives, having the numerical values
F i g. 2. The solutions of two layers for ri=2, Г2=2, Dı=0.01, 02=0.02
Современные информационные технологии и ИТ-образование
Том 16, № 1. 2020 ISSN 2411-1473 sitito.cs.msu.ru
H. Kalis,
I. Kangro
THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES
Example 4. Using the transformation Ui(z)= exp(-r*(z-Zi·) Vi(z) for equation (3) in every layer we obtain the following problem:
(z) - rf.V;(z) + Fi,exp(ri,(z — zi,) = 0, (5)
zE [zi-1, zi ], i=1,N, zo =0, zn=L
D^gf* _ (a + rı*)vı(0) + a uo=0, (BCs for z=0),
Dn dVg(L'> + (β-rN*) vn(L) - β u l exp(rN*L) =0, (BCs for z=L), ri vi(Hi)= ri+1vi+1 (Hi) exp((ri* -ri+i* )Hi) ,
i=1,...,N-1, (continuous conditions),
Ft, = Ft /Dt, n. = rt/(2 Di) .
For this problem, we can obtain the analytical solution in the following form:
vt(z) = mt sinh(a\ (z — zt,)) + eicosh{al (z — zt,)) + gt(z), zt, = (г— + )/2, al = rt,·
where gt(z) = Fit exp(rt,(z — zit)/(4r?)(1 — 2rlt.(z — zt,)) is the particular solution of (5).
From BCs and the continuous conditions, one can obtain N linear algebraic equations for the determination of unknown coefficients ei and mi, using integral hyperbolic type splines (1) in the following form:
л sinh(0.5ai1Hi)
Ai------------,
Q.5ai1Hf
where Viz-—JZl vt(z)dz are the mean integral values of Vi(z),
zi-i
ац = Ц,
Similarly, from BCs and the continuous conditions we can determine the unknown coefficients eiz and miz depending on the mean integral values Viz. By use of the mean integral values of differential equation, we obtain N linear algebraic equations to determine Viz:
0.5 eizüiicoth(0.25 anHi)/Hi -ri2,Viz+2Fi, sinh(0.5 r* Hi)/ (г^Щ ) -0. We can see, that the hyperbolic type splines give exact solution for the previous boundary-value problem in case of Fi-0, then mi-0.5mizHi/sinh(0.5 aiHi), ei-0.125eiz/sinh2 (0.25 aiHi), Viz - eiAi .
In case of Fi nonzeros andg iz-Viz - eiAi, we obtain the approximate solution
g iz- -7§z' gt(z)dz=Fi*/(4Hiri*2)((2/n*-2Hi)sinh(0.5 rH)
Hi zi-i
+4cosh(0.5 n*Hi))
A particular case. For N-2, Lz-3, Hi-1.8, H2-I.2, ri-0.4, Г2-0.1, Fi-0, α=100,β=1000, U0-1,ul-10, D-0.2, D2-0.1 we have discontinuous solutions U1(H1) -U2(HJ- -7.413, V1H1) -V2(HJ--9.362, V1z-6.347, V2z-33.560 (Figs 3 and 4).
If Г1- Г2-0.1 and D1-0.5, D2-0.1 then we have the following u-continuous solutions, v-discontinuous solutions and discontinuous u-derivatives:
U1H1 ) -U2(H1)-0, V1H1 ) -V2H1)- -5.605, V1z-3.150, V2z-27.036 (Fig.5.)
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HyjwEet^b» olic typ e s]f>lin es for 1-D nonstationary problem in N layered domains
Example 5. We consider the following parabolic type PDEs 1-D initial-value problem in IN layers with continuous conditions and BCs in z-d irection:
where ze [z;_г, z;], t Є [0, tf], i=1, N, zo =0, zn=L; r, i 0 is a
constant convection velocity in every layer, Ui(z,0)=uoo are the initial conditions. By using transformation u(z,t) = exp(-r*(z-z*) Vi(z,t) for equation (6) in every layer we obtain following PDEs:
ze [zi_v zt11 Є [0, tf ].
Using BCs (5) and hyperbolic type spline (1) we determine the unknown functions etz(t) and miz(t) depending on the mean integral values vtz(t). By using the mean integral values of differential equation (7), we obtain N linear ODEs equations for determination of vtz(t) in the following form:
0.5 eiz(t) r*coth(0.25 r*H )/H t-r?vtz(t) =^^, t=1N.
A particular case. Let us have parameters N=2, L=3, Hi=1.8, H2=1.2, ri=0.4, Г2=0.1, vtz(0) = v2z(0) =0, α=100, β=1000, uo=l, ul=10, Di=0.2, Ü2=0.l, tf=30. Then we have the following discontinuous solutions:
viz(Hi,tf)=6.344, v2z(Hi,tf) =33.555,
ui(Hittf)=2.47, U2(Hittf)=9.88,
vi(Ht,tf)= 14.94, v2(Hi,tf)=24.30 (see Figs 6 a)d 7).
t о 0
u(t,z)
F i g. 6. The nonstationary v-solutions
Conclusion
The 3-D mass transfer problem in multi-layered domain is reduced to 2-D and 1-D problems using the special integral parabolic, hyperbolic and exponential type splines. These splines are obtained from the general spline with two fixed functions. The parameters of these functions are the characteristic values for the corresponding homogeneous ODEs of second order in fixed direction. These parameters are the best parameters for minimal error The 1-D differential and discrete problems are solved analytically. For the corresponding diffusion-convection problem the discontinuous solutions are obtained. The solutions for the corresponding averaged non-stationary 3-D initial-boundary value problem are obtained also numerically. The numerical solution is compared with the analytical solution. The maximum absolute value of difference between corresponding numerical and averaged data was in the range of 2-3 percent.
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F i g. 7. The nonstationary u-solutions
References
[1] Sneps-Sneppe M. Emanuel Grinberg - Outstanding Achievements in Applied Mathematics: Radio Filters, Hulls of Tankers, Graphs and Integral Circuits. International Journal of Open Information Technologies. 2018; 6(7):21-31. Available at: https://elibrary.ru/item.asp?id=35232780 (accessed 05.02.2020).(In Russ.,abstractinEng.)
[2] Shelkovnikova E.A., Antonov G.A., Vanag A.A., Greenberg E.Ya., Katznelson L.Z. Analytical Coordination of the Hull of the Ship. The Works of ZNIIS. 1964; 52:3-40. (In Russ.)
[3] Buikis A.A. Interpolation of Integral Mean of Piecewise Smooth Function by Means of Parabolic Spline. Latvian Mathematical Yearbook. 1985; (29):194-197. (In Russ.)
[4] Buikis A.A Modelling of Filtration Processes in Layered Porous Media by the Conservative Averaging Method: dis. ... Dr.Sci. (Phys.-Math.). Kazan; 1987. (In Russ.)
Том 16, № 1. 2020 ISSN 2411-1473 sitito.cs.msu.ru
H. Kalis,
I. Kangro
THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES
[5] Kneser A. Belastete integralgleichungen. Rendiconti del Cir-colo Matematico diPalermo. 1914; 37(1):169-197. (In Eng.) DOI: https://doi.org/10.1007/BF03014816
[6] Kalis H., Kangro I. Analytical solution for 3-D model of peat blocks. In: Proceedings of 14th International Scientific Conference "Engineering for Rural Development”. Jelgava, Latvia; 2005. p. 155-161. Available at: http://www.tf.llu.lv/ conference/proceedings2015/Papers/026_Kalis.pdf (accessed 05.02.2020). (In Eng.)
[7] Kalis H., Buikis A., Aboltins A., Kangro I. Special Splines of Hyperbolic Type for the Solutions of Heat and Mass Transfer 3-D Problems in Porous Multi-Layered Axial Symmetry Domain. Mathematical Modelling and Analysis. 2017; 22(4):425-440. (In Eng.) DOI: https://doi.org/10.3846/13 926292.2017.1318796
[8] Buike M., Buikis A. Modelling of Three-Dimensional Transport Processes in Anisotropic Layered Stratum by Conservative Averaging Method. WSEAS Transactions on Heat and Mass Transfer. 2006; 1(4):430-437. (In Eng.)
[9] Buikis A. Definition and calculation of a generalized integral parabolic spline. In: Proceedings of the Latvian Academy of Sciences. Section B. 1995; 7/8(576/577):97-100. (In Eng.)
[10] Buikis A., Kalis H. Creation of Temperature Field in a Finite Cylinder by Alternated Electromagnetic Force. In: A. Buikis, R. Ciegis, A.D. Fitt (ed.) Progress in Industrial Mathematics at ECMI 2002. The European Consortium for Mathematics in Industry. Vol 5. Springer, Berlin, Heidelberg; 2004. p. 247-251. (In Eng.) DOI: https://doi.org/10.1007/978-3-662-09510-2_31
[11] Buikis A., Kalis H. Calculation of electromagnetic fields, forces and temperature in a finite cylinder. Mathematical Modelling and Analysis. 2002; 7(1):21-23. (In Eng.)
[12] Buikis A., Kalis H., Kangro I. Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains. Mathematical Modelling and Analysis. 2016; 21(4):450-465. (In Eng.) DOI: https://doi.org/10.3846/13 926292.2016.1182594
Submitted 05.02.2020; revised 27.04.2020; published online 25.05.2020.
bout the authors:
Harijs Kalis, Senior Researcher of the Department of Mathematics, Institute of Mathematics and Computer Sciences, University of Latvia (19 Raina bulv, Riga LV-1586, Latvia), Dr.Sci. (Phys.-Math.), Professor, ORCID: http://orcid.org/0000-0002-9438-2614, Scopus ID: 6602387930, harijs.kalis@lu.lv
Ilmars Kangro, Associate Professor of the Faculty of Engineering, Rezekne Academy of Technologies (115 Atbrivosanas aleja, LV-4601, Rezekne, Latvia), Dr.Sci. (Pedagogy), Associate Professor, ORCID: 0000-0001-6413-5308, ilmars.kangro@rta.lv
All authors have read and approved the final manuscript.
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[3] Буйкис, A. A. Интерполяция интегральных средних кусочно-гладкой функции параболическим сплайном / А. А. Буйкис // Латвийский математический ежегодник.
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Поступила 05.02.2020; принята к публикации 27.04.2020; опубликована онлайн 25.05.2020.
|0б авторах:|
Калис Харий, ведущий научный сотрудник отделения математики, Институт математики и компьютерных наук, Латвийский университет (1586, Латвия, г. Рига, Бульвар Райниса, д. 19), доктор физико-математических наук, профессор, ORCID: http://orcid.org/0000-0002-9438-2614, Scopus ID: 6602387930, harijs.kalis@lu.lv
Кангро Илмарс, доцент инженерного факультета, Резекнен-ская академия технологий (4601, Латвия, г. Резекне, аллея Освобождения, д. 115), доктор педагогических наук, доцент, ORCID: 0000-0001-6413-5308, ilmars.kangro@rta.lv
Все авторы прочитали и одобрили окончательный вариант рукописи.
Современные информационные технологии и ИТ-образование
Том 16, № 1. 2020 ISSN 2411-1473 sitito.cs.msu.ru
