=Paper=
{{Paper
|id=Vol-2744/short9
|storemode=property
|title=The Variational Iterations Method for the Three-dimensional Equations Analysis of Mathematical Physics and the Solution Visualization with its Help (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short9.pdf
|volume=Vol-2744
|authors=Aleksey Tebyakin,Irina Papkova,Vadim Krysko
}}
==The Variational Iterations Method for the Three-dimensional Equations Analysis of Mathematical Physics and the Solution Visualization with its Help (short paper)==
https://ceur-ws.org/Vol-2744/short9.pdf
The Variational Iterations Method for the Three-
dimensional Equations Analysis of Mathematical Physics
and the Solution Visualization with its Help*
Aleksey Tebyakin, Irina Papkova[0000−0003−4062−1437], and
Vadim Krysko[0000−0002−4914−764X]
Yuri Gagarin State Technical University of Saratov,
77 Politechnicheskaya street, Saratov, Russia, 410054
prototype9235@mail.ru, ikravzova@mail.ru, tak@san.ru
Abstract. The aim of the work is to use the variational iterations method to
study the three-dimensional equations of mathematical physics and visualize the
solutions obtained on its basis and the 3DsMAX software package. An
analytical solution of the three-dimensional Poisson equations is obtained for
the first time. The method is based on the Fourier idea of variables separation
with the subsequent application of the Bubnov-Galerkin method for reducing
partial differential equations to ordinary differential equations, which in the
Western scientific literature has become known as the generalized Kantorovich
method, and in the Eastern European literature has known as the variational
iterations method. This solution is compared with the numerical solution of the
three-dimensional Poisson equation by the finite differences method of the
second accuracy order and the finite element method for two finite element
types: tetrahedra and cubic elements, which is a generalized Kantorovich
method, based on the solution of the three-dimensional stationary differential
heat equation. As the method study, a set of numerical methods was used. For
the results reliability, the convergence of the solutions by the partition step is
checked. The results comparison with a change in the geometric parameters of
the heat equation is made and a conclusion is drawn on the solutions reliability
obtained. Solutions visualization using the 3Ds max program is also
implemented.
Keywords: Heat Equation, Method of Variational Iterations, Analytical
Solution, Finite Difference Method.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
* This work was supported by the Russian Science Foundation project RNF № 19-19-00215.
The Variational Iterations Method for the Three-dimensional… 2
1 Introduction. The history of the issue
In the work, to solve the three-dimensional equations of mathematical physics
(Poisson equation), a method is used that is a generalization of the Fourier variables
separation method and is known in the literature as the variational iterations method.
This method is based on the idea of reducing partial differential equations to ordinary
differential equations for elliptic equations - this is the Kantorovich method with the
subsequent refinement of the function with respect to the desired variables - the
variational iterations method. In Western European literature, this method is called the
extended Kantorovich method, in Eastern European literature it is called the method
of variational iterations (MVI). It allows you to get an analytically accurate solution at
every step of the iterative procedure.
This method was first proposed and used in 1933 by T.E. Shunk [1] for
calculating the bending of cylindrical panels. Unfortunately, the work was not
noticed, and the method was rediscovered again in 1964 E.E. Zhukov [2], who used it
to calculate thin rectangular plates. In the future, MVI was widely used by many
researchers in solving problems of the shells and plates theory (a bibliography on this
subject is presented in [3]). The method justification for the class of equations
described by positive definite operators is given in [4].
The variational iterations method (extended Kantorovich method) over the past
half century has been used to solve problems of statics, stability, natural frequencies
determination and dynamics. A fairly complete review of Western publications in this
area can be found in [5, 6]. In the USSR and Russia, this method was mainly used in
the works of V.A. Krysko and his students. For the first time, this scientific group
used the approach in 1968 to study the bending of flexible orthotropic plates [7], and
approach got its name the variational iterations method in the work [8] 1970, devoted
to the numerical study of flexible plates and comparison with experimental data.
Subsequently, scientists of this group used the variational iterations method to solve
geometrically and physically nonlinear problems in the theory of shells and plates [9,
10], in problems of designing optimal plates [11–12], and on other topics [13–16].
In this paper, this method is first used for the three-dimensional Poisson equation,
which shows the relevance of this work. To implement these methods and obtain the
final solution, the MATLAB application package is used. As a visualization of the
obtained data, a software utility made for the Autodesk 3Ds Max platform is used.
3Ds Max has extensive tools for creating a diverse in form and complexity of three-
dimensional computer models, real or fantastic objects of the surrounding world,
using a variety of techniques and mechanisms.
2 Variational iterations method
We apply MVI to find an analytical solution for the differential equation. Consider
the stationary differential heat equation for a three-dimensional body (1).
3 A. Tebyakin, . I. Papkova and V. Krysko
T ( x, y, z ) + f ( x, y, z ) = 0 (1)
T T T
2 2 2
T = + + ,
x 2 y 2 z 2
where T(x,y,z) – temperature field, f(x,y,z) – the internal heat source density, which is
considered to be given, - thermal diffusivity,
= Tmin T ( x, y, z ) Tmax ,
( x, y, z ) = (0, a) ( 0, b ) ( 0, c ) - temperature distribution area; - border area.
Edge conditions of the 1st kind
𝑇|∂Ω = 𝑡Г = 𝑐𝑜𝑛𝑠𝑡 (2)
Edge conditions of the 2nd kind
T
− = qr (3)
n n =−0
where q r - is the given value of the heat flux on the surface. The notation n=-0
emphasizes that the values are calculated inside the object infinitely close to its
surface.
Edge conditions of the 3rd kind
T
− = ( tw − t f ) (4)
n n =−0
The proportionality coefficient α in (4) is called the heat transfer coefficient and is
a measure of the convective heat transfer intensity between the surface and the heat
carrier (for the sake of brevity, this is called heat transfer), tw – tf is the temperature
thrust, the temperature difference between the wall (surface) and the environment.
We apply the method of variational iterations in a first approximation to the heat
equation taking into account the boundary conditions (2).
In the variational iterations method, it is assumed that the function consists of the
one-dimensional functions product with respect to each variable:
T ( x, y, z ) = A( x) B ( y )C ( z ) (5)
We introduce a three-dimensional bounded Hilbert space H (0,1) (0,1) (0,1)
where each space function has a continuous derivative of at least second order.
We introduce a three-dimensional bounded Hilbert space - where each function of this
space has a continuous derivative of at least second order. To find a solution, we use
condition (3) so that:
(T ( x, y, z ) + f ( x, y, z ) ) T ( x, y, z )dxdydz = 0
H
(6)
Substituting (10) into (11) we get
a b c
2 A( x) 2 B( x)
0 0 0 x2
B ( y )C ( z ) +
x 2
A( y )C ( z ) + (7)
2C ( z )
+ A( x) B( y) + f ( x, y, z ) A( x) B( y)C ( z )dxdydz = 0
x 2
The Variational Iterations Method for the Three-dimensional… 4
Subsequently, from equation (7), the functions A(x), B(y), C(z) are found. Since
the method implements an iterative process, it is required to set the initial
approximation. For example, if you want to find the function C(z), then you need to
specify any functions A(x) and B(y) satisfying the conditions of the Hilbert space H.
The result is an inhomogeneous ordinary differential equation with constant
coefficients with respect to the function C(z).
R R
C ( z ) + 2 C ( z ) = 3 , (8)
R1 R1
where
a b
R1 = A2 ( x) B 2 ( y )dxdy
0 0
A( x)
a b 2
2 B( y )
R2 = B( y ) + A( x) A( x) B( y )dxdy (9)
0 0 x 2
y 2
a b
R3 = f ( x, y, z ) A( x) B ( y )dxdy .
0 0
When solving an inhomogeneous differential equation, the boundary conditions
(2) are taken into account. The result of the solution is the function C(z). The
functions A(x) and B(y) are found in a similar way, but already using the found
functions. After all the functions for each variable are found, we obtain an
approximate analytical solution. Only the first iteration was carried out. Next you can
carry out this procedure as many times as you like and get an increasingly accurate
solution to equation (1). Further, this problem will be considered using the numerical
method.
3 Finite difference method
To compare and analyze the results using the methods described above, the finite
difference method with second-order approximation is used. We write the difference
heat equation:
T ( s )i +1, j , k + T ( s )i −1, j , k T ( s )i , j +1, k + T ( s ) i , j −1, k
T ( s +1) i , j , k = + +
hx2 hy2
(10)
T ( s )i , j , k +1 + T ( s )i , j , k −1 2 2 2
+ + f ( x, y , z ) / 2 + 2 + 2
hz2 h
x hy hz
max T ( s +1)i , j , k − T ( s )i , j , k , где 0 (11)
We attach the boundary conditions of the 1st kind (2). When the iteration process
starts, the array of the function T(0)i,j,k is filled with zeros. After using formula (10)
over the entire array, we obtain a new scalar field T(1)i,j,k. We carry out this procedure
while (11) is satisfied, where e is the maximum difference between the values of
5 A. Tebyakin, . I. Papkova and V. Krysko
adjacent iterations. A sufficiently small number is chosen as in order to approach
the exact solution. We apply MVI to find an analytical solution for the differential
equation.
4 Results analysis
For the heat equation (1), taking into account the boundary conditions (2), we
compare the solutions that were obtained by the numerical method (finite difference
method) and the analytical method (variational iteration method in a first
approximation). The results are presented in table 1. Three cases were considered: a =
b = c = 1; a = 3, b = c = 1; a = 5, b = c = 1. As the results, the maximum values of
the functions obtained in the solution process were taken. The difference between the
results obtained by analytical and numerical methods is 5.34%.
Table 1. Numerical experiment results.
MVI a=b=c=1 a = 3, b = c = 1 a = 5, b = c = 1
Maximum { А(х) } 0.493542 0.595027 0.594314
Maximum { B(y) } 0.409915 0.396978 0.395888
Maximum { C(z) } 29.377056 3.536978 1.281865
Maximum
{T(x,y,z)=А(х)B(y)C 5.9432295 0.835481 0.301600
(z) }
Finite difference
21x21x21 21x21x21 21x21x21
method
Maximum
{T(x,y,z)=А(х)B(y)C 5.599181 0.814254 0.294009
(z) }
For MVI, table 1 presents a data set for each of the functions obtained by the
method lying on its axis. The solution obtained by this method is shown in fig. 1, 2, 3.
Here is shown the surface of the temperature distribution in the middle of the region
relative to the Oz axis. To build the surface, we used a special program written on the
3Ds Max platform.
The Variational Iterations Method for the Three-dimensional… 6
Fig. 1. Visualization of the numerical solution for the median plane
a=b=c=1
Fig. 2. Visualization of the numerical solution for the median plane
a = 3; b = c = 1
Fig. 3. Visualization of the numerical solution for the median plane
a = 5; b = c = 1
Conclusion
1. For the first time, to obtain an analytical solution of a three-dimensional partial
differential equation, the variational iterations method is used, which is based on
the Kantorovich method — reduction of partial differential equations to ordinary
differential equations.
7 A. Tebyakin, . I. Papkova and V. Krysko
2. This method has high accuracy even in the first approximation, as evidenced by
the solution of the same equations obtained by the finite differences method of the
second accuracy order for a the first kind boundary value problem.
3. The results of a numerical solution using the variational iteration method were
visualized using a software package written on the 3Ds Max platform.
References
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394 – 414, (1933).
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calculation of thin rectangular plates. – V kn.: Raschet tonkostennyh prostranstvennyh
konstrukcij. Stroyizdat. Moscow. (1964).
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University. Saratov. (1976)
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