Revista
de la
Universidad
del Zulia
Fundada en 1947
por el Dr. Jesús Enrique Lossada
DEPÓSITO LEGAL ZU2020000153
ISSN 0041-8811
E-ISSN 2665-0428
Ciencias del
Agro,
Ingeniería
y Tecnología
Año 14 N° 39
Enero - Abril 2023
Tercera Época
Maracaibo-Venezuela
REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 14, N° 39, 2023
Y. Kostikov & A. Romanenkov /// On solution of pseudohyperbolic equation… 225-232
DOI: http://dx.doi.org/10.46925//rdluz.39.12
225
On solution of pseudohyperbolic equation with constant
coefficients
Yury Kostikov*
Aleksandr Romanenkov**
ABSTRACT
The paper proposes the method of forming the exact solution of the first initial boundary
value problem for one-dimensional linear pseudohyperbolic equation with constant
coefficients. To obtain the solution type, the modification of partition method (Fourier
method) is used, when the type of one of the solution functional factors is considered to be
known. At the same time, the initial problem is reduced to parameterized family of Cauchy
problems for ordinary differential equations. The paper presents explicitly calculated
formulas, which specify the solution. The qualitative research of the solution properties has
been conducted. The conditions for coefficients in the form of inequalities have been obtained
that is indicative of boundedness and variability of the solutions. Several examples
confirming the results obtained have been considered.
KEYWORDS: Pseudohyperbolic equation, exact problem-solving solutions, solution
properties.
*Candidate of Science in Physics and Mathematics, Head of Department 916, Moscow Aviation
Institute (National Research University) Volokolamskoe shosse, 4, Moscow, 125993. E-mail:
jkostikov@mail.ru
** Candidate of Technical Science, Associate Professor of Department 916, Moscow Aviation
Institute (National Research University) Volokolamskoe shosse, 4, Moscow, 125993 Federal
Research Center “Informatics and Control”, Russian Academy of Sciences Vavilova St., 44, bld. 2,
Moscow, 119333. ORCID: https://orcid.org/0000-0002-0700-8465. E-mail: romanaleks@gmail.com
Recibido: 13/09/2022 Aceptado: 16/11/2022
REVISTA DE LA UNIVERSIDAD DEL ZULIA. 3ª época. Año 14, N° 39, 2023
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DOI: http://dx.doi.org/10.46925//rdluz.39.12
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Sobre la solución de la ecuación pseudohiperbólica con coeficientes
constantes
RESUMEN
El artículo propone el método para formar la solución exacta del primer problema de valor
límite inicial para una ecuación pseudohiperbólica lineal unidimensional con coeficientes
constantes. Para obtener el tipo de solución se utiliza el método de modificación de la
partición (método de Fourier), cuando se considera conocido el tipo de uno de los factores
funcionales de la solución. Al mismo tiempo, el problema inicial se reduce a la familia
parametrizada de problemas de Cauchy para ecuaciones diferenciales ordinarias. El
documento presenta fórmulas calculadas explícitamente, que especifican la solución. Se ha
llevado a cabo la investigación cualitativa de las propiedades de la solución. Se han obtenido
las condiciones para los coeficientes en forma de desigualdades que son indicativas de
acotación y variabilidad de las soluciones. Se han considerado varios ejemplos que confirman
los resultados obtenidos.
PALABRAS CLAVE: Ecuación pseudohiperbólica, soluciones exactas de resolución de
problemas, propiedades de la solución.
Introduction
Let us consider the first initial boundary value problem for pseudohyperbolic
equation. It is necessary to find function
, which
satisfies the linear homogeneous equation (1):
where
to boundary conditions (2):
and initial conditions (3):
Let us point out that the equation considered is found in the liquid filtration theory.
In (Chudnovsky, 1976). the equation (1) emerges as the mathematical model of evaporation
and infiltration. When , the equation (1) is called Aller-Lykov moisture transfer
equation. Steklov-type problem for this equation is considered in (Lafisheva, Kerefov,
Dyshekova 2017), apriori estimates and numerical algorithm for forming the approximate
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solution are obtained in the assumption of solution existence. In modern publications the
significant attention is also paid to the research of inverse problems. Thus, for example, the
renewal of the right part of one-dimensional equation is discussed in (Kenzhebai, 2021), and
inverse problems for pseudohyperbolic equation are studied in (Kurmanbaeva, 2016), and the
existence and uniqueness conditions for the solution of a special boundary value problem are
obtained for constant coefficients of linear one-dimensional equation.
Generalized Aller equation of fractional order is considered in (Gekkieva, Karmokov,
Kerefov, 2020; Kerefov, Gekkieva, 2019), for which the exact solution is written out in the
form of finite integral formula. The integral operator kernel is explicitly written out in
(Kerefov, Gekkieva, 2019), with the help of which the exact solution of the second boundary
value problem is specified.
The aim of the work is to form the exact solution in the form of Fourier series of the
first initial boundary value problem for pseudohyperbolic equation. The application of
Fourier method (method of variable separation, for example, in (Ewans, (2003) is usually
successful; due to the mixed derivative availability the variables cannot be separated directly.
Therefore, referring to the type of boundary conditions and type of the differential operator
in the equation (1), we will search for the solution in the following form:
where parameter, which can be a complex number.
The aim of the work is to obtain an exact solution of the initial boundary value
problem in the form of a Fourier series using a special representation of the solution in the
form of formula (4).
Obtaining of exact solution. Then we apply the expression for the required function
from (4) to the equation (1):
After grouping the summands in (5) and simplifying by , we have
The equation (6) is a linear equation with constant coefficients, the solution of which
can be easily written out:
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where
and
The boundary conditions (2) allow specifying the set of values of parameter :
Further we introduce the designations of magnitude
Due to the linearity of the problem (1)-(3), the formula (8) specifies the solution for
each
from (9) and then we have Fourier series:
Let us specify coefficients
and
For this we use the initial condition (3):
Further we have the system to find the required coefficients:
where
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From the system (13) we have the expression for the required constants:
Then we obtain Fourier series for the required function:
or with the use of hyperbolic functions the formula (14) is as follows:
Practical results. On properties of solutions. The monotonous tend of magnitude
to infinity with the increase in index n is an obvious fact. It should be also pointed out that
at . This fact follows from (11). Indeed, after elementary transformations we
have
From (16) we have that if
, we have the inequality:
i.e. in (12) only a finite number of summands will not be limited with the increased . At the
same time, not all
can be real numbers but, again, only their finite number can be complex.
Let us formulate these remarks as the lemma.
Lemma. i) Summands unlimited by in (12) will correspond to indices , which satisfy the
following inequality:
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ii) Summands varying by in (12) will correspond to indices , which satisfy the
following inequality:
Proof: i) Let us require the fulfillment of
and with the help of (16) we obtain
the necessary result. ii) In (11) we will require the negativeness of the radical expression.
After that we obtain the required inequality.
Examples. Practical results and examples. Let
Example 1. Let us consider the initial boundary value problem (1)-(3) at
Then the equation (1) is as follows:
Boundary conditions (2):
And initial conditions (3):
For this case by the formula (9) we have that
where and by the formula (10)
we have
. The initial conditions correspond to . According to point ii) of the
lemma we should obtain the summands varying by in the solution. It is not difficult to write
out the solution for these parameters by the formula (14):
Example 2. Let us consider the initial boundary value problem (1)-(3) at
Then the equation (1) is as follows:
Boundary conditions (2):
And initial conditions (3):
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Y. Kostikov & A. Romanenkov /// On solution of pseudohyperbolic equation… 225-232
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As in the example 1,
where and by the formula (10) we have
. The
initial conditions correspond to and
, and condition ii) of the lemma is not
fulfilled. Let us write out the problem solution by the formula (14):
We note that there are no coefficients varying by time here. At the same time, there
are no values unlimited by time as well that is an expected factor since the inequality i) from
the lemma is not fulfilled.
Example 3. Let Let us consider the initial boundary value
problem (1)-(3) at
Then the equation (1) is as follows:
Boundary conditions (2):
And initial conditions (3):
As in the example 1 we have that
where and by the formula (10) we have
. The initial conditions correspond to and
. The inequality from point i) of
the lemma is fulfilled. Let us write out the exact solution of the problem
and we note that
is unlimited with the increasing .
Conclusion
The paper considers the linear equation with constant coefficients of
pseudohyperbolic type. The first initial boundary value problem for this equation is set. The
method of forming the exact solution in the form of Fourier series by countable trigonometric
system consisting only of sinuses is proposed. The exact formulas for defining the solution
coefficients, for the case of initial conditions from the corresponding functional space are
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obtained. The quantitative behavior of solutions at some correlations on the coefficients of
the initial differential equation is analyzed.
References
Chudnovsky, A.F. (1976). Thermal physics of soil. Moscow: Nauka, 352 p.
Ewans, L.K. (2003). Equations with partial derivatives. Translated from English. –
Novosibirsk: Tamara Rozhkovskaya, 562 p.
Gekkieva, S.Kh.; Karmokov, M.M.; Kerefov, M.A. (2020). On one boundary value problem for
generalized Aller equation // Bulletin of Samara State Technical University. Natural science
series. V. 26, № 2. P. 7–14. DOI: http://doi.org/10.18287/2541-7525-2020-26-2-7-14.
Kenzhebai, Kh. (2021). Inverse problem of the renewal of the right part of one-dimensional
pseudoparabolic equation. Bulletin of Kazan National University. Series: Mathematics,
mechanics. informatics, [S.l.], v. 111, n. 3, p. 28-37, oct. ISSN 2617-4871.
Kerefov, М.А.; Gekkieva, S.Kh. (2019). Second boundary value problem for generalized Aller-
Lykov moisture transfer equation. Bulletin of Samara State Technical University. Series:
Physical and mathematical sciences. 23:4, 607–621.
Kurmanbaeva, A.K. (2016). Linear inverse problems for pseudohyperbolic equations.
Educational resources and technologies. №2 (14). URL:
https://cyberleninka.ru/article/n/lineynye-obratnye-zadachi-dlya-psevdogiperbolicheskih-
uravneniy (reference date: 22.08.2022).
Lafisheva, M.M.; Kerefov, M.A.; Dyshekova, R.V. (2017). Difference schemes for Aller-Lykov
moisture transfer equation with nonlocal condition. Vladikavkaz mathematical journal. V.
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