Coefficient Identification Problem for the System of Heat and Wave
Equations Associated with a Non-Characteristic Type Change Line
DURDIMUROD DURDIEV
Bukhara Branch of Romanovskii Institute of Mathematics,
Uzbekistan Academy of Sciences,
UZBEKISTAN
also with
Differential Equations, Bukhara State University,
11 M. Ikbal St. Bukhara 200100, Bukhara,
UZBEKISTAN
Abstract: The solvability of the inverse problem associated with the search for an unknown coefficient at the
lowest term of a mixed parabolic-hyperbolic type equation with a non characteristic line of type change is
studied. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal
condition on the characteristics in the hyperbolic part and initial-boundary conditions in the parabolic part of
the domain. To determine unknown coefficient, with respect to the solution, defined in the parabolic part of the
domain, the integral overdetermination condition is specified. The unique solvability of the inverse problem in
the sense of the classical solution is proved.
Key-Words: - Parabolic-hyperbolic equation, characteristic, Green’s function, direct problem, inverse problem,
contraction principle mapping.
1 Formulation of the Problem
Let be a finite open domain, bounded for
by segments where
is fixed positive number,
and for - by the characteristics :
and : of the following equations:
(1)
Equation (1) is of mixed parabolic-hyperbolic
type, and its type change line is not a
characteristic (parabolic degeneration of the first
kind, [1]). In this case the parabolic boundary of
equation (1) at is
Direct problem. Find in the domain the solution
of the equation (1) satisfying the following
boundary conditions:
(2)
(3)
where are given functions.
By a solution (classical) of the direct problem (1)-
(3) we mean the function from the class
which
satisfies equation (1) and conditions (2), (3).
Let us formulate the inverse problem as the
problem of finding a pair of functions
that satisfies the equation (1), boundary conditions
(2), (3) and the following overdetermination
condition:
(4)
where in (4) are given sufficiently
smooth functions.
Direct and inverse problems for mixed type
equations are not as well studied as similar problems
for classical equations. Nevertheless, such problems
are relevant from the point of view of applications.
The importance of considering equations of mixed
type, where the equation is of parabolic type in one
part of the domain and hyperbolic in the other, was
first pointed out in the work, [2]. Another example
is the following phenomenon in electrodynamics: a
Received: May 9, 2023. Revised: October 11, 2023. Accepted: December 2, 2023. Published: December 31, 2023.
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mathematical study of the tension of an
electromagnetic field in an inhomogeneous medium
consisting of a dielectric and a conducting medium
leads to a system consisting of a wave equation and
a heat equation, [3]. There are many such examples.
The first results on the study of an analogue of
the Tricomi problem for a hyperbolic-parabolic
equation were obtained in [4]. Further, such
problems with different classical and non-local
boundary conditions for parabolic-hyperbolic
equations with both characteristic and non-
characteristic type change lines are formulated and
studied in [5], [6], [7], [8].
Methods for solving direct problems of finding
the solution of an initial-boundary value problem for
equations of the parabolic-hyperbolic type and
inverse source problems for these equations in a
rectangular domain were proposed in the
monograph, [9].
Note that with various inverse problems for
classical differential equations of hyperbolic and
parabolic types of the second order, the reader can
get acquainted in works, [10], [11], [12], [13], [14].
This article continues the study of the author
[15], in which the local unique solvability of the
inverse problem of determining the variable
coefficient at the lowest term of a hyperbolic
equation for a mixed hyperbolic-parabolic equation
with a noncharacteristic line of type change is
investigated.
Throughout this paper, with respect to the given
ones, we will assume that the following conditions
are satisfied:
(B1)
(B2)
(B3)
for all
The study of inverse problems requires studying
the differential properties of solutions of direct
problems. This is most clearly seen in coefficient
inverse problems (nonlinear problems), where, to
obtain solvability theorems, one must carefully
analyze the exact dependence of the differential
properties of solutions of the direct problem on the
smoothness of the coefficients and other data of the
problem. That is why, let us study the direct
problem first.
2 Investigation of the Direct Problem
Assume that the function is known.
Theorem 1. Let conditions (B1), (B2),
be satisfied.
Then, in the domain there exists an unique
solution to the direct problem (1)-(3).
Let there be a solution of the direct problem
(1)-(3). Let us introduce the notation:
,
Then, due to the
unique solvability of the Cauchy problem for the
wave equation, the solution to the equation (1) in the
domain can be written using the d’Alembert
formula
(5)
Taking into account condition (3) and equalities
(a consequence of the definition of the
classical solution), this implies the equality:
Further it follows from (3) at :
Then, comparing this with (5) at
we have
Using this equality we eliminate
in (6) and we find
Thus the function becomes known.
Introduce the notations
Using the Green’s function of
the first initial-boundary value problem for the heat
equation in the domain the solution of equation
(1) with the conditions (2), and
represent in the form of integral equation:
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Equations (8) represents a linear integral
equation Volterra type of the second kind to
determine the unknowns functions ( is
known). It is known from the general theory of
integral equations that under the conditions of
Theorem 1 this equation is solvable in the class of
continuous in functions and determines the
function
that is, the solution of
problem (1), (2) in the domain
Note that the functions
have equivalent representations:
and are infinitely differentiable in [3].
Since the functions and are known in
, let us now begin to find For this, we
calculate the derivatives of the first two terms on the
right side of (8) using the obvious relations
Integrating by parts, we get:
Using (10) and integrating by parts, we
calculate the derivative with respect to of the
following term in formula (8):
Taking into account the form of the function
according to the formula (7), we finally have:
Using the above equalities, we now differentiate
(8) with respect to and set . Since
, taking into account the
matching conditions (B2), we obtain:
(11)
Eliminating the function in (8) using
equality (7), we obtain integral equation for the
function
Note that the following holds for the function
equality:
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Equations (11) and (12) represent a system of
linear integral equations Volterra type of the second
kind to determine the unknowns functions
and By virtue of formula (13), integral
equation (11) has a weak polar singularity. It is
known from the general theory of integral equations
that the system of equations (11) and (12) is
solvable in the class of continuous in functions.
This solution can be found, for example, by the
method of successive approximations and
due to
for
Considering equality:
using integration by parts, based on conditions (B1),
(B2), we find
Assuming now the existence of a derivative of
the solution taking into account conditions
(B1), (B2) and (14), we obtain for the
equation
which is also solvable in the class of continuous
functions instead of with equation (12). Thus,
From the found function
the function is found from formulas (8).
Due to conditions (B2) and we
have And the function
, constructed as solution of equation (1) with
conditions (2), and when conditions
(B1), (B2) are met and inclusion
belongs to the class
Thus, found in solution and
function (6) in together determine the classical
solution to the direct problem (1)-(3) in the domain
Theorem 1 is proved.
3 Study of the Inverse Problem
Assume that conditions (B3) are satisfied.
Multiplying the equation (1) in the domain by
the function and integrating over the segment
in view of (4), we find
(15)
Now, substituting the right side of (15) instead
of in (8), we write the resulting equation in the
operator form:
(16)
where the operator is defined by the equality:
and in (17) denotes the sum of terms of integral
equation (8) which are free from unknown function:
Recall that the function is defined by the
formula (7).
The main result of this section is the following
assertion:
Theorem 2. Let conditions (B1)-(B3) be satisfied.
Then, there are positive numbers such that
equation (16) has a unique continuous solution in
the domain for
Proof. It is clear from (17) that under the conditions
of the theorem the operator translates the
functions into functions also
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belonging to the space The norm in
we define as follows:
For brevity, we also introduce notations
Let us now show that, for sufficiently small ,
the operator performs a contraction mapping of
the ball:
with radius ( is a known number) and
centered at the point of the functional space
into itself. Thus, we will prove that equation
(16) has in the domain an unique continuous
solution satisfying the inequality
It is obvious that for the element there
holds an estimate:
where denotes a known positive number.
Let us estimate
To do this, we need
estimates for integrals involving the functions
in the definitions of the function In
this case, we use the equality:
which follows from the definition of the function
Taking this into account, the first term of
can be easily estimated in modulo:
Based on (9), we have the equalities:
which are checked directly. Using these relations,
we transform the following integral:
Here, in intermediate calculations, we used the
relation is
the Dirac’s delta function and the main property of
the function :
which is valid for any continuous function on
the interval
From these relations for easily
it follows the estimate
Then, inequalities (18), (19) imply the estimate:
We now turn to obtaining conditions for under
which it is possible to apply the fixed point theorem
to the operator Let then, for all
, we obtain the inequalities:
Condition (that is
) will be valid if is chosen from the
condition This condition is satisfied by
all where
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Let us now to show that the operator contracts
the distance between elements of the ball
To prove this, we take any two elements
and estimate the norm of the difference
between their images For this purpose,
using (17) we have the inequality
Here to estimate the expression
we use
inequality
which holds for arbitrary
Continuing the estimate (20), we get
We choose so that the inequality
holds, then the operator U contracts the distance
between elements of the ball This
condition is satisfied by where
Let Since
then, it is easy to see that
Hence, the operator for
performs a contraction mapping of
the ball to itself. Hence, according to
the contraction mapping principle, equation
(16) defines a unique solution
Theorem 2 is proved.
After finding the function the functions
is determined by the formula (15).
Thus the following assertion is valid:
Theorem 3. Let conditions (B1)-(B3) be satisfied
and Then, the formula (15) defines
on any fixed segment
4 Conclusion
In this paper, the solvability of the inverse problem
associated with the search for an unknown
coefficient at the lowest term of a mixed parabolic-
hyperbolic type equation with a non-characteristic
line of type change is investigated. In the direct
problem, an analog of the Tricomi problem for this
equation with a nonlocal condition on the
characteristics in the hyperbolic part and initial-
boundary conditions in the parabolic part of the
domain is considered. To determine the unknown
coefficient, with respect to the solution of the direct
problem, defined in the parabolic part of the
domain, the integral overdetermination condition is
specified. The unique solvability of the inverse
problem in the sense of the classical solution is
proved.
Note that the zero-coefficient of the parabolic
equation is defined here. Many applied problems
require consideration of more general equations than
(1) and determination of other coefficients in both
parabolic and hyperbolic equations. Similar
problems and the numerical study of the inverse
problem considered in this article are open
problems.
Acknowledgement:
The author thanks the anonymous reviewers
who, after reading the article, made very useful
comments.
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Policy)
The author contributed to the present research, at all
stages from the formulation of the problem to
the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
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