Volume 04 Issue 06-2024
58
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
This paper addresses the formulation of boundary value problems (BVPs) for second-order differential equations.
Boundary value problems are essential in various scientific and engineering applications where solutions must satisfy
specific conditions at the boundaries of the domain. The study outlines a systematic approach to defining the
differential equation, determining the domain, and specifying the appropriate boundary conditions. The discussion
includes different types of boundary conditions such as Dirichlet, Neumann, and mixed conditions. An example is
provided to illustrate the formulation process, demonstrating how to combine the differential equation with
boundary conditions to define a complete BVP. Methods for solving these problems, including analytical and
numerical techniques, are also reviewed, highlighting their importance in obtaining accurate solutions for complex
systems.
KEYWORDS
Boundary Value Problems (BVPs), Second-order Differential Equations, Dirichlet Boundary Condition, Neumann
Boundary Condition, Mixed Boundary Condition, Analytical Methods.
INTRODUCTIO
Consider the equation:
Research Article
ON THE FORMULATION OF BOUNDARY VALUE PROBLEMS FOR ONE
SECOND-ORDER EQUATION
Submission Date:
June 12, 2024,
Accepted Date:
June 17, 2024,
Published Date:
June 22, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue06-11
Muminov F.M.
Almalyk Branch of Tashkent State Technical University Almalyk, Uzbekistan
Dushatov N.T.
Almalyk Branch of Tashkent State Technical University Almalyk, Uzbekistan
Miratoev Z.M.
Almalyk Branch of Tashkent State Technical University Almalyk, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 06-2024
59
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
( ; y) U
( ; y) U
( ; )
( ; )
yy
xx
y
LU
K x
U
x
b x y U
m u U
f x y
=
+
+
+
+
=
(1)
where
К
(
х
,
у
)-continuously differentiable function, and
( )
,
0
К х у
at
(
)
0
0,
,
у
К х у
at
( )
1
0,
,
, b ,
C( ) (
)
C (D) ,
0,
0
у
а х у
D
x y
m
р
The region D-which consists at
у
>
0 of a
rectangle with vertices at points
А
(0;0),
В
(1;0),
А
1
(0;1),
В
1
(1;1), and at
𝑦 < 0
is bounded by the characteristics
of equation (1)
1
( ; ) :
, (0)
0,
0
dx
S
x y
K y
y
dy
=
= − −
=
2
( ; ) :
, (0)
0,
0
dx
S
x y
K y
y
dy
=
= − −
=
Let's put
1
2
S
S
S
=
Boundary value problem. Find a solution of
equation (1) in the region D such that
( )
( )
0;
1;
0
U
y
U
y
=
=
(2)
U(x;1)
(x) U(x; y) /
S
=
(3)
Everywhere below it is assumed that
,
у
∈
S.
(x)
exp
( 1
) ,
0, y
S.
2
y
p
=
− +
+
Where y>0 Let
1
2
( )
W D
denote the space of functions
from
1
2
( )
W D
that satisfy the boundary conditions
(2)-
(3)
Definition
1.
The functions
1
2
(x; y)
( )
W D
)
are
called the generalized solution of the problem
(1)-(3),
if
the integral identity holds.
(
)
( ; )
y
Y
x
x
y
d
D
U
KV
U V
a x y U V
bUY
m U UV dD
fVdD
−
−
+
+
+
=
for any function
V from
1
0
2
(D)
W
.
The existence of a generalized solution to the
boundary value problem
(1)-(3)
will be established
using the Galerkin method. Let
)}
,
{
(
n
x y
–
be the set
of functions from the space
1
2
( )
W D
possessing the
property that all elements of
)
,
(
n
x y
are linearly
independent, and their linear combinations are dense
in this space. Such a set, as known from
[1], [6]
exists
.
Volume 04 Issue 06-2024
60
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Let's consider an auxiliary problem
(5)
W ( ;1)
( )
( ; )
n
n
x
x W x y
S
=
(6)
The solution to the problem
(5)-(6)
(
)
1
,
( ; )
( ; )
1
y
t
n
n
n
s
S
W
x y
e
x
dt
e
x t dt
=
+
−
It is clear that
( )
,
n
W
х у
is linearly independent.
Indeed, if
1
0
N
n
n
n
C W
=
=
is for any set of
W
1
, W
2
,
…
, W
n
,
then by applying the operator
L
to this sum, we have
1
(x; y)
0
0,
n
N
n
n
n
n
C
C
=
=
=
It is clear that
1
2
(
)
W ( )
;
n
W x y
D
is easily obtainable an estimation
(
)
(
)
p
D
p
p
p
n
n
L
L
D
W
m
Moreover,
W
n
(x,y)
satisfies the conditions (6) for any
𝑛
. We will seek an approximate solution to the problem
(1)-(3) in the form of
1
( ; y)
( ;
)
N
N
n
n
n
U
x
C W
x y
=
=
where
n
С
are constants determined from a system of
nonlinear algebraic equations in the form
,
0
,
0
(
)
(
) ,
1,
N
n
n
LU U
f U
n
N
=
=
(7)
The solvability of this system of algebraic equations
follows from the a priori estimates obtained for the
approximate solutions and Lemma
«
acute angle
»
from
[8]
(
)
( ; )
;
y
n
ny
n
lW
e
W
x y
x y
−
=
=
Volume 04 Issue 06-2024
61
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Lemma 1. Suppose the conditions
( )
;1
0
К х
are satisfied and the inequalities
(
)
(
)
(
)
2
;
;
;
0
y
а х у
К
х у
К х у
−
−
Then the estimate holds true
1
2
2
2
W (
)
(
)
p
p
N
N
D
L
D
U
U
K
+
(8)
𝐾
2
does not depend on
n.
Proof. Multiplying (7) by
C
n
and summing over
n
from 1 to
N
, we obtain the identity
y
N
N
y
N
y
y
D
D
e U LU dD
e U fdD
=
(9)
Integrating the left-hand side of equation (9) by parts, we obtain
2
2
2
1
1
1
2
2
2
2
2
2
1
2
0
0
0
2
(
)
(2
)(
)
(
)
1
(
)
( ;1)(
)
(
)
((
)
(
)
(
) )
2(
)(
)
2
2
2
2
P
N
N
N
N
y
x
p
N
N
N
y
N
N
N
N
N
N
x
y
y
y
x
y
S
m
U
a
K
Ky U
U
U
dD
p
e
e
e
U
dx
K x
U
dx
U
dx
e
U
K U
m U
U
n
U
U
n ds
+
−
−
+ +
+
−
−
+
−
+
−
+
+
−
where
n=(n
1
;n
2
)
–
is the unit vector of the inward normal to
𝜕
D.
Using conditions (3) and the conditions of the lemma,
we obtain inequality (8). Let's return to the question of the solvability of the system of equations (7).
If we write it in
the form
( )
0
m
F C
=
, where
1
(
.....
)
m
m
n
C
C
C
=
then as we have just seen multiplying
0
(
( ), )
m
F C C
we get the
estimate
1
2
2
0
0
1
(
)
(
( ), )
N
m
w
D
F C C
K U
K
−
Since the linear envelope
1
2
(W , W ,......W )
m
L
is a finite-dimensional space, there exists
К
2
(𝑚)
such that,
therefore, the inequality is satisfied
2
0
2
1
1
(
( ), )
( )
0
N
m
S
S
F C C
K m
C
K
=
−
If
С
⃗
is large enough
And this is the “acute angle” condition sufficient
for solvability of the system of equations (7).
Theorem. Let the conditions of lemma.
Then for any function
2
(
)
;
( )
f x y
L D
there
exists a generalized solution of the problem (1)
–
(3).
Volume 04 Issue 06-2024
62
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Proof. By virtue of the estimate (8), the
sequence
p
N
N
U
U
is bounded in the space
q
L
where
1
1
1
p
q
+ =
. Then, based on (8), from the
sequence
{𝑈
𝑁
}
we can choose a subsequence
converging weakly in
1
2
W ( )
D
to some function
𝑈
(
х
,
у
)
and the sequence
p
N
N
U
U
converges weakly in
L
q
(𝐷)
to the function
q(
)
;
x y
( ; )
N
N
U
U
q x y
→
in
q
L
However, by the theorem, the embedding of
1
2
W ( )
D
in
2
( )
L
D
is quite continuous. Hence, we can
assume that the subsequence
( ; )
N
U
x y
is strong in
2
( )
L
D
and almost everywhere. Now let us apply
lemma 1 of [9],[11] on the limit transition in the
nonlinear term in the case where it follows that
( )
,
p
q х у
U U
=
Then, passing to the limit at
N
→
in (7) at fixed n, we have the equality
(
)
( ; )
p
y
n
y
x
n
n
n
n
n
D
D
U
K
U
U
x y U
m U
U
dD
f
dD
−
−
+
+
+
=
where the function
( ; y)
U x
belongs to
1
2
W ( )
D
.
Hence, in view of the density
n
in the space
1
2
W ( )
D
it follows that the integral identity
4
is valid
for any
1
0
2
V(x; y)
( )
W
D
The theorem is proved.
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уравнений
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Volume 04 Issue 06-2024
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American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
58-63
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
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Oriental renaissance: Innovative, educational,
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