American Journal of Applied Science and Technology
30
https://theusajournals.com/index.php/ajast
VOLUME
Vol.05 Issue01 2025
PAGE NO.
30-34
10.37547/ajast/Volume05Issue01-08
Approximate solution of the galerkin method for one
non-classical problem of parabolic type
Mamatov A. Z.
Tashkent Institute of Textile and Light Industry, Tashkent city, Uzbekistan
Raxmonov J.T.
Senior teacher of Gulistan state University, Uzbekistan
Xamzakulov E.A.
Intern teacher of Gulistan state peadagogical institute, Gulistan town, Uzbekistan
Sulaymanova N.O.
Intern teacher of Guliston state pedagogical institute, Gulistan town, Uzbekistan
Received:
25 October 2024;
Accepted:
28 December 2024;
Published:
30 January 2025
Abstract:
The article considers one boundary value problem of parabolic type with a divergent main part, when
the boundary condition contains the time derivative of the desired function. Such non-classical problems arise in
a number of applied problems, for example, when a homogeneous isotropic div is placed in the inductor of an
induction furnace and an electromagnetic wave falls on its surface. Such problems have been little studied, so the
study of problems of parabolic type, when the boundary condition contains the time derivative of the desired
function, is relevant. The work defines a generalized solution to the problem under consideration in the space
Н
1,1
̃ (𝑄
𝑇
).
The purpose of the study is to prove the theorem of the existence and uniqueness of an approximate
solution of the Bubnov-Galerkin method for the considered non-classical parabolic problem with a divergent main
part, when the boundary condition contains the time derivative of the desired function.
Keywords:
Mixed problems, quasilinear equation, boundary condition, Galerkin method, generalized solution,
parabolic type, approximate solution, error estimate, a priori estimates, coordinate system, monotonicity,
inequalities, boundary, domain, scalar product.
Introduction:
When studying a number of current
technical problems, it becomes necessary to study
mixed parabolic problems, when the boundary
condition contains a time derivative of the desired
function. Problems of this type arise, for example,
when a homogeneous isotropic div is placed in the
inductor
of
an
induction
furnace
and
an
electromagnetic wave falls on its surface. Some
nonlinear problems of parabolic type with a boundary
condition containing the time derivative of the desired
function were considered, for example, in works [1-3].
Many scientists have been involved in constructing an
approximate solution using the Galerkin method and
obtaining a priori estimates of the approximate
solution for parabolic classical quasilinear problems
without a time derivative in the boundary condition:
Mikhlin S.G., Douglas J. Jr., Dupont T., Dench J. E., Jr.,
Jutchell L., and others [4-7]. And quasilinear problems,
when the boundary condition contains the time
derivative of the desired function using the Galerkin
method, are studied in works [8-12].
Statement of the problem
. In this paper, we consider a
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
quasilinear problem of parabolic type, when the
boundary condition contains the time derivative of the
desired function:
{
𝑢
𝑡
−
𝑑
𝑑𝑥
𝑖
𝑎
𝑖
(𝑥, 𝑡, 𝑢, ∇𝑢) + 𝑎(𝑥, 𝑡, 𝑢, ∇𝑢) = 0 ,
𝑎
0
𝑢
𝑡
+ 𝑎
𝑖
(𝑥, 𝑡, 𝑢, ∇𝑢) cos(𝜈, 𝑥
𝑖
) = 𝑔(𝑥, 𝑡, 𝑢), (𝑥, 𝑡) ∈ 𝑆
𝑡
,
𝑢(𝑥, 0) = 𝑢
0
(𝑥) , 𝑥 ∈ 𝛺
(1)
𝑤ℎ𝑒𝑟𝑒 𝛺 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑑𝑜𝑚𝑎𝑖𝑛 𝑖𝑛 Е
2
, 𝑎
0
= 𝑐𝑜𝑛𝑠𝑡 > 0, 𝑄
𝑇
= 𝛺 × [0, Т]
,
𝑆
𝑇
= 𝑆 × [0, Т]
, S=
𝜕𝛺
Definition.
A generalized solution from the space
𝑊
2
1,1
̃ (𝑄
𝑇
) = {𝑈 ∈ 𝑊
2
1,1
(𝑄
𝑇
): 𝑎
0
𝑈
𝑡
∈ 𝐿
2
(𝑆
𝑇
)}
of problem (1) is
a function from
𝑊
2
1,1
̃ (𝑄
𝑇
),
satisfying the following identity
∫ (𝑢
𝑡
𝜂 + 𝑎
𝑖
(𝑥, 𝑡, 𝑢, ∇𝑢)𝜂
𝑥𝑖
+ 𝑎(𝑥, 𝑡, 𝑢, ∇𝑢)𝜂)
𝑄
𝑇
𝑑𝑥𝑑𝑡 + ∫ (𝑎
0
𝑢
𝑡
+ +𝑔(𝑥, 𝑡, 𝑢)))𝜂)
𝑆
𝑇
𝑑𝑥𝑑𝑡 = 0
(2)
∀ 𝜂 ∈ 𝑊
2
1
(𝛺)
Let us assume that the following conditions are satisfied:
A.
𝑎𝑡 (𝑥, 𝑡, 𝑢, 𝑝) ∈ {𝛺
̅ × [𝑂, 𝑇] × 𝐸
1
× 𝐸
2
}
functions
𝑎
𝑖
(𝑥, 𝑡, 𝑢, 𝑝) , 𝑎(𝑥, 𝑡, 𝑢, 𝑝)
are measurable
in
(𝑥, 𝑡, 𝑢, 𝑝)
, continuous in (t,u,p) and satisfy the inequalities
|𝑎
𝑖
(𝑥, 𝑡, 𝑢, 𝑝)| ≤ 𝐶(|𝑃| + |𝑈|
𝑘
) + 𝜑
1
(𝑥, 𝑡) ,
𝜑
1
∈ 𝐿
2
(𝑄
𝑇
) , 𝑖 = 1,2
(2.1)
|𝑎(𝑥, 𝑡, 𝑢, 𝑝)| ≤ 𝐶(|𝑃|
2−𝜖
+ |𝑈|
𝑘
) + 𝜑
2
(𝑥, 𝑡) , 𝜑
2
∈ 𝐿
𝑞
(𝑄
𝑇
),
(3)
where
|𝑃| = (∑
𝑝
𝑖
2
𝑚
𝑖=1
)
1
2
,
𝑘 < ∞, 𝜀 > 0, 𝑞 > 1
B
. The functions
𝑎
𝑖
(𝑥, 𝑡, 𝑢, 𝑝)
have the form:
𝑎
𝑖
(𝑥, 𝑡, 𝑢, 𝑝) = 𝑎̅
𝑖
(𝑥, 𝑡, 𝑢, 𝑝) + 𝑎̿
𝑖
(𝑥, 𝑝)
(4)
here
𝑎̅
𝑖
(𝑥, 𝑡, 𝑢, 𝑝) =
𝜕𝑎̅(𝑥,𝑡,𝑢,𝑝)
𝜕р
𝑖
,
|
𝜕𝑎̅
𝜕𝑡
| ≤ 𝐶(|𝑢|
2𝑟
+ |𝑝|
2
) + 𝜑
3
(𝑥, 𝑡) , 𝜑
3
∈ 𝐿
1
(𝑄
𝑇
)
|
𝜕𝑎̅
𝜕𝑢
| ≤ 𝐶(|𝑢|
𝑟
+ |𝑝|) + 𝜑
4
(𝑥, 𝑡) , 𝜑
4
∈ 𝐿
2
(𝑄
𝑇
)
(5)
𝑟 ≥ 0 , ∫ 𝑎̅
𝛺
(𝑥, 𝑡, 𝑢, ∇𝑢)𝑑𝑥 |
𝑡
0
≥ 0
C
. For any smooth function
𝑈(𝑥, 𝑡)
the inequality holds.
∫ 𝑎̿
𝑖
𝑄
𝑇
(𝑥, ∇𝑈)𝑈
𝑡𝑥
𝑖
dxdt ≥ ν‖∇U‖
L
2(𝛺)
2
(6)
where
ν
- positive constant.
D
. Monotonicity condition. For any functions u
, 𝑣 ∈ 𝑊
2
1
(𝛺)
(𝑎
𝑖
(𝑥, 𝑡, 𝑢, ∇𝑢) − 𝑎
𝑖
(𝑥, 𝑡, 𝑣, ∇𝑣), 𝑢
𝑥
𝑖
− 𝑣
𝑥
𝑖
)
𝛺
+
+(𝑎(𝑥, 𝑡, 𝑢, ∇𝑢) − 𝑎(𝑥, 𝑡, 𝑣, ∇𝑣), 𝑢 − 𝑣)
Ω
≥ 0
(7)
E
. At
(𝑥, 𝑡, 𝑢) ∈ {𝛺̅ × [𝑜, 𝑇] × 𝐸
1
} 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑔(𝑥, 𝑡, 𝑢)
(𝑡, 𝑢)
is continuous in
(𝑡, 𝑢)
and satisfies the inequality:
|𝑔(𝑥, 𝑡, 𝑢) − 𝑔(𝑥, 𝑡, 𝑣)| ≤ 𝑔
0
|𝑢 − 𝑣|, 𝑔(𝑥, 𝑡, 0) ∈ 𝐿
2
(𝑆
𝑇
)
(8)
Main results
. Let us construct an approximate solution according to Galerkin [13-17]. Let's take a coordinate system
from the space
𝑊
2
1
(𝛺)
. We will seek an approximate solution
𝑈(𝑥, 𝑡)
in the form
𝑈(𝑥, 𝑡) = ∑ 𝐶
𝑘
𝑛
𝑛
𝑘=1
(𝑡)𝜑
𝑘
(𝑥) (9)
where
С
𝑘
𝑛
(𝑡)
are determined from the system of ordinary differential equations
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
(𝑈
𝑡
, 𝜑
𝑗
)
𝐿̂
2
+ (𝑎
𝑖
(𝑥, 𝑡, 𝑈, ∇𝑈), 𝜑
𝑗𝑥
𝑖
)
𝛺
+ (𝑎(𝑥, 𝑡, 𝑈, ∇𝑈), 𝜑
𝑗
)
𝛺
=
= (𝑔(𝑥, 𝑡, 𝑈), 𝜑
𝑗
)
𝑆
, 𝑗 = 1, 𝑛
̅̅̅̅̅̅̅
(10)
and initial conditions
(𝑈(𝑥, 𝑂) − 𝑢
0
, 𝜑
𝑗
)
𝑊
2
1
(𝛺)
= 0
Here
𝐿̂
2
(𝛺)
–
space of functions with scalar product
(𝑢, 𝑣)
𝐿̂
2
= (𝑢, 𝑣)
𝛺
+ (𝑢, 𝑣)
𝑠
,
(𝑢, 𝑣)
𝐾
= ∫ 𝑢𝑣𝑑𝑥
𝐾
If the system {
𝜑
𝑘
} is orthonormal in the metric
𝐿̂
2
(𝛺)
, then system (10) takes the form
С̇
𝑖
𝑛
= 𝑓
𝑗
𝑛
(𝑡, 𝐶
1
𝑛
, . . , 𝐶
𝑛
𝑛
),
(11)
𝑤ℎ𝑒𝑟𝑒 𝑓
𝑗
𝑛
(𝑡, 𝐶
1
2
, . . , 𝐶
𝑛
𝑛
) = −(𝑎
𝑖
(𝑥, 𝑡, 𝑈, ∇𝑈), 𝜑
𝑗𝑥
𝑖
)
𝛺
− (𝑎(𝑥, 𝑡, 𝑈, ∇𝑈), 𝜑
𝑗
)
𝛺
+ (𝑔(𝑥, 𝑡, 𝑈), 𝜑
𝑗
)
𝑆
Theoreme
.. If conditions A-E are satisfied, then there is a unique generalized solution to problem (1) in the space
𝑊
2
1,1
̃ (𝑄
𝑇
)
.
Proof.
Condition A ensures the existence and continuity of the function
𝑓
𝑗
𝑛
(𝑡, 𝐶
1
𝑛
, . . , 𝐶
𝑛
𝑛
)
with respect to
𝑡
and
С
𝑘
𝑛
.
Therefore, for the existence of at least one solution to problem (11) on the entire interval
[𝑂, 𝑇]
, it is sufficient to
know that all possible solutions are uniformly bounded. This limitation follows from the a priori assessment
max
0≤𝑡≤𝑇
‖𝑈(𝑥, 𝑡)‖
𝐿
̂2
2
+ ‖𝑈
𝑡
(𝑥, 𝑡)‖
𝐿
2(𝑜,𝑇,𝐿
̂2)
2
+ max
0≤𝑡≤𝑇
‖∇𝑈(𝑥, 𝑡)‖
𝐿̂
2
2
≤ 𝑁
(12)
where is a constant that does not depend on
𝑛.
From here we obtain the inequality [18-19]
max
0≤𝑡≤𝑇
‖С
𝑛
(𝑡)‖
2
= max
0≤𝑡≤𝑇
‖𝑈(𝑥, 𝑡)‖
𝐿
2(𝛺)
2
≤ 𝑁, С
𝑛
= {𝐶
𝑘
𝑛
(𝑡)}
𝑘=1
𝑛
Let us now proceed to the limit transition with respect to
𝑛 → ∞
. From estimate (12) it follows that there exists a
function
𝑢(𝑥, 𝑡) ∈ 𝑊
2
1,1
̅̅̅̅̅̅̅(𝑄
𝑇
)
and a subsequence
𝑈(𝑥, 𝑡)
, such that the functions
𝑈(𝑥, 𝑡)
converge to u(x,t) weakly
in the norm
𝑊
2
1,1
̅̅̅̅̅̅̅(𝑄
𝑇
)
and the functions
𝑈
𝑡
converge to
𝑢
𝑡
in
𝐿
2
(𝑆
𝑡
)
. Since the embeddings
𝑊
2
1,1
̅̅̅̅̅̅̅(𝑄
𝑇
) ∈
𝐿
2
(𝑄
𝑡
), 𝐿
2
(𝑆
𝑡
)
are compact, then
𝑈(𝑥, 𝑡) → 𝑢(𝑥, 𝑡)
strongly in
𝐿
2
(𝑆
𝑡
)
and in
𝐿
2
(𝑄
𝑡
)
. From this convergence it
follows that
𝑈(𝑥, 𝑡)
converges to
𝑢(𝑥, 𝑡)
in
𝐿
2
(𝛺)
and in
𝐿
2
(𝑆)
for almost all
𝑡
in
[𝑂, 𝑇]
and almost everywhere in
𝑄
𝑡
𝑈𝑆
𝑡
.
Further, from condition A it follows that the functions
𝑎
𝑖
(𝑥, 𝑡, 𝑈, ∇𝑈) 𝑖 = 1,2
converge weakly in
𝐿
2
(𝑄
Т
)
and the
elements
𝐴
𝑖
(𝑥, 𝑡)
of the space
𝐿
2
(𝑄
Т
)
and the functions
𝑎(𝑥, 𝑡, 𝑈, ∇𝑈)
converge weakly
𝐴(𝑥, 𝑡) ∈ 𝐿
1
(𝑄
𝑇
)
in the
space
𝐿
1
(𝑄
𝑇
)
.
Let us denote by
𝑃
𝑙
the set of linear combinations of the form
𝑉(𝑥, 𝑡) = ∑ 𝑑
𝑘
𝑙
𝑘=1
(𝑡)𝜑
𝑘
(𝑥)
where
𝑑
𝑘
(𝑡) −
are arbitrary smooth functions on the interval
[𝑂, 𝑇]
. Multiplying relations (10) by
𝑑
𝑘
(𝑡)
, summing
over
𝑘
from
1
to
𝑙
and integrating from
0
to
𝑡
, we find that for any function
𝑉(𝑥, 𝑡) ∈ Р
е
the equality
∫ (𝑈
𝑡
, 𝑉)
𝐿̂
2
𝑡
0
𝑑𝑡 + ∫ 𝑎
𝑖
(𝑥, 𝑡, 𝑈, ∇𝑈)𝑉
𝑥
𝑖
+
𝑄
𝑡
𝑎(𝑥, 𝑡, 𝑈, ∇𝑈)𝑉]𝑑𝑥𝑑𝑡 == ∫ g(x, t, U)
S
t
𝑉𝑑𝑥𝑑𝑡
(13)
holds true.
Let's move on to the limit in n→∞.
As a result we get:
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
∫ (𝑈
𝑡
, 𝑉)
𝐿̂
2
𝑑𝑡 + ∫ [𝐴
𝑖
(𝑥, 𝑡)𝑉
𝑥
𝑖
+ 𝐴(𝑥, 𝑡)𝑉]
𝑄
𝑇
𝑑𝑥𝑑𝑡 = ∫ 𝑔(𝑥, 𝑡, 𝑢)
𝑆
𝑇
𝑉𝑑𝑥𝑑𝑡
Т
0
(14)
Since
⋃
𝑃
𝑒
∞
𝑙=1
is dense in
𝑊
2
1,0
(𝑄
𝑇
)
, then by performing the closure over
𝑉
in (14), we obtain that equality (14) is
valid for any function
𝑉 ∈ 𝑊
2
1,0
(𝑄
𝑇
)
.
From equality (14) we obtain that the function
𝑈(𝑥, 𝑡)
is the desired generalized solution.
Let's prove the uniqueness of the solution.
Let
𝑈
1
(𝑥, 𝑡), 𝑈
2
(𝑥, 𝑡)
be two solutions to problem (9), then their difference
𝑈
1
− 𝑈
2
satisfies the relation
∫
𝜕(𝑈
1
− 𝑈
2
)
𝜕𝑡
𝑄
𝑡
(𝑈
1
− 𝑈
2
)𝑑𝑥𝑑𝑡 + 𝑎
0
∫
𝜕(𝑈
1
− 𝑈
2
)
𝜕𝑡
𝑆
𝑡
(𝑈
1
− 𝑈
2
)𝑑𝑥𝑑𝑡
+ ∫ {[𝑎
𝑖
(𝑥, 𝑡, 𝑈
1
, ∇U
1
) − 𝑎
𝑖
(𝑥, 𝑡, 𝑈
2
, ∇U
2
)](𝑈
1
− 𝑈
2
)
𝑥
𝑖
𝑄
𝑇
+ [𝑎(𝑥, 𝑡, 𝑈
1
, ∇𝑈
1
) − 𝑎(𝑥, 𝑡, 𝑈
2
, ∇𝑈
2
)](𝑈
1
− 𝑈
2
)}𝑑𝑥𝑑𝑡
= ∫ [𝑔(𝑥, 𝑡, 𝑈
1
) − 𝑔(𝑥, 𝑡, 𝑈
2
)]
𝑆
𝑡
(𝑈
1
− 𝑈
2
)𝑑𝑥𝑑𝑡
Using conditions (5) and (7), we obtain
∫ (𝑈
1
− 𝑈
2
)
2
𝛺
𝑑𝑥 + 𝑎
0
∫ ∫ (𝑈
1
− 𝑈
2
)
2
𝛺
𝑑𝑥 ≤ 2 𝑔
0
𝑆
∫ ∫ (𝑈
1
− 𝑈
2
)
2
𝛺
𝑑𝑥𝑑𝑡
𝑆
𝑇
Therefore
𝑈
1
≡ 𝑈
2
. Thus, the theorem is proved.
CONCLUSION
In this paper, a generalized solution to the problem
under consideration is defined in the space
Н
1,1
̃ (𝑄
𝑇
)
when the dimension of the domain in spatial variables
is equal to two. Further, the existence and uniqueness
theorem of an approximate solution of the Bubnov-
Galerkin method for the considered non-classical
parabolic problem with a divergent principal part is
proved, when the boundary condition contains a time
derivative of the desired function.
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