SOLUTION OF PARABOLA EQUATION BY
USING REGULAR ,BOUNDARY AND CORNER
FUNCTIONS
Dr. Hayder Jabbar Abood, Dr. Iftichar Mudhar Talb
Department of Mathematics, College of Education,
Babylon University.
Abstract:we solve convergent sequence by using the parabola equation
which have a small positive parameter ξ and find a unique solution for a
given convergent sequence.
1-Introduction :
Differential equations a very important tool for solving many
phenomenas .There are many authers which studied the applied
mathematics as physical mathematics .Levenshtam [4] studied the
ordinary differenatiol equations of the first order and the first degree
.Dieudonne [1] studied the ordinary equation of the second degree with
initial and boundary conditions .Techanoff and Samarcki [5] studied only
the partial differenatiol equations of the first order and the first degree
with boundary conditions .Levenshtam [3] studied the parabola equation
(partial differential equation of the second order and the first degree )
.They put initial and boundary conditions to solve this equation .They
used converge series from uniform function with two boundary functions
.They proved the existence and uniquness of the solution .
One of the mathematical branch which take care by studying
phenomena's which comes from the the environment we live in. Several
mathematical models are formed for many researchers to solve these
phenomena's and really most of these models have a high ability to study
see (Kreysig [2] and Smith [6]) .
Some effects on the accuracy of the solution may be small and
some researchers don’t take care to study it and about the importance of
these effect may be have an effect on the result, of the phenomena's and
inversely, on the other side some researchers are emphatics that these
effects must be studied and one of them is (Dieudonne J.[1]) , we also
like him.
In this work ,we devlope the reserch of [5] ,we study the converge
series which is uniform function ,two boundary functions and two corner
functions ( see section 4 equation (6)).We prove the converge function is
unique solution of parabola equation .
2-Main problem
In this research, we study the following problem :١
ξ 2(
∂u ∂ 2 u
)=
−
∂t ∂x 2
∞
∑ξ
i =1
i
, 0<x< l , 0<t< ∞ ……… (1)
f i ( x, t , u )
∂u
∂u
………… (2)
(0, t , ξ ) =
(l, t , ξ )
∂x
∂x
where u is an arbitrary vector and ξ a very small positive parameter .We
u ( x,0, ξ ) = ϕ (x ) ,
will discuss the problem in the region ℜ = { 0 ≤ x ≤ l} × { 0 ≤ t ≤ Τ}
We will prove that the analytic convergent series
u (x,t,ξ ) =
∞
∑
i =1
ξ i u i ( x , t ) is a solution for the problem (1) with
boundary conditions and initial condition in (2).In order to prove the
existence of a solution for (1) and (2)., we find an estimation for the
convergent series. In other meaning, for construct like this convergence
and for more accurate this contains two functions:
-The first is the function, on the boundary of ℜ when t = 0, x = 0, x = l .
-The corner function at the points (0,0), (0, l ) in the region ℜ .
t
0≤ x≤l
0 ≤ t ≤ T
(l,0)
(0,0)
x
3-Some Special conditions to Solve The Main Problem
We can find the solution of our main problem (1),and by using the
following conditions :∞
Ι − The functions f ( x, t , u , ξ ) = ∑ ξ i f i ( x, t , u ) and ϕ (x) which are (n+2)
i =1
differentiable to construct a convergent series of order n and satisfy (2) at
(0, 0), ( l , 0) ∋ ϕ (0) = ϕ (l) = 0 .
From (1) and when ξ = 0 we get the following equation
٢
f(x, t, u ,0) = 0 ………
…. (3)
ΙΙ − Equation(3) at the region ℜ has an arbitrary solution assume it
u 0 = u 0 ( x, t )
ΙΙΙ − When x is a parameter (0 ≤ x ≤ l ) then:dB0
= f ( x,0, u 0 ( x,0 ) + B0 ,0 )
, τ >0
dτ
………………….. (4)
Ιν − By using of the conditions in (2), the solution of equation (4)
becomes
B0 ( x,0) = ϕ ( x ) − u 0 ( x,0)
…………………....………. (5)
4The convergent series which is analytic with respect to ξ to solve (1) and
(2) it give as follow:u ( x, t , ξ ) = u ( x, t , ξ ) + B( x,τ , ξ ) + Q(ζ , t , ξ ) + Q ∗ ζ ∗ , t , ξ + P(ζ ,τ , ξ ) + p ∗ (ζ ∗ ,τ , ξ ) .
(6)
(
Such that τ =
t
ξ
2
,ζ =
x
ξ
,ζ ∗ =
)
l−x
ξ
Where u is the regular function Q, Q ∗ , B boundary functions and p,
p ∗ the corner functions which are series raised to powers with respect to ξ ,
for example
∞
u ( x, t , ξ ) = ∑ ξ i u i ( x , t ) .
i =0
To find the coefficients of these series ,we put (6) in (1), (2) and at
the function f(x, t, u, ξ ) as follows:-
(
∞
)
(
f ( x, t , u , ξ ) = ∑ ξ i f i ( x, t , u i ( x, t ) + Bi (x,τ ) + Qi (ζ , t ) + Q ∗ ζ ∗ , t + Pi (ζ ,τ ) + Pi ζ ∗ ,τ
∗
)
i =0
we can write it as below
f = f + Bf + Qf + Q ∗ f + Pf + P ∗ f
f ( x, t , ξ ) = f ( x, t , u (x, t , ξ ), ξ ) ;
such that
Bf ( x,τ , ξ ) = f (x, ξ 2τ , u (x, ξ 2τ , ξ ) + B( x,τ , ξ ), ξ ) _ f ( x, ξ 2τ , ξ ) ;
Qf (ζ , t , ξ ) = f (ξζ , t , u (ξζ , t , ξ ) + Q(ζ , t , ξ ), ξ ) − f (ξζ , t , ξ ) ;
Pf (ζ , t , ξ ) = f (ξζ , ξ 2τ , u (ξζ , ξ 2τ , ξ ) + B(ξζ ,τ , ξ ) + Q (ζ , ξ 2τ , ξ ) + P(ζ ,τ , ξ ), ξ )
(
) (
− Bf (ξζ ,τ , ξ ) − Qf ζ , ξ 2τ , ξ + f ζ , ξ 2τ , ξ
)
;
By the same way , we can define Q f , P ∗ f .
By the standard procedures for the analysis of series with respect to ξ
powers and by putting the coefficients for the equal powers with respect
to ξ ,we get an equation for any convergent series, for example when ξ Ο :
∗
٣
(a) the function u 0 has The following equation:f ( x, t , u 0 ,0) = 0
Which can be found by the correspondence with the equation (3) ,thus
we get u 0 = u 0 (x, t ) and for the functions u i (x, t ) , i ≥ 1 .
We have the following linear equations f u (x, t )u i (x, t ) = f i (x, t ) where f i (x, t )
represented by u j (x, t ) , j< i and from this
u i ( x, t ) = f u
−1
( x , t ) f i ( x, t ) .
(b) Equation of the function B0 (x,τ ) is
∂B0
= B0 f ≡ f ( x,0, u 0 ( x,0 ) + B0 (x,τ ) − f ( x,0, u 0 ( x,0 ),0 )) ,this correspond with
∂τ
the equation (4) added to the boundary condition in (5) from the
conditions (I – IV), the function B0 (x,τ ) have the following expontioal
estimation (see [1] page 57).
B0 ( x, τ ) ≤ C exp(−bτ ) , 0 ≤ x ≤ l , τ > 0 ……………………… (7)
Where b>0, c>0 are arbitrary constants .The functions Bi (x,τ )
when i ≥ 1 is linear with the formula
∂Bi
= f u ( x,0, u 0 ( x,0 ) + B0 ( x,τ ),0 )B j + π i ( x,τ )
∂τ
We define B j (x,0) = −ui (x,0 ) and π i ( x,τ ) represented by
B j ( x,τ ) where
j<i .
The solution for these linear equations which possesses the expontioal
estimation is as in (7).
5- The Main Resaults
We find the function Q0 (ζ , t ) ,where t is as a parameter from the
following equation
−
∂ 2ϕ 0
∂ζ 2
= Q0 f ≡ f (0, t , u 0 (0, t ) + Q0 (ζ , t ),0 ) − f (0, t , u 0 (0, t ),0 )
From the equation (2) for Qi (ζ , t ) ,we get the two conditions
− ∂u i −1
∂ϕ
∂ϕ 0
(0, t ) = 0 , i (0, t ) =
,
∂x
∂ζ
∂ζ
i
≥
……………………………. (8)
1
The boundary condition in (2) gives Qi (ξ ,t ) → 0 when ζ → ∞ , i ≥ 0 so
Q0 (ζ , T ) ≡ 0 while the functions Qi (ζ , t ) , i ≥ 1 are defined from the linear
equations ,( ζ constant coefficient )
-
∂ 2 Qi
∂ζ 2
= f u (0, t )Qi + qi (ζ , t )
…………………………….. (9)
٤
We can find qi (ζ , t ) through the functions Q j (ζ , t ) , j < i The solutions of
(8) and (9) and from condition III with the boundary condition ,we get
the single values which have the following expontioal estimation
Qi (ζ , t ) ≤ C exp(− bζ ) , ζ ≥ 0 , 0 ≤ t ≤ T …………………………….(10)
The function P(ζ ,τ , ξ ) has a role to make the function B active with
the boundary conditions and the function Q with its initial condition
also, the linear differential equation for P0 (ζ ,τ ) is
∂p0 ∂ 2 p0
−
= p 0 f ≡ f (0,0, u 0 (0,0 ) + B0 (0,τ ) + P0 (ζ ,τ ),0 ) − f (0,0, u 0 ) + B0 (0,τ ),0),
∂τ
∂ζ 2
ζ >0 , τ >0
By substitution (6) in the conditions of (2) and for the function Pi (ζ ,τ ) we
get
∂p 0
(0 , τ ) = 0 …………………… (11)
∂ζ
∂p
∂B
pi (ζ ,0) = −Qi (ζ ,0 ), i (0,τ ) = − i −1 (0, t ), i ≥ 1 ……………… (12)
∂ζ
∂x
p 0 (ζ , 0 ) = 0 ,
From the conditions (11) and (12), the function p is boundary function for
both variables ,that is pi (ζ ,τ ) → 0 when ζ 2 + τ 2 → ∞ …………. (13)
At it p0 (ζ ,τ ) ≡ 0 ,while the functions pi (ζ ,τ ) can be defined from the
linear parabola equations
∂pi ∂ 2 pi
−
− A(τ ) pi ≡ H i (ζ ,τ ) …………… (14)
∂τ ∂ζ 2
where A(τ ) = Pu (0,0, u 0 (0,0) + B0 (0,τ ),0) .
The function H i (ζ ,τ ) represents the functions Qi , B j , j ≤ i and
p j , i > j . The solutions of (12) and (14) can be defined by (Techanoff ,
Samarcki [4])
τ
∞
0
0
pi (ζ ,τ ) = g (ζ ,τ ) + ∫ dτ 0 ∫ G (ζ ,τ , ζ 0 ,τ 0 )h(ζ 0 ,τ 0 )dζ 0 ;
here g (ζ ,τ ) is an arbitrary function which is differentiable (smooth)
satisfy the conditions (12) and (13)
h(ζ ,τ ) = H i (ζ ,τ ) −
∂g ∂ 2 g
+ A(τ )g
+
∂τ ∂ζ 2
,
G (ζ ,τ , ζ 0 ,τ 0 ) _ (Greena function),
G (ζ ,τ , ζ 0 ,τ 0 ) = φ (τ )φ
−1
⎧⎪ ⎡ − (ζ − ζ )2 ⎤
⎡ − (ζ + ζ )2 ⎤ ⎫⎪
0
0
(τ 0 )
⎥⎬
⎥ + exp ⎢
⎨exp ⎢
(
)
(
4
a
τ
τ
4
a
τ
τ
−
−
2 πa(τ − τ 0 ) ⎪⎩ ⎢⎣
⎢⎣
0 ⎥⎦
0 ) ⎥⎦ ⎪
⎭
1
, where φ (τ ) is fundamental matrix have the following expontioal
estimation [1]:
٥
φ (τ )φ −1 (τ 0 ) ≤ C exp[− b(τ − τ 0 )]
, thus the expontioal estimation for the
function p becomes :-
p i (ζ , τ ) ≤ C exp[− C (ζ + τ )], ζ ≥ 0, τ ≥ 0 ………………………. (15)
To find the estimation of the functions P ∗ , Q ∗ by same method used for
P and Q and having the same expontioal estimation as in (10) and (15)
U n represent to the convergent series of order n for the series(6)
Un =
n
[
( )]
( )
∑ ξ i ui (x, t ) + Bi (x,τ ) + Qi (ζ , t ) + Qi ∗ ζ ∗ , t + Pi (ζ , t ) + Pi ∗ ζ ∗ ,τ
i =0
Therefore the following statement is valid.
Theorem (5-1):
If the conditions (I – IV) are valid then u (x,t , ξ ), (ξ small parameter) have
a unique solution to the problems (1) ,(2) and for the uniform convergent
series U n in R converges to the estimation Ο(ξ n+1 ) i.e.
max u − U
n
(
= Ο ξ n +1
)
R
Proof:Let w= u – U
U = Un +ξ
(Q
;
)
, where U n the partial convergent
series defined in (6). Thus we have the following equation:n +1
n +1
+ Qn∗+1 + Pn +1 + Pn∗+1
⎛ ∂w ∂ 2 w ⎞
− 2 ⎟⎟ − f u ( x, t , ξ )w = h(w, x, t , ξ ) ,………………………… (17)
⎝ ∂t ∂x ⎠
ξ 2 ⎜⎜
f u ( x, t , ξ ) = f u (u 0 ( x, t ) + B0 ( x, t / ζ ), x, t ,0)
⎛ ∂u ∂ 2 u ⎞
h(w, x, t , ξ ) = f (U + w, x, t , ξ ) − ξ 2 ⎜⎜ − 2 ⎟⎟ − f u ( x, t , ξ )w
⎝ ∂t ∂x ⎠
,
where
The function h(w, x, t , ξ ) have the following two properties:1- when w = 0 ,it is clear that the equation
( )
⎛ ∂U ∂ 2U ⎞
⎟ = Ο ξ n +1
h(0, x, t , ξ ) = f (u, x, t , ξ ) − ξ 2 ⎜
−
2 ⎟
⎜ ∂t
∂x ⎠
⎝
w i ( x , t , ξ ) ≤ c 1ξ , i = 1, 2 then there exist c 2 > 0, ξ 0 > 0 such that
2- If
0 < ξ ≤ ξ 0 satisfy the following inequality
sup h(w , x, t , ξ ) − h(w , x, t , ξ ) ≤ C sup w
2
1
2
R
2
− w1 .
R
By using (Greena matrix), equation (1) with conditions (2) ,(3)
becomes
t l
w( x, t , ξ ) = ∫ ∫ G ( x, t , x0 , t 0 , ζ )h( w( x0 , t 0 , ζ ), x0 , t 0 , ζ )d x0 d t 0 ≡ L( w, x, t , ζ )
0 0
……(18)
٦
From condition III, Greena matrix have the estimation (Techanoff
, Samarcki [4]):
// G (x, t , x0 , t 0 , ξ ) // ≤
c
1
ξ2
t − t0
exp[−c
t − t0
ξ2
] exp[−c
(x − x0 )2
t − t0
].
By applying the theorem of the convergence of sequences for the
equation (18) ,we get w0 = 0, wi +1 = L(wi , x, t , ξ ), i = 1,2,.....
From above, we conclued that w(, x, t , ξ ) is a unique solution for (2)
when ξ is a small parameter and have the estimation
n +1
max w ( x , t , ξ ) = Ο ( ξ ) , thus we get
R
max
u − U
n
= Ο (ξ
n +1
) .
R
References:-
[1].Dieudonne J.(1984). Foundations of Modern Analysis.
Naok.Moscow .
[2] .Kreysig E. (1988).Introductory Functional
Analysis with
Applications.John Wiley and Sons . New York .
[3] .Levenshtam V.B. ,Abood H.D.(2003). Asymptotic Integration of
The Problem on Heat Distribution in a Thin Rod with Rapidly
Varying Sources of Heat //contemporary mathematics and its
applications . vol.7. 137-149.
[4] .Levenshtam V.B. Asymptotic Integration for Differential
Equations with Quick Oscillatory Composed .Math. Cb. 2003 .T .81 N
:1357
[5]. Techanoff A.N. Samarcki A.A.(1988). Mathematical-Physicist
Equation Naok Moscow .
[6] .Smith D.R. (1985).Singular –l'erturbatin Theory .an Introduction
with Application. Cambridge Univ .press ,Cambridge,
-:‫اﻟﺨﻼﺻﺔ‬
‫ﻓ ﻲ ه ﺬا اﻟﺒﺤ ﺚ ﻧﺤ ﻞ ﻣﺘﺘﺎﺑﻌ ﺔ ﻣﺘﻘﺎرﺑ ﺔ ﺑﺎﺳ ﺘﺨﺪام ﻣﻌﺎدﻟ ﺔ ﻗﻄ ﻊ ﻣﻜ ﺎﻓﺊ واﻟﺘ ﻲ ﺗﻤﺘﻠ ﻚ ﻣﻌﻠﻤ ﺔ‬
.‫ﺻﻐﻴﺮة وإﻳﺠﺎد ﺣﻞ وﺣﻴﺪ ﻟﺘﻠﻚ اﻟﻤﺘﺘﺎﺑﻌﺔ اﻟﻤﺘﻘﺎرﺑﺔ‬
٧