approximate solution of a dirichlet problem for …solution for a dirichlet boundary value problem...
TRANSCRIPT
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IJRRAS 31 (1) ● April 2017 www.arpapress.com/Volumes/Vol31Issue1/IJRRAS_31_1_04.pdf
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APPROXIMATE SOLUTION OF A DIRICHLET PROBLEM FOR
GENERAL SECOND ORDER ELLIPTICAL LINEAR PDEs WITH
CONSTANT COEFFICIENTS IN THE UNIT DISK OF ℝ𝟐
Tchalla Ayekotan Messan Joseph, Djibibe Moussa Zakari & Tcharie Kokou University of Lome, Departement of Mathematics, , 01 PO BOX: 1515 Lome 01-Togo
Tel: 0022890288327, 0022899869119
E.mail: [email protected], [email protected], [email protected]
ABSTRACT
In this paper, we give, in each fix point of unit disk of the space ℝ𝟐, a generalized analytic approximate solution of a Dirichlet problem for general second order elliptical partial differential equation with constant coefficients. This
approximate solution is constructed by using Bubnov-Galerkin method. The present work is the prolongation of the
work published in [1] and [2].
Keywords: Elliptic equation, Dirichlet problem, Green’s fonction, Bubnov Galerkin method, approximate solution.
1. INTRODUCTION
For ordinary differential equation (EDO) or partial differential equation (PDE) which has no analytic solution (or such
a solution has not yet been found), it is often possible to develop approximate methods for finding analytical
approximate solution or to establish error estimate of the problem. In the work [1] and [2] we establish pointwise error
estimate for a Dirichlet boundary value problem for a general partial linear elliptic equation of the second order with
constant coefficients. The idea of the method used to estimate this error is based on a proposal which is originally
develops by N. J. Lehmann in [3] for ODEs. Then this idea was used in the works [4] and [5] respectively to establish
error estimate of a Dirichlet boundary value problem for Schrodinger’s steady state equation.
In This paper, according to the results we obtain in [1] and [2], we construct a sequence of analytic approximate
solution for a Dirichlet boundary value problem for a general partial linear elliptic equation of the second order with
constant coefficients. Indeed, the results of this paper, which are found mainly in Theorems 4.2, 5.3, 5.4 rely on
theorems 4.1 obtain in [1] and 5.1 and 5.2 obtain in [2].
The remainder of this paper is organised as follows. After this introduction, in section 2, we state the problem, in
section 3, we present some preliminaries and basic definitions. In section 4 and 5, we have done, in three different
cases, the approximate study of the problem state, by using the Bubnov-Galerkin method to construct a generalized
analytic approximate solution nu which converges, in each fix point ),( yx of unit disk , to the generalized
solution u . In section 5, we have showed that the analytic approximate solution nu depend on the conformal map
which transforms the interior of an ellipse to the unit disk . Hence, in section 6, we establish an algorithm in computer system Mathematica, to define an approximate conformal map which transforms the interior of an ellipse to
the unit disk.
2. THE STATE OF THE PROBLEM
Let consider the general linear second order equation:
),(=2=)( 02
22
2
2
yxfuay
ug
x
ud
y
uc
yx
ub
x
uauL
(GE)
where gdcba ,,,, and 0a are real constants and f a real function. In this paper, we deal with this work when
equation (GE) is of elliptic type; it means when 0<2 acb and (0,0,0).),,( cba Let consider the
homogeneous Dirichlet’s problem of the equation (GE) in the unit disk Ω = {(𝑥, 𝑦) ∈ ℝ2: 𝑥2 + 𝑦2 < 1}.. The goal of this problem is to determine the function u which satisfies the equation (GE) in the domain under the boundary condition
0.=| u (BCs)
For ),( yx , by setting
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
26
,)2(
exp,=),(2
acb
yagbdxbgcdyxvyxu (2.1)
the Dirichlet boundary value problem (GE),(BCs) becomes:
),,(=22
22
2
2
yxRvy
vc
yx
vb
x
va
(GE')
0,=| v (BC's)
).,(
)2(exp=),(and
)4(
2)(4=where
22
222
0 yxfacb
yagbdxbgcdyxR
acb
agbdgcdacba
3. PRELIMINARIES AND SOME BASIC DEFINITION
Next designations are used in the present work:
is bounded domain in ℝ𝟐, is the boundary of the domain .
)(2 L is the Hilbert space of square integrable reals functions (in the sense of Lebesgue) on . Scalar product on
)(2 L will be designate by )(2(.,.) L and the norm by
∥ 𝑢 ∥𝐿2(Ω)= (∫Ω |𝑢(𝑥, 𝑦)|2𝑑𝑥𝑑𝑦)
1
2, ∀𝑢 ∈ 𝐿2(Ω).
)(12 W is the Hilbert space, consisting of the elements of )(2 L having generalized derivatives of first order
which are square summable on . The scalar product in this space is defined by
dxdyy
v
y
u
x
v
x
uyxvyxuvu
),(),(=),( 2,1
and the norm by ∥ 𝑢 ∥2,1= √(𝑢, 𝑢)2,1.
)(cC is the set of infinitely differentiable functions with compact support laying in .
)(12 W is the closure of )(
cC in the space )(1
2 W .
)(22 W is the Hilbert space consisting of elements of )(2 L having first and second order generalized derivatives
in )(2 L . Its scalar product is defined by:
dxdyy
v
y
u
x
v
x
u
y
v
y
u
x
v
x
uyxvyxuvu
2
2
2
2
2
2
2
2
2,2 ),(),(=),(
and the norm is given by: ∥ 𝑢 ∥2,2= √(𝑢, 𝑢)2,2.
)(22,0 W is the closure, in )(2
2 W , of functions of )(2 C which vanish on .
)()(=)( 221
2
2
2,0 WWW if 2C .
Definition 1 (Classical solution) A classical solution u of the problem (GE),(BCs) is a function u from
)()(2 CC which satisfies the problem (GE),(BCs).
Definition 2 (Generalized Solution) A function u from )(12 W is called generalized solution of the problem
(GE),(BCs) if it satisfies the integral identity
dxdyuay
ug
x
ud
yy
uc
xy
u
yx
ub
xx
uauL
0),(
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
27
dxdyf = (3.1)
for all )(12 W . Any generalized solution of the problem (GE),(BCs) belongs to the space )(2
2,0 W and
hence, is a continuous function in .
Definition 3 (Characteristic equation of (GC), (BCs) ). We call characteristic equation of (GE),(BCs), the equation
given by 0;=)(det EA E is the unit matrix of ℝ2,, the unknown variable of the equation and
.=
cb
baA
In the extended form, the characteristic equation is given by
0.=)( 22 bacca (CE)
It’s well known that the solutions of (CE) are eigenvalues of matrix A and are given by
.2
))(4(=;
2
))(4(=
22
2
22
1
cabcacabca
4. APPROXIMATE STUDY OF THE PROBLEM (GE),(BCs) IN THE CASE OF EQUALITY
SOLUTIONS 1 , 2 OF THE CHARACTERISTIC EQUATION (CE) In this section we shall establish error estimate of generalized approximate solution of the problem (GE),(BCs) and
construct, by using Bubnov-Galerkin method, analytic approximate solution of this problem when the solutions 1
and 2 of the characteristic equation (CE) are equal.
4.1 Error Estimate of Approximate Solution of Problem (GE),(BCs) When 1 = 2
Let’s notice that 21 = if, and only if, 0=b and ca = . In this case, the Dirichlet problem (GE'),(BC's) takes the form
),(2
exp1
=44
=)(2
2
2
2
00 yxf
a
ygxd
av
a
g
a
d
a
avvL
(4.1)
0.=| v (4.2)
Theorem 4.1 Let’s assume 0=b , ca = and ).(2 Lf If the number
2
2
2
2
0
44 a
g
a
d
a
a does not
belong to the spectrum of the problem
vv = (4.3) 0,=| v (4.4)
then the unique generalized solution u of the problem (GE),(BCs) belongs to the space )(2
2,0 W and for any
function *u belong to )(
2
2,0 W , for any ),( yx , the following posterori estimation is satisfied:
|𝑢(𝑥, 𝑦) − 𝑢∗(𝑥, 𝑦)| ≤ exp (−𝑥𝑑+𝑦𝑔
2𝑎) ∥ 𝐺𝐿0(𝑥, 𝑦, . ) ∥𝐿2(Ω)×
√∫Ω|1
𝑎exp (
𝜉𝑑+𝜂𝑔
2𝑎) 𝑓(𝜉, 𝜂) − 𝐿0(𝑣∗)(𝜉, 𝜂)|
2
𝑑𝜉𝑑𝜂 (4.5)
where .2
exp),(=),(0
** problemtheoffunctionsnGreetheisGanda
ygxdyxuyxv L
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
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Proof. Let’s set .),(),,(2
exp1
=),(
yxyxf
a
ygxd
ayxF
If the real
2
2
2
2
0
44 a
g
a
d
a
a does not belong to the spectrum of the problem (4.3),(4.4), then the
operator 0L defined a homeomorphism from the space )(2
2,0 W to the space )(2 L . Hence the unique
generalized solution v of problem (4.1),(4.2) belongs to )(2
2,0 W and we have
ddFyxGyxFLyxvyx L ),(),,,(=),)((=),(,),(0
1
0
where 1
0
L is the inverse of the operator 0L . ),(),(2
2,0
* yxWv , we have
.)],)((),()[,,,(=),(),(*
00
* ddvLFyxGyxvyxv L
In particular, for
a
ygxdyxuyxv
2exp),(=),( ** with )(
2
2,0
* Wu ; we get
.)],)((),()[,,,(2
exp=),(),( *00
* ddvLFyxGa
ygxdyxuyxu L
Since then, By using Hölder inequality we have:
|𝑢(𝑥, 𝑦) − 𝑢∗(𝑥, 𝑦)| ≤ exp (−𝑥𝑑+𝑦𝑔
2𝑎) ∥ 𝐺𝐿0(𝑥, 𝑦, . , . ) ∥𝐿2(Ω)
×√∫Ω|1
𝑎exp (
𝜉𝑑+𝜂𝑔
2𝑎) 𝑓(𝜉, 𝜂) − 𝐿0(𝑣∗)(𝜉, 𝜂)|
2
𝑑𝜉𝑑𝜂.
Finally we have the result of theorem 4.1.
4.2 Analytic Approximate Solution of the Problem )(,)( BCsGE When 1 = 2
Here we shall construct a sequence of approximate solution nu of )(2
2,0 W which converges to the unique
generalized solution u of )(),( BCsGE when the eigenvalues 1 and 2 are equal. According to the theorem
4.1, it follows that we can get an approximate solution of the problem )(),( BCsGE from an approximate solution
of the boundary value problem (4.1),(4.2). Let ),,,(, yxG be the Green’s function of homogeneous Dirichlet
problem for Poisson’s equation on the unit disk . Then the problem (4.1),(4.2) is equivalent to the integral Fredholm equation of the second kind:
,),(),,,(=),(),,,(||),( ,, ddFyxGddvyxGyxv
(4.6)
with .44
=2
2
2
2
0
a
g
a
d
a
a In this part we suppose that 𝜇 ≤ 0. Then the equation (4.6) is uniquely solvable
because || is not a characteristic number of this equation. Let’s find a sequence of approximate solutions of
integral equation (4.6) by using the Bubnov-Galerkin method. It’s well known that the space )(2 L on the unit disk
is equipped with a complet orthonormal set of functions }{ n which vanishing on the boundary of .
These functions ......,,, 21 n are generalized eigenfunctions of the Laplace operator which vanishing on .
In polar coordinates, they are expressed by means of cylindrical Bessel functions. For all n , n is the corresponding
eigenvalue of n and we have ...>>...>>>0 21 n Let’s find this approximate solution of the equation
(4.6) in the form:
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
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),,(),(=),(1=
yxAyxSyxv jjj
n
j
n (4.7)
.determined be totscoefficien reals areand),(),,,(=),(where jAddFyxGyxS
,),(),,,(||),(=),)((by defineoperator thesettingBy ddvyxGyxvyxvTT
we obtain, for all ),( yx , the residual
),(),(||||=),)((),)((=),( 11=
yxSyxAyxvTyxvTyx jjj
n
j
nn (4.8)
with
.),(),,,(|=|),(1 ddSyxGyxS
According to the Bubnov-Galerkin method, the required coefficients jA are determined, for all 𝑛 ≥ 1, from the
orthogonality condition of the residual n with the functions n ...,,, 21 in the space )(2 L :
.1,2,...,=0,=),(),( nkdxdyyxyx kn (4.9)
By using the expression (4.8), we get the coefficients kA from the condition (11) by the formula
.),(),(=where1,2,...,=||||
= 1 dxdyyxSyxnkforA kkk
kk
Thus the sequence }{ nv of functions of )(2
2,0 W defined for all ),( yx in the unit disk by
),(||||
),(=),(1=
yxyxSyxv jj
jjn
j
n
is an approximate solution of equation (4.6) in )(2 L . It’s also an approximate solution of the problem (4.1),(4.2)
in )(2 L because the equation (4.6) is equivalent to (4.1),(4.2). Hence we obtain the following theorem:
Theorem 4.2 Let’s 0=b , ca = and )(2 Lf in (GE),(BCs) and 𝜇 = (𝑎0
𝑎−
𝑑2
4𝑎2−
𝑔2
4𝑎2) ≤ 0. For
all ,*Nn for all ),( yx belongs to the unit disk , let nu defined by
),(2
exp=),( yxva
ygxdyxu nn
where
.),(2
exp1
),,,(=),(,),(||||
),(=),( ,1=
ddf
a
gd
ayxGyxSyxyxSyxv j
j
jjn
j
n
Then nu is a sequence of functions of )(2
2,0 W which converge in each fix point ),( yx of the unit disk to
the unique generalized solution u of the problem (GE),(BCs).
Proof. The result of this theorem is followed from the inequality (4.5) of theorem 4.1. According to this inequality,
we have, ∀𝑛 ≥ 1, ∀(𝑥, 𝑦) ∈ Ω,
|𝑢(𝑥, 𝑦) − 𝑢𝑛(𝑥, 𝑦)| ≤ exp (−𝑥𝑑+𝑦𝑔
2𝑎) ∥ 𝐺𝐿0(𝑥, 𝑦, . ) ∥𝐿2(Ω)×
√∫Ω|1
𝑎exp (
𝜉𝑑+𝜂𝑔
2𝑎) 𝑓(𝜉, 𝜂) − 𝐿0(𝑣𝑛)(𝜉, 𝜂)|
2
𝑑𝜉𝑑𝜂.
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
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So we have the result because
.as0),)((),(2
exp1
2
0
nddvLfa
gd
an
5. APPROXIMATE STUDY OF THE PROBLEM (GE),(BCs) IN THE CASE OF DISTINCT SOLUTIONS
1 , 2 OF THE CHARACTERISTIC EQUATION (CE)
5.1 Error Estimate of Approximate Solution of the Problem (GE),(BCs) when 21 with 0=b
With the condition 0=b , the equation (GE'),(BC's) becomes
),(2
exp=44
=)(22
02
2
2
2
0 yxfac
agycdxv
c
g
a
da
y
vc
x
vavL
(5.1)
0=| v (5.2)
Hence the solutions of the characteristic equation (CE) of the elliptic problem (GE),(BCs) are reals a and c . So
without lost the generality, suppose that 0>a and 0.>c Let’s apply in the problem (5.1),(5.2), the change of
function
c
y
a
xwyxv ,=),( with independent variables
a
x= and
c
y= . So we get the following
Dirichlet problem
,),(),,(2
exp=44
=22
00 Dcafac
cagacdw
c
g
a
dawwL
(5.3)
),[0,20,=)(sin1
),(cos1
caw (5.4)
.1<11
:),(=where2
2
2
22
ca
D
R
Theorem 5.1 Let’s assume 0=b , 0>0,> ca and )(2 Lf in problem (GE),(BCs). If the number
c
g
a
da
44
22
0 does not belong to the spectrum of the problem
ww = (5.5)
),[0,20,=)(sin1
),(cos1
caw (5.6)
then the unique generalized solution u of the problem (GE),(BCs) belongs to space )(2
2,0 W and for any function
*u belongs to )(2
2,0 W , for any point ),( yx of the unit disk , the following posteriori error estimate is
realized:
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
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|𝑢(𝑥, 𝑦) − 𝑢∗(𝑥, 𝑦)| ≤ exp (−𝑐𝑑𝑥+𝑎𝑔𝑦
2𝑎𝑐)‖𝐺𝐿′0 (
𝑥
√𝑎,𝑦
√𝑐, . , . )‖
𝐿2(𝐷)×
√(∫𝐷|exp (
𝑐𝑑√𝑎𝜏+𝑎𝑔√𝑐𝜎
2𝑎𝑐) 𝑓(√𝑎𝜏, √𝑐𝜎) − 𝐿′0(𝑤∗)(𝜏, 𝜎)|
2
𝑑𝜏𝑑𝜎) ,
(5.7)
where 0
LG is the Green’s function of the problem (5.3),(5.4) on the domain D , and *w a function of )(
2
2,0 DW
given by
ac
cagacdcauw
2exp),(=),( **
for all .),( D
Proof. The proof of this theorem is similar to that of the theorem 4.1. But it can be found in [2].
5.2 Error Estimate of Approximate Solution of the Problem (GE),(BCs) when 21 with 0.b
As the equation (GE) is elliptic, the solutions 21 and of characteristic equation (CE) satisfy 0>21 and
according to the Viet theorem 2
21 = bac . Without lost the generality, suppose that 0>> 21 .
Let’s transform the equation (GE') to the canonical form. Let );(= 211 ppP and ),(= 212 ppP be, according to
the usual norm of 2R , the normalized eigenvectors corresponding respectively to the eigenvalues 1 and 2 of
the symetric matrix
cb
baA = deduced from (GE). By simple calculations, we have
𝑝1 =1
√1+
((𝑐−𝑎)+√4𝑏2+(𝑎−𝑐)2)
2
4𝑏2
𝑎𝑛𝑑 𝑝2 =(𝑐−𝑎)+√4𝑏2+(𝑎−𝑐)2
2𝑏√1+
((𝑐−𝑎)+√4𝑏2+(𝑎−𝑐)2)
2
4𝑏2
;
𝑝′1 =1
√1+
((𝑐−𝑎)−√4𝑏2+(𝑎−𝑐)2)²
4𝑏2
𝑎𝑛𝑑 𝑝′2 =(𝑐−𝑎)−√4𝑏2+(𝑎−𝑐)2
2𝑏√1+
((𝑐−𝑎)−√4𝑏2+(𝑎−𝑐)2)²
4𝑏2
.
Let’s make a change of function of independent variables in the equation (GE') in the form
),,(=),( yxvw (5.8)
.=and=where 2121 ypxpypxp It means
2
22
21
22
1 2
))(4)((=,
2
))(4)((=
bk
ycabac
k
x
bk
ycabac
k
x
.
4
)(4)(1=,
4
)(4)(1=where
2
222
22
222
1b
caback
b
caback
Then the problem (GE'),(BC's) becomes
𝜆1∂2𝑤(𝜉,𝜂)
∂𝜉2+ 𝜆2
∂2𝑤(𝜉,𝜂)
∂𝜂2+ (
4𝑎0(𝑏2−𝑎𝑐)+𝑐𝑑2−2𝑏𝑑𝑔+𝑎𝑔2
4(𝑏2−𝑎𝑐))𝑤(𝜉, 𝜂) = 𝒢(𝜉, 𝜂), (5.9)
0,=|w (5.10)
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
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Where 𝒢(𝜉, 𝜂) = exp [𝜆2[𝑑𝑝1+𝑔𝑝2]𝜉+𝜆1[𝑑𝑝′1+𝑔𝑝′2]𝜂
2(𝑎𝑐−𝑏2)] 𝑓(𝑝1𝜉 + 𝑝′1𝜂, 𝑝2𝜉 + 𝑝′2𝜂).
Now we shall transform the equation (5.9) to the canonical form. For this purpose let’s introduce a new unknown
function which is given by the formula ),,(=),( w (5.11)
.
)(4
2==and
)(4
2==where
222
221 cabcacabca
Thus, from the problem (GE'),(BC's), the Dirichlet problem (GE),(BCs) is reduced to the equivalent boundary value
problem
𝐿′′0(𝜅) = Δ𝜅 + (4𝑎0(𝑏
2−𝑎𝑐)+𝑐𝑑2−2𝑏𝑑𝑔+𝑎𝑔2
4(𝑏2−𝑎𝑐)) 𝜅 = ℋ(𝛼, 𝛽), (𝛼, 𝛽) ∈ 𝐷, (5.12)
𝜅 (√2cos𝜃
√𝑎+𝑐+√4𝑏2+(𝑎−𝑐)2,
√2sin𝜃
√𝑎+𝑐−√4𝑏2+(𝑎−𝑐)2) = 0, 0 ≤ 𝜃 < 2𝜋, (5.13)
where 𝐷 =
{
(𝛼, 𝛽) ∈ ℝ2:𝛼2
(1
√𝜆1)
2 +𝛽2
(1
√𝜆2)
2 < 1
}
and ℋ(𝛼, 𝛽) = 𝒢(√𝜆1𝛼,√𝜆2𝛽).
Theorem 5.2 If
)4(
2)(42
222
0
acb
agbdgcdacba does not belongs to the spectrum of the spectral
problem
,),(),,(=),( D (5.14)
𝜅 (√2cos𝜃
√𝑎+𝑐+√4𝑏2+(𝑎−𝑐)2,
√2sin𝜃
√𝑎+𝑐−√4𝑏2+(𝑎−𝑐)2) = 0, 0 ≤ 𝜃 < 2𝜋, (5.15)
and )(2 Lf in the problem (GE),(BCs) then for any function *u of )(
2
2,0 W , for any fix point ),( yx of
the unit disk , the following posteriori error estimate is realized
|𝑢(𝑥, 𝑦) − 𝑢∗(𝑥, 𝑦)| ≤ exp [(𝑐𝑑−𝑏𝑔)𝑥−(𝑏𝑑−𝑎𝑔)𝑦
2(𝑏2−𝑎𝑐)] ∥ 𝐺𝐿′′0(𝛼, 𝛽, . , . ) ∥𝐿2(𝐷)×
√∫𝐷|ℋ(𝜎, 𝜏) − 𝐿′′0(𝜅∗)(𝜎, 𝜏)|2𝑑𝜎𝑑𝜏,
(5.16)
where u is the unique generalized solution of the problem (GE),(BCs), * is a function of )(22,0 DW such that for
),( in ,D
,)2(
exp,=,2
21212112
22211211
**
bac
pgpdgpdpppppu
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
33
,
)(4)(4
)(42
)(4)(2=
)(=
22
22
22
22
1
21
cabcaccaba
cabb
ycabacbxypxp
.
)(4)(4
)(42
)(4)(2=
)(=
22
22
22
22
2
21
cabcaccaba
cabb
ycabacbxypxp
Proof. The proof of this theorem is similar to the proof of theorem 4.1 of this work.
5.3 Analytic Approximate Solution of the Problem )(),( BCsGE When 21
Here we shall give an approximate solution of the problem )(),( BCsGE when the solutions of its characteristic
equation are distinct. According to the work did in the section 4 (see theorems (5.1) and (5.2) , it follows that the
approximate solution of the problem )(),( BCsGE is obtained from an approximate solution of a boundary value
problem of the form
ℒ0𝜗 = Δ𝜗(𝜉, 𝜂) + 𝜇𝜗(𝜉, 𝜂) = 𝔐(𝜉, 𝜂), (𝜉, 𝜂) ∈ 𝐷, with𝜇 ≤ 0 (5.17) 𝜗| ∂𝐷 = 𝜗(𝜌cos𝜃, 𝜏sin𝜃) = 0, 𝜃 ∈ [0,2𝜋[, (𝜌 > 𝜏 > 0) (5.18)
where 𝐷 = {(𝑥, 𝑦) ∈ ℝ2:𝑥2
𝜌2+𝑦2
𝜏2< 1} and 𝔐 ∈ 𝐿2(𝐷).
Thus we shall show by a one more mean (except of variational, specified in the work [1]) a construction of an
approximate solution of the boundary value problem (5.17),(5.18).
Let’s, for ),( , D),( with ii =and= ,
,)()(1
)()(ln
2
1=),,,(,
DG
be the Green’s function of the homogeneous Dirichlet boundary value problem
,1>(),[0,20,=)sin,cos(=| ww D
We remind that is the complex conformal map which maps domain D into unit disk ; denotes its complex conjuguate map. Hence, for all ),( belong to D , the problem (5.17),(5.18) is reduced to the next
integral Fredholm equation of the second kind:
𝜗(𝜉, 𝜂) − |𝜇| ∫𝐷𝐺Δ,𝐷(𝜉, 𝜂, 𝛼, 𝛽)𝜗(𝛼, 𝛽)𝑑𝛼𝑑𝛽 = ∫𝐷 𝐺Δ,𝐷(𝜉, 𝜂, 𝛼, 𝛽)𝔐(𝛼, 𝛽)𝑑𝛼𝑑𝛽. (5.19)
The equation (5.19) is uniquely solvable because || is not a characteristic number of this equation. Let’s copy the
equation (5.19) in an other form. For this purpose let’s designate by )(z the inverse function of the conformal
function )( and by )(z the complex derivative of )(z . Let’s introduce in integral equation (5.19) the
following changes of functions:
),,(=)))(()),(((=),( yxViyxImiyxRe (5.20)
𝔐(𝛼, 𝛽) = 𝔐(𝑅𝑒(Ψ(𝑟 + 𝑖𝑠)), 𝐼𝑚(Ψ(𝑟 + 𝑖𝑠))) = 𝔪(𝑟, 𝑠), (5.21) where )(= iyxi , )(= isri and )(zRe and )(zIm are respectively the real part and
imaginary part of a complex z . Then the equation (5.19) is equivalent to the integral equation
;|)(|),(),,,(=|)(|),(),,,(||),( 2,2
, drdsisrsrsryxGdrdsisrsrVsryxGyxV
m (5.22)
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
34
where z
zzG
1ln
2
1=),(, is the Green’s function of homogeneous Dirichlet boundary value problem
for Poisson equation in the unit disk. Let’s )(2, L be the Hilbert space of square integrable function with respect
to the measure .|)(=| 2 dxdyiyx We denote K the integral operator defines on the Hilbert space
)(2, L of which kernel is the Green’s function ),,,(, sryxG . We recall that the application |)(| iyx is
continuous and strictly positif on . So the norm ∥. ∥𝐿2,𝜑(Ω) over )(2, L is equivalent to the norm ∥. ∥𝐿2(Ω)
over )(2 L . Then the equation (5.22) takes the form
(𝕀 − |𝜇|𝐾)𝑉(𝑥, 𝑦) = 𝑀(𝑥, 𝑦), (5.23) where 𝕀 is the identity operator on )(2 L and
.|)(|),(),,,(=),(2
, drdsisrsrsryxGyxM m
Let’s find an approximate solution of the integral equation (5.22) or (5.23) by using the Bubnov-Galerkin method.
It’s well known that the squarre summable space of functions on the unit disk , )(2 L , is endowed with a complet
and orthonormal set of functions }{ n which vanishing on the boundary of the unit disk. These functions
......,,, 21 n are orthonormal eigenfunctions in )(2 L of Laplace’s operator. In polar coordinates, they are
expressed by means of cylindrical Bessel functions. Their corresponding eigenvalues ......,,, 21 n satisfy
...>>...>>>0 21 n The approximate solution of (5.22) or (5.23) that we look for takes the form
2
1= |)(|
),(),(=),(
iyx
yxAyxMyxV
jj
j
n
j
n
(5.24)
where jA are coefficients that we shall look for. Let’s denote by n the residual .)||( MVK n I
),(),(|||)(|
||=),(,),(,get We
21=
yxyxiyx
Ayxyx jj
j
n
j
n
(5.25)
where
.|)(|),(),,,(|=|),(2
, drdsisrsrMsryxGyx
According to the Bubnov-Galerkin method, the required coefficients jA are determined from the orthogonality
condition of n with all functions n ...,,, 21 in the space )(2, L . It leads to the set of equations
.1,2,...,=0,=|)(|),(),(2 njdxdyiyxyxyx jn (5.26)
By using (5.25), the set (5.26) becomes
,1,2,...,=,=)|||(|1=
njA jkjkjkk
n
k
(5.27)
where jk is the Kronecker symbol and
;|)(|),(),(=2 dxdyiyxyxyx kjjk
𝛾𝑗 = ∫Ω 𝜓𝑗(𝑥, 𝑦)𝒩(𝑥, 𝑦)|Ψ′(𝑥 + 𝑖𝑦)|2𝑑𝑥𝑑𝑦.
The quadratic form of the matrix of the set (5.27) is positive define for any .n Therefore for any 𝑛. the set (5.27) has a unique solution ),...,,( 21 nAAA and the sequence nV , define in (5.24), satisfies
((𝕀 − |𝜇|𝐾)𝑉𝑛 −𝑀,𝜓𝑗)𝐿2,𝜑(Ω) = 0, for 𝑗 = 1,2, … , 𝑛.
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
35
Lemma The sequence }{ n defined for all 𝑛 ≥ 1 by
,),()),(()),(((=),( DiImiReVnn is a set of functions of )(2
2,0 DW such that
00 n converges to zero in )(2 DL .
Proof. For all ),( belongs to D , we have
))(),()](()([|))((|
1=),)((
200
iImiReVVVV
innn
))).(()),(((|))((|
1=
2
iImiRe
in
(5.28)
The module || is strictly positive and continous on the closed unit disk de .C So there exist 0>C such
that 1
|Ψ′(𝜔(𝜉+𝑖𝜂))|2≤ 𝐶.
Since then. We have ∥ (ℒ0𝜗𝑛 − ℒ0𝜗) ∥𝐿2(𝐷)2 ≤ 𝐶 ∥ Δ𝛿𝑛 ∥𝐿2(Ω)
2 . (5.29)
To get the result of the lemma, It is enough to show that n converge to zero in ).(2 L Let’s consider the set
||
j of functions of the space ).(2, L This set is an orthonomal complet basis of )(2, L because j
is an orthonomal complet basis of )(2 L .
.)|||(|==),(,,1,=1=
)(2,
kjkjkk
n
k
jLj Anj
So we have, .||
)|||(|=||
)||
,|(|1=1=
)(2,
1=
j
kjkjkk
n
k
n
j
j
L
jn
j
A
Hence the serie )(|| toconverges||
)|||(| 2,1=1=
LinA
j
kjkjkk
kj
and
),(||||
|||=|
||)|||(|=||
21=1=1=
yxAA jj
j
j
j
kjkjkk
kj
Since then
).,(||||
||=
21=
yxA jj
j
j
So 𝛿𝑛 = (𝕀 − |𝜇|𝐾)𝑉𝑛 −𝑀 = ∑𝑛𝑗=1 𝐴𝑗 (
|𝜇𝑗|
|Ψ′|2+ |𝜇|)𝜓𝑗(𝑥, 𝑦) −𝒩 converges to zero in 𝐿2(Ω).
From 𝛿𝑛 = (𝕀 − |𝜇|𝐾)𝑉𝑛 −𝑀 = (𝕀 − |𝜇|𝐾)(𝑉𝑛 − 𝑉), we have ∥ 𝑉𝑛 − 𝑉 ∥𝐿2(Ω)≤∥ (𝕀 − |𝜇|𝐾)
−1 ∥∥ 𝛿𝑛 ∥𝐿2(Ω).
Hence, the sequence 𝑉𝑛(𝑥, 𝑦) = 𝑀(𝑥, 𝑦) −∑
𝑛
𝑗=1
𝐴𝑗𝜇𝑗𝜓𝑗(𝑥, 𝑦)
|Ψ′(𝑥 + 𝑖𝑦)|2 convergesto 𝑉 in 𝐿2(Ω).
).(in toconverges|)(|
),(),(=),( sequence the,Hence 22
1=
LViyx
yxAyxMyxV
jj
j
n
j
n
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
36
So there exist a function )(2 Lh such that ),(1=
yxA jjj
n
j
converges to h in )(2 L and
hMV2||
1=
. From the definition of V and M , we have )(
2
2,0 WV and )(2
2,0 WM . So
)(22,0 Wh . Therefore the serie ),(=),(=1=
)(2
1=
yxAhh jjjj
jLj
j
converge to h in )(22 W
because the boundary of unit disk is of class 2C (see [6], theorem 7.4). So
hhVV jLjj
n )(2
1=2
),(||
1= converges to zero in ).(22 W It follows that n converges to
zero in )(2 L . Hence from the inequality (5.29), ℒ0𝜗𝑛 − ℒ0𝜗 converges to zero in )(2 DL . We establish the following theorems which give analytic approximate sequences of the problem (GE),(BCs) in each
fix point of the unit disk when the solutions of characteristic equation are distinct.
Theorem 5.3 Let 0=b , 0>a , 0>c and )(2 Lf in (GE),(BCs) and 𝜇 = (𝑎0 −𝑑2
4𝑎−𝑔2
4𝑐) ≤ 0..
For all 𝑛 ∈ ℕ∗,, for ),( yx belongs to the unit disk , let
c
y
a
xw
ac
agycdxyxu nn ,
2exp=),(
where
𝑤𝑛(𝜉, 𝜂) = 𝑉𝑛(𝑅𝑒(𝜔(𝜉 + 𝑖𝜂)), 𝐼𝑚(𝜔(𝜉 + 𝑖𝜂)) 𝑓𝑜𝑟 (𝜉, 𝜂) ∈ 𝐷 = {(𝜉, 𝜂) ∈ ℝ2:
𝜉2
(1
√𝑎)2 +
𝜂2
(1
√𝑐)2 < 1}
with nV a sequence of functions of )(2
2,0 W previously constructed, under the Bubnov-Galerkin
condition((𝕀 − |𝜇|𝐾)𝑉𝑛 −𝑀,𝜓𝑗)𝐿2,𝜑(Ω) = 0, for nj ,1,= ,in the form
21= |)(|
),(),(=),(
iyx
yxAyxMyxV
jj
j
n
j
n
and the conformal map transforming D into unit disk .
Here drdssrsryxGyxM2
, |)(|),(),,,(=),( m
))].(()),(([2
))(())((exp=),(with isrImcisrReaf
ac
isrImcagisrReacdsr
m
Then (𝑢𝑛)𝑛≥1 is a sequence of functions of )(2
2,0 W which converges in each fix point ),( yx of to the
unique generalized solution u of the problem (GE),(BCs).
Theorem 5.4 Let(4𝑎0(𝑏2 − 𝑎𝑐) + 𝑐𝑑2 − 2𝑏𝑑𝑔 + 𝑎𝑔2) ≤ 0, )(2 Lf in the problem (GE),(BCs)
and the real
)4(
2)(4=
2
222
0
acb
agbdgcdacba . Let }{ nu be a sequence of functions defined on
unit disk by
2
21
1
21
2
)(,
)(
)2(exp=),(
ypxpypxp
acb
yagbdxbgcdyxu nn
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
37
where ,10,> ca and )(2 Lf in the problem (GE),(BCs), the problem (GE),(BCs) is equivalent to
,),(),,(2
exp=44
=22
00 Dcafac
cagacdw
c
g
a
dawwL
),[0,20,=)(sin1
),(cos1
caw
where 𝐷 = {(𝜉, 𝜂) ∈ ℝ2:𝜉2
(1
√𝑎)2 +
𝜂2
(1
√𝑐)2 < 1}.
if the number
c
g
a
da
44=
22
0 is less than zero, then it can’t belong to the spectrum of the problem
ww =
).[0,20,=)(sin1
),(cos1
caw
Hence, according to the theorem 5.1, we have for all ),( yx belongs to ,
|𝑢(𝑥, 𝑦) − 𝑢𝑛(𝑥, 𝑦)| ≤ exp (−𝑐𝑑𝑥+𝑎𝑔𝑦
2𝑎𝑐)‖𝐺𝐿′0 (
𝑥
√𝑎,𝑦
√𝑐, . , . )‖
𝐿2(𝐷)×
× √(∫𝐷|exp (
𝑐𝑑√𝑎𝜉+𝑎𝑔√𝑐𝜂
2𝑎𝑐) 𝑓(√𝑎𝜉, √𝑐𝜂) − 𝐿′0(𝑤𝑛)(𝜉, 𝜂)|
2
𝑑𝜉𝑑𝜂) ;
According to the lemma 5.3, the sequence }{ nw defined in the theorem 5.3 is a sequence such that )(0 nwL
converges to )(0 wL in )(2 DL . It means
ddwLcaf
ac
cagacdn
D
2
0 ),)((),(2
exp
converges to zero when n tends to . It’s followed that nu converges to u in each fix point ),( yx of the unit disk . The proof of theorem 5.4 is similar to the proof of theorem 5.3.
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
38
6. ALGORITHM FOR APPROXIMATE DETERMINATION OF THE CONFORMAL MAP
TRANSFORMING THE INSIDE OF AN ELLIPSE INTO THE UNIT DISK IN THE SYSTEM
MATHEMATICA
We propose here an algorithm to determine an approximation of conformal map which Transforms the interior D of
an ellipse to the unit disk of 2R . This approximation is based on a method of decomposing a function in a non-
orthogonal basis in the computer system Mathematica. It is based on the method of decomposing a function in a
database of non-orthogonal functions. The algorithm we establish is free from any limitation on semi-axes of an ellipse
can be modified easily in the case of anyone connected region with enough smooth boundary. At the end of the
algorithm, we give an approximate of the Green’s function of homogeneous Dirichlet problem for Poisson equation
in interior D of an ellipse of semi-axes a and b : ,),(),,(=),( Df (6.1)
[.[0,20,=)sin,cos(=| ba (6.2)
In this section, data a and b of the semi-axes of the ellipse are not those defined in problem (GE),(BCs) of the previous sections.
We suppose that the conformal map which transforms the simply connected area D to the unit disc transforms the ellipse to the unit circle and the center O of D to the center O of ( 0=(0) ); O is the origin of the orthonormal coordinate of the complex plane. In this case, the Green’s function of the Laplace operator on
domain D is given by
),(ln2
1|=)(|ln
2
1=),(=,0,0),( 22,,
gGG DD (6.3)
where g is a harmonic function on D satisfying the problem
,),(0,=),( Dg (6.4)
.),(,ln2
1=, 22
g (6.5)
We will express the boundary condition (6.5) as a function of the polar coordinates ),( r of the ellipse . The
polar equation of the ellipse, relative to its center O , is given by
.
1
=
2
2
22
sinb
ba
ar
(6.6)
If ),( sc hh is a point in the ellipse of which polar coordinates is ),( r , then
.=
1
=,
1
= 22
2
2
222
2
22rhhand
sinb
ba
asinh
sinb
ba
acosh scsc
Hence the problem (6.4),(6.5) is written
0,=),( g (6.7)
[.[0,2,)(ln2)(sin11ln4
1=, 2
2
2
a
b
ahhg sc (6.8)
Let’s determine the solution g of the problem (6.7),(6.8) in the form
.)()(ln=),( 221=
kkk
M
k
yxcg (6.9)
The unknowns coefficients kc are real decomposition coefficients of function g in the non-orthogonal basis
Mkyx kk 1,2,...,=,)()(ln 22 ; The M pairs );( kk yx are collocation points which are everywhere dense on a contour which envelops completely the ellipse and has no common point
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
39
with it. In our algorithm we choose as contour a confocal ellipse to the ellipse .From the equality (6.3) and
according to the expression (6.9) of g , if the coefficients kc are known, one can then write
,)),(),((2exp)(=),( ihgi (6.10)
.)()(=),(where1=
kkkM
k
yixArgch
It is known that, for all k , the functions 22 )()(ln kk yx are harmonic with respect to the couple ),( . Thus, by replacing the expression (6.9) of g in the problem (6.7), (6.8), we get the following equation to
determine the coefficients Mkck 1,2,...,=, :
[.[0,2,)(ln2)(sin11ln4
1=)()(ln 2
2
222
1=
a
b
ayhxhc kskck
M
k
For the determination of the coefficients Mcc ,,1 we will use, in the algorithm, the least squares method. The
algorithm is realized in the Mathematica system with an ellipse of semi-axes 4=a and 3=b and a confocal ellipse whose Values of the semi-axes are 1,1 from those of :
mReIaAlgebrIn
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
40
;},1,{,
21
2
=:=[15]2
2
22
ptsn
npts
Sinb
ba
npts
aCos
TableuXIn
;},1,{,
21
2
=:=[16]2
2
22
ptsn
npts
Sinb
ba
npts
aSin
TableuYIn
The matrix of coordinates of the collocation points is given by
}];,1,{]]},[[]],[[[{=:=[17] ptsnnuYnuXTablescollopointIn
Let us recall in a figure the two confocal ellipses and the points of collocations.
];},,0,2]}{,,[],,,[[{=1:=[18] IdentityctionDisplayFunbaYbaXPlotParametricgrIn
];},,0,2{]},,,[],,,[[{=2:=[19] IdentityctionDisplayFunbaYbaXPlotParametricgrIn
]}];[],[[{=3:=[20] scollopointPointLargePointSizeGraphicsgrIn
],,3},2,1,[{=:=[21] ctionDisplayFunctionDisplayFunAutomaticoAspectRatiAllPlotRangegrgrgrShowshIn
[21]Out
];,,[:=[22] minsmatClearIn
Let us choose 150 points of collocations uniformly distributed on the ellipse
Let us use the least squares method for the determination of the coefficients Mccc ,...,, 21 . Let us introduce the
expression to minimize for the determination of the coefficients Mccc ,...,, 21 .
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
41
;
][1
][=][:=[23]
2
2
22
Sinb
ba
CosahIn c
;
][1
][=][:=[24]
2
2
22
Sinb
ba
SinahIn s
22
1=
]])[[][(]])[[][(]][[:=][:=[25] kuYhkuXhLogkucffunIn sc
pts
k
;][2][114
12
2
2
2
aLogSin
b
aLog
For the determination of the coefficients Mccc ,...,, 21 by least-squares method, it is necessary to minimize the
expression
;2
=:=[26]1=
j
ptsfunminIn
pts
j
From the sum "min", we get the following two matrices for the determination of the coefficients
]][[= kucfck .
;},1,{},,1,{,]][[2
]][[2
:=:=[27]
22
ptskptsjkuYj
ptshkuXj
ptshLogTablematIn sc
;},1,{,][22
114
1::[28]
2
2
2
ptsjaLogj
ptsSin
b
aLogTablesIn
The matrix of the coefficients sought is obtained thanks to the command "LeastSquare" of mathematica
];,[=:[29] smateLeastsquarucfIn
The minimum value corresponding to the coefficients Mccc ,...,, 21 found is:
][:[30] minPrintIn
.10*2.4020230
Let us introduce the important functions for the construction of the conformal application and of the Green function
of the Dirichlet problem for the Laplace operator in an ellipse
)]];]][[()]][[[([]][[=],[:=[31]1=
ykuYIxkuXAbsLogkucfyxanswerInpts
k
)];]][[()]][[[(]][[=],[:=[32]1=
ykuYIxkuXArgkucfyxhInpts
k
The approximate conformal map transforming an ellipse to the unit circle is given by
]])][],[[]][],[[(2[exp:=][:=[33] zImzReIhzImzReanswerzzIn ; Let us confirm this result by the following constructions which transforms an ellipse to the unit circle and the inside
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
42
of an ellipse to the unit disk.
oAspectRatiAllPlotRangeSinbCosaPlotParametricellipsIn ,},,0,2{]},[],[[{=:[34]
]Automatic ;
},,0,2{]]]},[][[[]]],[][[[[{=:[35] IbSinacosImSinIbCosaRePlotParametriccircleIn
];, AutomaticoAspectRatiAllPlotRange
],,},,[{=:[36] ctionDisplayFunctionDisplayFunAutomaticoAspectRatiAllPlotRangecircleellipsShowshIn
[36]Out
𝐼𝑛[37]:= 𝑑𝑖𝑠𝑐 = 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑟𝑖𝑐𝑃𝑙𝑜𝑡[{𝑅𝑒[𝜔[𝑟𝐶𝑜𝑠[𝜑] + 𝐼 𝑟𝑆𝑖𝑛[𝜑]]], 𝐼𝑚[𝜔[𝑟𝑐𝑜𝑠[𝜑] + 𝐼 𝑟𝑆𝑖𝑛[𝜑]]]}, {𝜑, 0,2𝜋},
],}
][1
,0,{
2
2
22AutomaticoAspectRatiAllPlotRange
Sinb
ba
ar
[37]Out
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
43
Let us evaluate the maximal and minimal deflections of the module |),(| yx with respect to the radius 1 of the
unit disk when the points ),( yx belong the ellipse . These evaluations are carried out for 2500 points taken on
the ellipse : ];[:=[38] mistakeClearIn
;)]]][][[[]]][][[[(:=][:=[39] 21
22 bsinIacosImbsinIacosRemistakeIn
2500;=1:=[40] ptsIn
Maximum deviation for the 2500 selected points;
1}]],1,{]],1
2[[[[:=[41] ptskk
ptsmistakeNTableMaxIn
1.00000002385.
Minimal deviation for the 2500 points choosen;
1}]],1,{]],1
2[[[[:=[42] ptskk
ptsmistakeNTableMinIn
0.99999998553.
The formula for the approximate determination of the Green’s function in a point C for the Dirichlet problem
for the Poisson equation inside the ellipse is given by
,)()(
)()(1ln
2
1=),(,
z
zzG D
where )],([)],([=),()],,([)],([=),( iImResiiImRe . This function in
point 1= I inside of ellipse is given in computer system mathematica by:
;][1][
][][11
2
1:=],1[:[43]
Iz
zIAbsLog
PiIzGIn
Plot of the Green’s function for homogeneous Dirichlet problem for the Poisson equation inside of ellipse of semi-
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
44
axes 4=a and 3=b for I1= .
𝐼𝑛 [44]: = 𝑃𝑙𝑜𝑡3𝐷 [𝐺[𝑟cos[𝜑] + 𝐼 𝑟𝑠𝑖𝑛[𝜑]], {𝜑, 0,2𝜋}, {𝑟, 0,𝑎
√1+𝑎2−𝑏2
𝑏2𝑆𝑖𝑛[𝜑]2
} , 𝑃𝑙𝑜𝑡𝑅𝑎𝑛𝑔𝑒 → 𝐴𝑙𝑙] Out[44]
7. CONCLUSION
In this paper, we construct, from the space )(2
2,0 W , a sequence of approximate solution nu of Dirichlet elliptic
problem (GE),(BCs) when the solutions of characteristic equation (CE) are identic and when they are distinct. We
show that the approximate solution nu converges, in each fix point of the unit disk , to the Generalized solution
u of the problem (GE),(BCs). When the solutions of characteristic equation are different, we show that the
approximate solution depends on the conformal mapping which maps the interior of an ellipse to the unit disk of .2R
By an algorithm, in the Mathematica system, we construct this approximate conformal map by using mean least
squarre method with collocations points. Hence we have give an approximate green’s function for homogeneous
Dirichlet problem for the Poisson equation inside of an ellipse . We give in particular, a plot of this green’s function for an ellipse of semi-axes 4=a and 3=b in a point i1= .
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IJRRAS 31 (1) ● April 2017 Tchalla et al. ● Approximate Solution of a Dirichlet Problem
45
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