approximate solution of a dirichlet problem for …solution for a dirichlet boundary value problem...

IJRRAS 31 (1) ● April 2017 www.arpapress.com/Volumes/Vol31Issue1/IJRRAS_31_1_04.pdf

25

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

mailto:[email protected]:[email protected]

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),(


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


28

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:


29

),,(),(=),(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.


30

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:


31

|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)


32

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


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)


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.


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


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


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.


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


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


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 .


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


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


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-


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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approximate solution of a dirichlet problem for …solution for a dirichlet boundary value problem...

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