APPROXIMATE SOLUTION OF A DIRICHLET

IJRRAS 31 (1) ● April 2017

www.arpapress.com/Volumes/Vol31Issue1/IJRRAS_31_1_04.pdf

APPROXIMATE SOLUTION OF A DIRICHLET PROBLEM FOR GENERAL SECOND ORDER ELLIPTICAL LINEAR PDEs WITH 𝟐 CONSTANT COEFFICIENTS IN THE UNIT DISK OF ℝ 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 u n which converges, in each fix point ( x , y ) of unit disk  , to the generalized

u . In section 5, we have showed that the analytic approximate solution u n 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 solution

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:

 2u  2u  2u u u  2 b  c d g  a0u = f ( x, y ) (GE) 2 2 x xy y x y where a, b, c, d , g and a0 are real constants and f a real function. In this paper, we deal with this work when L(u ) = a

equation (GE) is of elliptic type; it means when b  ac < 0 and ( a, b, c )  (0,0,0). Let consider the 2

homogeneous Dirichlet’s problem of the equation (GE) in the unit disk Ω = {(𝑥, 𝑦) ∈ ℝ2 : 𝑥 2 + 𝑦 2 < 1}.. The goal of this problem is to determine the function u which satisfies the equation (GE) in the domain  under the boundary condition (BCs) u| = 0. For ( x, y )   , by setting

25


Tchalla et al. ● Approximate Solution of a Dirichlet Problem

 cd  bg x  bd  ag y  u( x, y) = vx, y exp , 2(b 2  ac)  

(2.1)

the Dirichlet boundary value problem (GE),(BCs) becomes:

 2v  2v  2v a 2  2b  c 2  v = R( x, y ), x xy y v| = 0,

where  =

(GE') (BC's)

  cd  bg x  bd  ag y  4a0 (b 2  ac)  cd 2  2bdg  ag 2 and R( x, y) = exp  f ( x, y). 2 4(b  ac) 2(b 2  ac)  

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  .

L2 () is the Hilbert space of square integrable reals functions (in the sense of Lebesgue) on  . Scalar product on L2 () will be designate by (.,.) L2 (  ) and the norm by 1

∥ 𝑢 ∥𝐿2 (Ω) = (∫Ω |𝑢(𝑥, 𝑦)|2 𝑑𝑥𝑑𝑦)2 , ∀𝑢 ∈ 𝐿2 (Ω).

W21 () is the Hilbert space, consisting of the elements of L2 () having generalized derivatives of first order which are square summable on  . The scalar product in this space is defined by  u v u v  dxdy (u, v) 2,1 =   u( x, y)v( x, y)    x x y y   and the norm by

∥ 𝑢 ∥2,1 = √(𝑢, 𝑢)2,1 .

C () is the set of infinitely differentiable functions with compact support laying in .  c 1 2

W () is the closure of Cc () in the space W21 () . W22 () is the Hilbert space consisting of elements of L2 () having first and second order generalized derivatives in L2 () . Its scalar product is defined by:

 u v u v  2u  2 v  2u  2v  dxdy (u, v) 2,2 =   u ( x, y )v( x, y )      x x y y x 2 x 2 y 2 y 2   and the norm is given by: ∥ 𝑢 ∥2,2 = √(𝑢, 𝑢)2,2 . 2 W2,0 () is the closure, in W22 () , of functions of C 2 () which vanish on



2 W2,0 () =W 12()  W22 () if   C 2 .

Definition 1 (Classical solution) A classical solution

u of the problem (GE),(BCs) is a function u from

C ()  C () which satisfies the problem (GE),(BCs). 2

Definition 2 (Generalized Solution) A function (GE),(BCs) if it satisfies the integral identity

u from W 12() is called generalized solution of the problem

 u     u   u    u   u u dxdy c L(u,  )    a  b   d   g   a u  0     x  x  x  y  y  x  y  y  x  y     26

.



=   f  dxdy

(3.1)



for all

2 () and   W 12() . Any generalized solution of the problem (GE),(BCs) belongs to the space W2,0

hence, is a continuous function in

.

Definition 3 (Characteristic equation of (GC), (BCs) ). We call characteristic equation of (GE),(BCs), the equation given by det ( A  E ) = 0; E is the unit matrix of ℝ2 ,,  the unknown variable of the equation and

a b . A =  b c In the extended form, the characteristic equation is given by

2  (a  c)  ac  b 2 = 0.

(CE)

It’s well known that the solutions of (CE) are eigenvalues of matrix A and are given by

1 =

(a  c  4b 2  (a  c) 2 ) (a  c  4b 2  (a  c) 2 ) ; 2 = . 2 2

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 Let’s notice that the form

1 = 2

if, and only if,

b=0

and

1 = 2

a = c . In this case, the Dirichlet problem (GE'),(BC's) takes

a d2 g2  1  xd  yg  L0 (v) = v   0  2  2 v = exp  f ( x, y ) a 4 a 4 a a 2 a    

(4.1)

v| = 0.

(4.2)

 a0 d 2 g2    does not a = c    Theorem 4.1 Let’s assume b = 0 , and f  L2 (). If the number  2 2  a 4 a 4 a   belong to the spectrum of the problem

v = v v| = 0,

then the unique generalized solution *

function u belong to

(4.3) (4.4) 2 u of the problem (GE),(BCs) belongs to the space W2,0 () and for any

2 W2,0 () , for any ( x, y )   , the following posterori estimation is satisfied:

|𝑢(𝑥, 𝑦) − 𝑢∗ (𝑥, 𝑦)| ≤ exp (−

𝑥𝑑+𝑦𝑔 2𝑎

) ∥ 𝐺𝐿0 (𝑥, 𝑦, . ) ∥𝐿2 (Ω) × 2

√∫ |1 exp (𝜉𝑑+𝜂𝑔) 𝑓(𝜉, 𝜂) − 𝐿0 (𝑣 ∗ )(𝜉, 𝜂)| 𝑑𝜉𝑑𝜂 Ω 𝑎 2𝑎 where

 xd  yg  v* ( x, y) = u * ( x, y) exp  and GL0 is the Greens function of the problem.  2a  27

(4.5)


Proof. Let’s set


F ( x, y) =

1  xd  yg  exp  f ( x, y), ( x, y)  . a  2a 

a d2 g2    0  2  2  does not belong to the spectrum of the problem (4.3),(4.4), then the 4a   a 4a 2 operator L0 defined a homeomorphism from the space W2,0 () to the space L2 () . Hence the unique If the real

generalized solution

2 v of problem (4.1),(4.2) belongs to W2,0 () and we have

( x, y )  , v( x, y ) = L01 ( F )( x, y ) =  GL ( x, y,  , ) F ( , )dd 

where

1 0

L

is the inverse of the operator L0 .

0

v W (), ( x, y)   , we have *

2 2,0

v( x, y )  v* ( x, y ) =  GL ( x, y,  , )[F ( , )  L0 (v* )( , )]dd. 

In particular, for

0

 xd  yg  * 2 v* ( x, y) = u * ( x, y) exp  with u W2,0 () ; we get  2a   xd  yg  * u( x, y)  u * ( x, y) = exp  GL0 ( x, y,  , )[F ( , )  L0 (v )( , )]dd. 2a  

Since then, By using Hölder inequality we have:

|𝑢(𝑥, 𝑦) − 𝑢∗ (𝑥, 𝑦)| ≤ exp (− 1


× √∫Ω |𝑎 exp (

) ∥ 𝐺𝐿0 (𝑥, 𝑦, . , . ) ∥𝐿2 (Ω)

𝜉𝑑+𝜂𝑔 2𝑎

2

) 𝑓(𝜉, 𝜂) − 𝐿0 (𝑣 ∗ )(𝜉, 𝜂)| 𝑑𝜉𝑑𝜂.

Finally we have the result of theorem 4.1. 4.2 Analytic Approximate Solution of the Problem (GE ), ( BCs ) When Here we shall construct a sequence of approximate solution u n of

1 = 2

W () which converges to the unique 2 2,0

u of (GE ), ( BCs ) 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 (GE ), ( BCs ) from an approximate solution of the boundary value problem (4.1),(4.2). Let G, ( x, y,  , ) 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 generalized solution

equation of the second kind:

v( x, y ) |  |  G , ( x, y,  , )v( , )dd =  G , ( x, y,  , ) F ( , )dd , 



(4.6)

 a0 d 2 g2   2  2 . In this part we suppose that 𝜇 ≤ 0. Then the equation (4.6) is uniquely solvable with  =   a 4a 4a  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 L2 () on the unit disk  is equipped with a complet orthonormal set of functions { n } which vanishing on the boundary  of  . These functions  1 ,  2 , ..., 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 > 1 >  2 > ... >  n > ... Let’s find this approximate solution of the equation (4.6) in the form:

28



n

vn ( x, y ) = S ( x, y )  A j  j j ( x, y ),

(4.7)

j =1

where S ( x, y ) =  G ( x, y,  , ) F ( , )dd and A j are reals coefficients to be determined. 

By setting T the operator define by T (v)( x, y ) = v( x, y ) |  |  G ( x, y,  , )v( , )dd , 

we obtain, for all ( x, y )   , the residual

 n ( x, y ) = T (v)( x, y )  T (vn )( x, y ) = A j |  j |  |  | j ( x, y )  S1 ( x, y ) n

(4.8)

j =1

with

S1 ( x, y ) =|  |  G ( x, y,  , ) S ( , )dd . 

According to the Bubnov-Galerkin method, the required coefficients orthogonality condition of the residual

 

n n

A j are determined, for all 𝑛 ≥ 1, from the

 1 ,  2 , ..., n

with the functions

in the space

L2 () :

( x, y ) k ( x, y )dxdy = 0, k = 1,2,..., n.

(4.9)

By using the expression (4.8), we get the coefficients Ak from the condition (11) by the formula

Ak =

k

for k = 1,2,..., n where  k =   k ( x, y) S1 ( x, y)dxdy.

| k |  |  |



Thus the sequence {vn } of functions of

2 W2,0 () defined for all ( x, y ) in the unit disk  by

 j j

n

vn ( x , y ) = S ( x, y )   j =1

is an approximate solution of equation (4.6) in in

| j | |  |

 j ( x, y )

L2 () . It’s also an approximate solution of the problem (4.1),(4.2)

L2 () because the equation (4.6) is equivalent to (4.1),(4.2). Hence we obtain the following theorem:

Theorem 4.2 Let’s

b =0, a = c

and

𝑎

𝑑2

𝑔2

f  L2 () in (GE),(BCs) and 𝜇 = ( 𝑎0 − 4𝑎2 − 4𝑎2 ) ≤ 0. For

all n  N , for all ( x , y ) belongs to the unit disk  , let u n defined by *

 xd  yg  un ( x, y) = exp  vn ( x, y) where 2a   n    j j 1   d  g  vn ( x , y ) =  S ( x , y )    ( x, y ) , S ( x, y ) =  G , ( x, y,  , ) exp  f ( , ) dd. j     2a  j =1 |  j |  |  | a   

Then u n  is a sequence of functions of the unique generalized solution

2 W2,0 () which converge in each fix point ( x, y ) of the unit disk  to

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 (−


) ∥ 𝐺𝐿0 (𝑥, 𝑦, . ) ∥𝐿2 (Ω) × 2

√∫ |1 exp (𝜉𝑑+𝜂𝑔) 𝑓(𝜉, 𝜂) − 𝐿0 (𝑣𝑛 )(𝜉, 𝜂)| 𝑑𝜉𝑑𝜂. Ω 𝑎 2𝑎 29



So we have the result because 2

1  d  g   a exp 2a  f ( , )  L0 (vn )( , ) dd  0 as n  . 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 With the condition

1  2

with

b=0

b = 0 , the equation (GE'),(BC's) becomes  2v  2v  d2 g2   cdx  agy  L0 (v) = a 2  c 2   a0   v = exp  f ( x, y ) x y  4a 4c   2ac  v| = 0

(5.1) (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 a > 0 and c > 0. Let’s apply in the problem (5.1),(5.2), the change of function

x y  x y  and  = . So we get the following v( x, y) = w ,  with independent variables  = a c  a c

Dirichlet problem

 cd a  ag c   d2 g2   f ( a , c ), ( , )  D, L0 w = w   a0    w = exp  4a 4c  2ac    1  1  w cos( ), sin( )  = 0, [0,2 ), c  a 

   where D = ( , )  R 2    Theorem 5.1 Let’s assume

b=0

, a > 0,c > 0 and

:

2  1     a

2



2  1     c

2

(5.3)

(5.4)

   < 1.   

f  L2 () in problem (GE),(BCs). If the number

 d g    a0    does not belong to the spectrum of the problem 4 a 4c   (5.5) w = w 1  1  (5.6) w cos( ), sin( )  = 0, [0,2 ), c  a  2 then the unique generalized solution u of the problem (GE),(BCs) belongs to space W2,0 () and for any function 2

2

2 u * belongs to W2,0 () , for any point ( x, y ) of the unit disk  , the following posteriori error estimate is

realized:

30



|𝑢(𝑥, 𝑦) − 𝑢∗ (𝑥, 𝑦)| ≤ exp (− √(∫ |exp ( 𝐷

𝑐𝑑 √𝑎𝜏+𝑎𝑔√𝑐𝜎 2𝑎𝑐

𝑐𝑑𝑥+𝑎𝑔𝑦 2𝑎𝑐

𝑥

𝑦

√

√𝑐

) ‖𝐺𝐿′0 (

, 𝑎

, . , . )‖

𝐿2 (𝐷)

× (5.7)

2

) 𝑓(√𝑎𝜏, √𝑐𝜎) − 𝐿′0 (𝑤 ∗ )(𝜏, 𝜎)| 𝑑𝜏𝑑𝜎) , *

where G L is the Green’s function of the problem (5.3),(5.4) on the domain D , and w a function of 0

2 W2,0 ( D)

 cd a  ag c   for all ( , )  D.   2 ac  

given by w ( , ) = u ( a , c ) exp *

*

Proof. The proof of this theorem is similar to that of the theorem 4.1. But it can be found in [2].

1  2

b  0. As the equation (GE) is elliptic, the solutions 1 and 2 of characteristic equation (CE) satisfy 12 > 0 and 2 according to the Viet theorem 12 = ac  b . Without lost the generality, suppose that 1 > 2 > 0 . Let’s transform the equation (GE') to the canonical form. Let P1 = ( p1 ; p2 ) and P2 = ( p1, p2 ) be, according to 2 the usual norm of R , the normalized eigenvectors corresponding respectively to the eigenvalues 1 and 2 of a b  deduced from (GE). By simple calculations, we have the symetric matrix A =  b c   5.2 Error Estimate of Approximate Solution of the Problem (GE),(BCs) when

1

𝑝1 = √

𝑎𝑛𝑑

2 ((𝑐−𝑎)+√4𝑏2 +(𝑎−𝑐)2 )

1+

𝑎𝑛𝑑

𝑝′2 =

4𝑏2

(𝑐−𝑎)−√4𝑏 2 +(𝑎−𝑐)2

.

((𝑐−𝑎)−√4𝑏2 +(𝑎−𝑐)2 )²

√ 2𝑏 1+

4𝑏2

;

2 ((𝑐−𝑎)+√4𝑏2 +(𝑎−𝑐)2 )

√ 2𝑏 1+

((𝑐−𝑎)−√4𝑏2 +(𝑎−𝑐)2 )²

√ 1+

(𝑐−𝑎)+√4𝑏2 +(𝑎−𝑐)2

𝑝2 =

4𝑏2

1

𝑝′1 =

with

4𝑏2

Let’s make a change of function of independent variables in the equation (GE') in the form

w( , ) = v( x, y ),

(5.8)

where  = p1 x  p2 y and  = p1x  p2 y. It means

x ((c  a)  4b 2  (a  c) 2 ) y x ((c  a)  4b 2  (a  c) 2 ) y =  , =  k1 2bk1 k2 2bk2

where

k1 =

(c  a)  1





2



2

4b 2  (a  c) 2 (c  a)  4b 2  (a  c) 2 , k = 1  . 2 4b 2 4b 2

Then the problem (GE'),(BC's) becomes

𝜆1

∂2 𝑤(𝜉,𝜂) ∂𝜉 2

+ 𝜆2

∂2 𝑤(𝜉,𝜂) ∂𝜂 2

+(

4𝑎0 (𝑏 2 −𝑎𝑐)+𝑐𝑑2 −2𝑏𝑑𝑔+𝑎𝑔2 4(𝑏 2 −𝑎𝑐)

w| = 0,

) 𝑤(𝜉, 𝜂) = 𝒢(𝜉, 𝜂),

(5.9) (5.10)

31


Where

𝒢(𝜉, 𝜂) = exp [


𝜆2 [𝑑𝑝1 +𝑔𝑝2 ]𝜉+𝜆1 [𝑑𝑝′1 +𝑔𝑝′2 ]𝜂 2(𝑎𝑐−𝑏 2 )

] 𝑓(𝑝1 𝜉 + 𝑝′1 𝜂, 𝑝2 𝜉 + 𝑝′2 𝜂).

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)

where  =

 2 = 1 a  c  4b 2  (a  c) 2

 2 = . 2 2 2 a  c  4b  (a  c)

=

and

Thus, from the problem (GE'),(BC's), the Dirichlet problem (GE),(BCs) is reduced to the equivalent boundary value problem

𝐿′′0 (𝜅) = Δ𝜅 + ( 𝜅(

4𝑎0 (𝑏 2 −𝑎𝑐)+𝑐𝑑2 −2𝑏𝑑𝑔+𝑎𝑔2 4(𝑏 2 −𝑎𝑐)

√2cos𝜃

) 𝜅 = ℋ(𝛼, 𝛽), (𝛼, 𝛽) ∈ 𝐷,

(5.12)

) = 0, 0 ≤ 𝜃 < 2𝜋,

(5.13)

√2sin𝜃

,

√𝑎+𝑐+√4𝑏 2 +(𝑎−𝑐)2 √𝑎+𝑐−√4𝑏 2 +(𝑎−𝑐)2

where 𝐷 = (𝛼, 𝛽) ∈ ℝ2 :

problem

2+

1 ( ) √𝜆1

{ Theorem 5.2 If

𝛼2

𝛽2 1 ( ) √𝜆2

2

𝜏 > 0) (5.18) where

𝑥2

𝑦2

𝐷 = {(𝑥, 𝑦) ∈ ℝ2 : 𝜌2 + 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  =   i and  =   i ,

G , D ( , ,  ,  ) = 

1  ( )   ( ) ln , 2 1   ( ) ( )

be the Green’s function of the homogeneous Dirichlet boundary value problem

  x2 y2 2 w( , ) = h( , ), ( , )  D = ( x, y )  R : 2  2 < 1,     w|D = w(  cos , sin ) = 0,  [0,2 ), (  >  > 0). 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| ∫ 𝐺Δ,𝐷 (𝜉, 𝜂, 𝛼, 𝛽)𝜗(𝛼, 𝛽)𝑑𝛼𝑑𝛽 = ∫ 𝐺Δ,𝐷 (𝜉, 𝜂, 𝛼, 𝛽)𝔐(𝛼, 𝛽)𝑑𝛼𝑑𝛽. (5.19) 𝐷

𝐷

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:  ( , ) =  ( Re( ( x  iy)), Im ( ( x  iy))) = V ( x, y ), (5.20)

𝔐(𝛼, 𝛽) = 𝔐(𝑅𝑒(Ψ(𝑟 + 𝑖𝑠)), 𝐼𝑚(Ψ(𝑟 + 𝑖𝑠))) = 𝔪(𝑟, 𝑠), (5.21) where   i =  ( x  iy ) ,   i =  ( r  is) and Re(z ) and Im (z ) are respectively the real part and imaginary part of a complex z . Then the equation (5.19) is equivalent to the integral equation

V ( x, y ) |  |  G , ( x, y, r , s)V (r , s) | (r  is) |2 drds =  G , ( x, y, r , s)m(r , s) | (r  is) |2 drds; (5.22) 



33



1 z  is the Green’s function of homogeneous Dirichlet boundary value problem ln 2 1   z for Poisson equation in the unit disk. Let’s L2, () be the Hilbert space of square integrable function with respect where

G, ( z,  ) = 

to the measure

 =| ( x  iy) |2 dxdy.

We denote K the integral operator defines on the Hilbert space

L2, () of which kernel is the Green’s function G, ( x, y, r, s) . We recall that the application |  ( x  iy) | is continuous and strictly positif on  . So the norm

∥. ∥𝐿2,𝜑(Ω) over L2, () is equivalent to the norm ∥. ∥𝐿2 (Ω)

L2 () . Then the equation (5.22) takes the form (𝕀 − |𝜇|𝐾)𝑉(𝑥, 𝑦) = 𝑀(𝑥, 𝑦), where 𝕀 is the identity operator on L2 () and M ( x, y ) =  G , ( x, y, r , s)m(r , s) | (r  is) |2 drds. over

(5.23)



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  , L2 () , is endowed with a complet and orthonormal set of functions { n } which vanishing on the boundary of the unit disk. These functions

 1 ,  2 , ..., n ...

L2 () of Laplace’s operator. In polar coordinates, they are expressed by means of cylindrical Bessel functions. Their corresponding eigenvalues 1 ,  2 , ...,  n ... satisfy 0 > 1 >  2 > ... >  n > ... The approximate solution of (5.22) or (5.23) that we look for takes the form n  j j ( x, y) (5.24) Vn ( x, y) = M ( x, y)  Aj | ( x  iy) |2 j =1 where A j are coefficients that we shall look for. Let’s denote by  n the residual (I |  | K )Vn  M . are orthonormal eigenfunctions in

n | j |   We get, ( x, y )  ,  n ( x, y ) = A j   |  |  j ( x, y )   ( x, y ) 2 j =1  |  ( x  iy) | 

(5.25)

where

 ( x, y ) =|  |  G , ( x, y, r , s) M (r , s) |  (r  is) | 2 drds. 

According to the Bubnov-Galerkin method, the required coefficients condition of

n

with all functions

 

n

 1 ,  2 , ..., n

in the space

A j are determined from the orthogonality

L2, () . It leads to the set of equations

( x, y ) j ( x, y ) | ( x  iy) |2 dxdy = 0,

j = 1,2,..., n.

(5.26)

By using (5.25), the set (5.26) becomes n

(| 

k

|  jk  |  |  jk ) Ak =  j ,

j = 1,2,..., n,

(5.27)

k =1

where

 jk

is the Kronecker symbol and

 jk =   j ( x, y ) k ( x, y ) | ( x  iy) |2 dxdy; 

𝛾𝑗 = ∫Ω 𝜓𝑗 (𝑥, 𝑦)𝒩(𝑥, 𝑦)|Ψ′(𝑥 + 𝑖𝑦)|2 𝑑𝑥𝑑𝑦. n. Therefore for any 𝑛. the set (5.27)   V ( A , A ,..., A ) has a unique solution and the sequence n , define in (5.24), satisfies 1 2 n ((𝕀 − |𝜇|𝐾)𝑉𝑛 − 𝑀, 𝜓𝑗 )𝐿2,𝜑(Ω) = 0, for 𝑗 = 1,2, … , 𝑛. The quadratic form of the matrix of the set (5.27) is positive define for any

34



The sequence {n } defined for all

Lemma

𝑛 ≥ 1 by

n ( , ) = Vn ( Re( (  i )), Im ( (  i )),( , )  D,

is a set of functions of

2 W2,0 ( D) such that

 0 n   0 converges to zero in L2 ( D) . Proof. For all ( , ) belongs to D , we have

1 [(Vn  V )   (Vn  V )]( Re (  i ), Im (  i )) |  ( (  i )) | 2 1 =  n ( Re( (  i )), Im ( (  i ))). (5.28) | ( (  i )) |2

( 0n   0 )( , ) =

The module |   | is strictly positive and continous on the closed unit disk  de that

1 |Ψ′(𝜔(𝜉+𝑖𝜂))|2

C.

So there exist

C>0

such

≤ 𝐶.

∥ (ℒ0 𝜗𝑛 − ℒ0 𝜗) ∥2𝐿2 (𝐷) ≤ 𝐶 ∥ Δ𝛿𝑛 ∥2𝐿2 (Ω) . (5.29) To get the result of the lemma, It is enough to show that  n converge to zero in L2 (). Let’s consider the set Since then. We have

j    of functions of the space L2, (). This set is an orthonomal complet basis of L2, () because  j  |  |  is an orthonomal complet basis of L2 () . n

=  j = (|  k |  jk  |  |  jk ) Ak .

j = 1,, n, (  , j ) L

2, (  )

n

So we have,

(|  |  , j =1  

Hence the serie

(| 

k

j |  |

)L

k =1

j

2, (  )

n

|  |

j =1 k =1

j

|  jk  |  |  jk ) Ak

|  |

j =1 k =1

and  

|   |  = (|  k |  jk  |  |  jk ) Ak j =1 k =1

n

= (|  k |  jk  |  |  jk ) Ak

j |  |

.

converges to |   |  in L2, ()

j

  | j |  =|   | A j   |  |  j ( x, y ) 2 |  | j =1  |  | 

Since then   | j |   = A j   |  |  j ( x, y ). 2 j =1  |  | 

|𝜇 |

So 𝛿𝑛 = (𝕀 − |𝜇|𝐾)𝑉𝑛 − 𝑀 = ∑𝑛𝑗=1 𝐴𝑗 (|Ψ′|𝑗 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) . 𝑛

Hence, the sequence 𝑉𝑛 (𝑥, 𝑦) = 𝑀(𝑥, 𝑦) − ∑ 𝐴𝑗 𝑗=1 n

Hence, the sequence Vn ( x, y) = M ( x, y)  A j j =1

𝜇𝑗 𝜓𝑗 (𝑥, 𝑦) convergesto 𝑉 in 𝐿2 (Ω). |Ψ′(𝑥 + 𝑖𝑦)|2

 j j ( x, y)

|  ( x  iy) | 2

35

converges to V in L2 ().


So there exist a function


h  L2 () such that

n

A   j

j

j

( x, y ) converges to h in L2 () and

j =1

V =M

1 2 2 () and M W2,0 () . So h . From the definition of V and M , we have V W2,0 |   |2 

2 h W2,0 () . Therefore the serie h = (h, j ) L j =1

because

the

Vn  V = 

boundary

of

unit

disk



 j = A j  j j ( x, y )

2 ()

is

of

converge to

in W2 () 2

h

j =1

class

C2

(see

[6],

theorem

7.4).

So

 1   (h, j ) L (  ) j  h  converges to zero in W22 (). It follows that  n converges to 2   2 |   |  j =1  

zero in L2 () . Hence from the inequality (5.29), ℒ0 𝜗𝑛 − ℒ0 𝜗 converges to zero in L2 ( D) . 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 For all

b = 0 , a > 0, c > 0

and

𝑑2

𝑔2

f  L2 () in (GE),(BCs) and 𝜇 = (𝑎0 − 4𝑎 − 4𝑐 ) ≤ 0..

 cdx  agy   x y  𝑛 ∈ ℕ∗ ,, for ( x, y ) belongs to the unit disk  , let un ( x, y) = exp  ,  wn  2ac   a c  

where

𝑤𝑛 (𝜉, 𝜂) = 𝑉𝑛 (𝑅𝑒(𝜔(𝜉 + 𝑖𝜂)), 𝐼𝑚(𝜔(𝜉 + 𝑖𝜂)) 𝑓𝑜𝑟 (𝜉, 𝜂) ∈ 𝐷 = {(𝜉, 𝜂) ∈ ℝ2 : with Vn  a sequence of functions of

𝜉2 1 2 ( ) √𝑎

+

2 W2,0 () previously constructed, under the Bubnov-Galerkin

𝜂2 1 2 √𝑐

< 1}

( )

condition((𝕀 − |𝜇|𝐾)𝑉𝑛

− 𝑀, 𝜓𝑗 )𝐿2,𝜑(Ω) = 0, for j = 1, , n ,in the form n  j j ( x, y) and  the conformal map transforming D into unit disk . Vn ( x, y) = M ( x, y)  Aj | ( x  iy) |2 j =1

Here

M ( x, y ) =  G , ( x, y, r , s)m(r , s) |  ( ) | 2 drds 

 cd a Re( (r  is))  ag c Im ( (r  is))   f [ a Re( (r  is)), c Im ( (r  is))]. with m(r , s) = exp  2ac   2 (𝑢𝑛 )𝑛≥1 is a sequence of functions of W2,0 () which converges in each fix point ( x, y ) of  to the unique generalized solution u of the problem (GE),(BCs).

Then

Theorem 5.4 Let(4𝑎0 (𝑏 2 − 𝑎𝑐) + 𝑐𝑑 2 and

− 2𝑏𝑑𝑔 + 𝑎𝑔2 ) ≤ 0, f  L2 () in the problem (GE),(BCs)

 4a (b 2  ac)  cd 2  2bdg  ag 2   . Let {un } be a sequence of functions defined on  the real  =  0 4(b 2  ac)  

unit disk  by

 cd  bg x  bd  ag  y   ( p1 x  p2 y ) ( p1x  p2 y )  un ( x, y ) = exp ,  n   2(b 2  ac) 1 2    

36



 n ( , ) = Vn ( Re((  i )), Im((  i )) for ( , )  D = ( ,  ) R2 : 1 2  2  2 < 1, Vn  a sequence of functions of W2,02 () which is previously constructed, under the Bubnov-Galerkin

where with

((𝕀 − |𝜇|𝐾)𝑉𝑛 − 𝑀, 𝜓𝑗 )𝐿2,𝜑(Ω) = 0, for j = 1,, n, in the form n  j j ( x, y) and  the conformal map transforming D into . Here Vn ( x, y) = M ( x, y)  Aj | ( x  iy) |2 j =1

condition

M ( x, y ) =  G , ( x, y,  , )m( , ) |  (  i ) | 2 dd with 

𝔪(𝜉, 𝜂) = ℋ[𝑅𝑒(Ψ(𝜉 + 𝑖𝜂)), 𝐼𝑚(Ψ(𝜉 + 𝑖𝜂))]. 2 Then (𝑢𝑛 )𝑛≥1 is a sequence of functions of W2,0 () which converges in each fix point ( x , y ) of  to the unique generalized solution u of the problem (GE),(BCs). Proof of theorem 5.3 We’ve showed that for the problem (GE),(BCs) is equivalent to

b=0

, a > 0,c > 0 and

f  L2 () in the problem (GE),(BCs),

 cd a  ag c   d2 g2   f ( a , c ), ( , )  D, L0 w = w   a0    w = exp  4 a 4 c 2 ac     1  1  w cos( ), sin( )  = 0, [0,2 ), c  a 

where 𝐷 = {(𝜉, 𝜂) ∈ ℝ2 : if the number



 =  a0 

𝜉2 1 2 ( ) √𝑎

+

𝜂2 1 2 √𝑐

< 1}.

( )

d2 g2    is less than zero, then it can’t belong to the spectrum of the problem 4a 4c 

 w = w 1  1  w cos( ), sin( )  = 0, [0,2 ). c  a  Hence, according to the theorem 5.1, we have for all ( x , y ) belongs to , |𝑢(𝑥, 𝑦) − 𝑢𝑛 (𝑥, 𝑦)| ≤ exp (− 𝑐𝑑√𝑎𝜉+𝑎𝑔√𝑐𝜂

× √(∫𝐷 |exp (

2𝑎𝑐

𝑐𝑑𝑥+𝑎𝑔𝑦 2𝑎𝑐

) ‖𝐺𝐿′0 (

𝑥

𝑦

√

√𝑐

, 𝑎

, . , . )‖

𝐿2 (𝐷)

×

2

) 𝑓(√𝑎𝜉, √𝑐𝜂) − 𝐿′0 (𝑤𝑛 )(𝜉, 𝜂)| 𝑑𝜉𝑑𝜂) ;

According to the lemma 5.3, the sequence {wn } defined in the theorem 5.3 is a sequence such that L0 ( wn ) converges to L0 ( w) in

L2 ( D) . It means

2    exp cd a  ag c  f ( a , c )  L ( w )( , ) dd  0 n  D    2ac     converges to zero when n tends to   . It’s followed that u n  converges to u in each fix point ( x , y ) of the unit disk .

The proof of theorem 5.4 is similar to the proof of theorem 5.3.

37



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 R . This approximation is based on a method of decomposing a function in a nonorthogonal 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 :  ( , ) = f ( , ), ( , )  D, (6.1) 2

(6.2) | =  (a cos, b sin ) = 0, [0,2 [. 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

G , D ( , ,0,0) = G , D ( , ) = 

1 1 ln |  ( ) |=  ln  2   2  g ( , ) 2 2

(6.3)

where g is a harmonic function on D satisfying the problem

g ( , ) = 0, ( , )  D, 1 g  ,  = ln  2   2 , ( , )  . 2

(6.4) (6.5)

We will express the boundary condition (6.5) as a function of the polar coordinates ( r ,  ) of the ellipse polar equation of the ellipse, relative to its center

r=

 . The

O , is given by

a

.  a  b2  2 sin  1   2  b  If (hc , hs ) is a point in the ellipse  of which polar coordinates is ( r ,  ) , then acos asin hc = , hs = and 2 2 a2  b2 a  b 1 sin 2 1 sin 2 b2 b2

(6.6)

2

hc2  hs2 = r.

Hence the problem (6.4),(6.5) is written

g ( , ) = 0,

g hc , hs  = 

(6.7)

1 4

   a2  2   ln1   2  1 sin ( )   2 ln(a),   [0,2 [.    b   

Let’s determine the solution g of the problem (6.7),(6.8) in the form M





g ( , ) = ck ln (  xk ) 2  (  yk ) 2 .

(6.8)

(6.9)

k =1

The unknowns coefficients ck are real decomposition coefficients of function g in the non-orthogonal basis

ln (  x ) k

2



 (  y k ) 2 , k = 1,2,..., M

which are everywhere dense on a contour



; The M

pairs ( xk ; y k ) are collocation points

 which envelops completely the ellipse  and has no common point 38



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 ck are known, one can then write

( , ) = (  i ) exp 2 ( g ( , )  ih( , )),

(6.10)

M

where h( , ) = ck Arg ( xk   )  i( yk  ).





k =1

ln (  xk ) 2  (  yk ) 2 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 ck , k = 1,2,..., M : It is known that, for all

M

k,

the functions





ck ln (hc  xk )2  (hs  yk )2 =  k =1

1 4

   a2  2   ln1   2  1 sin ( )   2 ln(a),   [0,2 [.    b   

c1 ,, cM we will use, in the algorithm, the least squares method. The algorithm is realized in the Mathematica system with an ellipse  of semi-axes a = 4 and b = 3 and a confocal ellipse  whose Values of the semi-axes are 1,1 from those of  : For the determination of the coefficients

In[1] :