Asymptotic solution of the perturbed first boundary value problem with a non-smooth coefficient

Consider the partial differential equation of elliptic type sAu(p,ф,£) - ^p — 1u(p,ф,£) = f (р,ф),   (p, ф) e D ,(1)

with the inhomogeneous boundary conditions of the first kind

u(1,ф,s) = ^(ф), u(a,Ф,s) = ^(ф), фе [0,2п],(2)

, л                                 .    д2     1 д 1   д2    г.

where 0 < ε is a small parameter, Δ =    +     +        , f∈C∞( D ), D = {(ρ,ϕ)| 1 < ρ < a is a con- др2  p др  p2 дф2

stant, 0 ≤ ϕ ≤ 2 π }, ψ _k ∈ C ^∞ [0, 2 π ], k = 1, 2.

Mathematical models of many natural phenomena are described using boundary value problems for partial differential equations [1–2].

According to the theory of partial differential equations, the solution to considered first boundary value problem (2) for differential equation of elliptic type (1) exists and is unique [3]. We are interested in the behavior of the solution, i. e. dependence of this s olution on the small parameter ε , where ε → 0. We consider the question about the part of the domain D in which passage to the limit is performed.

The considered first boundary value problem has two features (bisingularity) [2]. The first singularity is the fact that the solution to the limit equation ( ε = 0) cannot satisfy the boundary conditions, since the limit equation is not a differential equation. The second feature states that the solution to the limit equation is not a smooth function in the domain D :

u ( p , ф , 0) = - f ) ( p , ф ) / 7 p - 1.

In order to show how this nonsmooth solution affects the asymptotic behavior of the solution to the Dirichlet problem, we consider the classical outer asymptotic expansion of the solution to the first boundary value problem:

U ( p , ф , s ) = £ ^ U k ( p , ф ), £ ^ 0.                            (3)

k =0

Substitute series (3) into differential equation of elliptic type (1) and equate the coefficients at the same powers of s , then we obtain:

- 4 p - 1 u 0 ( p , ф ) = f ( p , ф ) 4 p - 1 ^uk ( p , ф ) = A u k -1 ( p , ф ), k e N.

J                                     Z           f ( P , ф )       Z X   ^A uk -1⁽ p , ф ) , _

Here we determine all u k ( р , ф ) as follows: u ₀( p , ф) =--.     ,    u_k ( p , ф ) = — ^k. ¹   —, k e N .

4 p - 1              4- - 1

In the general case, all these functions u k ( р , ф ) have increasing singularities of the form U k ( p , ф ) = O ( ( p - 1 )^{- (} 5 k ^{+ 1)/ 2} ) , p ^ 1, k e N о = 0,1,2,...

Therefore, asymptotic solution (3) can be represented as follows:

,    »(          Ak

£ ^ 0,

U ( p ^, ф ^{, £} ) =-== £            F k ( p , ф ) ,

4p -1 k=o( 4( p -1)5 J where Fk e CTO (D),k = 0,1,2,...

Hence, series (3) is an asymptotic solution to the first boundary value problem only in the domain {( p , ф )| 1+ s ²/⁵ <  p <  a , 0 <  ф < 2 n }, and does not satisfy the boundary condition u ( a , ф , s ) = ^ ₂( ф ) on the circle p - a , and solution (3) loses its asymptotic character in the domain {( p, ф )| 1 <  p <  1+ s ²², 0 <  ф <  2 n }.

Let us prove the following theorem.

Theorem . The asymptotic solution to first boundary value problem (1) and (2) can be represented in the following form:

u ( p , ф , s ) = £ s ^k V k ( p , ф ) + £ ^ Z k ( t , ф ) + 1 £ ^ ^k W k ( т , ф ) , s ^ 0, k =0               k =0               ^p k =0

where t = (a - p )/ X , X = ^£ , т - ( p - 1)/ p 2 , p = Ps .

The functions v_k ( p , ф ),z_k( t , ф ), w_k ( т , ф ) are specified below, in the proof of Theorem.

Proof . First, construct a formal asymptotic solution to the first boundary value problem. As usual, we look for such a solution in the form [4–6]

u ( p , ф , s ) = V ( p , ф , s ) + Z (t, ф , X ) + W( т , ф , p ) ,                        (4)

where

TO

V ( p , ф , s ) = r s ^k V k ( p , ф ) , k =0

TO

Z ( t , ф , 2 ) = r 2 k Z k ( t, ф ) , k =0

t = (a - p )/ X , X = V F ,

W ( т ,ф, p ) = 1 rr P w k ( т , ф ) , т - ( p - 1)/ p =, p = ^SP.

P k =0

Taking into account conditions (2), we obtain:

W(0,ф, p) - ^1(ф) - V(1,ф,р) Vk: lim wk(т,ф) = 0, фе[0,2п];(5)

Z(0,ф, X) - ^2(ф) - V(a,ф, X2), Vk: lim zk(t,ф) = 0, фе[0,2п].(6)

t ^to

We write differential equation of elliptic type (1) in the form [4–6]

sAu(p,ф,£) - 4p -1 u(p,ф,e) -/(p,ф,s) - h(p,ф,e) + h(p,ф,s),(7)

where

(

ZZ       A V   2 k, Z X    2 k +1   h 2 k +1,1 ⁽А ^{ф )}

h(p,ф, s) = r s  h2k (P,ф) + s     ---,---- k=0 (                  (  Vp -1

h 2 k +1,3 ^{( ф )}

4 ( p - 1)³

, h 2 k +1,1 ( p, ^ф ), h 2 k +1,з ( ф ) are un-

known functions for the present.

Substituting formal series (4) into differential equation of elliptic type (7), we have sAV(p, ф, s) - 4p-1V(p, ф, s) = f (p, ф, s) - h(p, ф, s), (p, ф) e D,               (8)

д 2 W   2 d W _z 2x2 0 ² W    Г™ ,A 2    5\ 7 X тч            zax

Ц —+ + Ц c— + (ц C) ----JtW = h(1 + тц ,ф,ц ), (т,ф)еDo,(9)

(дт2        дт           дф2        J ''

^-Z-XC — + (XC)2^-Z-Ta-1 /1 —— Z = 0, (t,ф)еD1,(10)

д t ²        д t          д ф>          4 a - 1

where c =------,— < c < 1,   D0 = {(т,ф)| 0 < т< (a - 1)/ц2, 0 < ф< 2л},   Di = {(t,ф)| 0 < t <(a-

1 + тЦ a

1)X 0 < ф < 2 n}, C = —1—,1 < C < 1, W = W(т,ф,ц), Z = Z(t,ф,Х). a - Xt a to

From (8), taking into account that V ( р , ф , е ) = ^ e^k v_k ( р , ф ) and equating the coefficients at the k =0

same degrees of e , we obtain:

v o ( P , Ф ) = - f^^ L h 0 ^ф

4 р - 1

if h ₀( ф ) = f( 1, ф ), then v ₀ e C ( D ) .

In this case, v ₀ ( р , ф) = - 4 р - 1 v ₀ ( р , ф ), v ₀ e C ” ( D ) and

A ^v o =

____¹____ v -  '   д о 4 ( р - 1 )      V р - 1 ⁰ р

^- V ^{р - 1}

д ² v o д р ²

—

1 (    v о

р ( 2 4р- 1

+V ^{р -1} - v ⁰- ^- V ^{р - 1 д} р J

02^ д ф²

Let us determine v 1 ( р , ф ):

^{A v} o ⁽ P ^, Ф ^{) + h} 1 ⁽ P ^, Ф ^{) v} 1 ^{( р} , ^{ф )} =          i-----        ,

V ^{р - 1}

i_t,z х       ^v o,o ⁽ ф ⁾  ^^{з v} o,1 ⁽ ф ⁾ р ^{- 2 v} o,o ⁽ ф ⁾   .     -                X -  / X  д ^ о (1, ф )

^{let h}1 ⁽ р , ф ) = ^-   ’      + —’——==^----, ^{where v} 0,0 ⁽ ф ) = ^v 0 ^(1, ф ), ^v 0,1 ⁽ ф ) = -Л---.

4( р - 1)^3/2        4 рр - 1              ^{,               ,         д} р

Then

A v 0 ( р , ф ) + h 1 ( р , ф ) = V р - 1 v 0 ( р , ф ), V 0 e C ^да (D) .

Hence, z x  Av0 (р,ф) + h (р,ф)  ~

V 1 ⁽ р , ф ) = —---J——¹-----= V 0 ⁽ р , ф ) .

4 р ^{- 1}

Continuing this process in a similar way, for v _{2 k +1}( р , ф ) we have:

	1;   (nmA_ A V 2 k(p , ф ) l h 2 k ± 1 ( £ , ф ) v 2 k + 1( р , ф )                                ,
let	,     ,    x       V 2k,o ( ф )    3 v 2 k 0,1 ( ф ) р - 2 v 2 k ,0 ( ф ) h2 k +1 ( р , ф ) =  л<    1л3/2+         л   /----7        , 4( р - 1)3/2        4 р/р - 1
where	-   Z Л    -71   . -    z x    д р 2 А(1, ф ) v 2 k ,0 ( ф ) = v2k (1 ф ), v2k ,1 ( ф ) =     I      , ’                       ’             д р
then	A v 2 k ( р , ф ) + h 2 k +1 ( р , ф ) = х/ р - 1v2k ( р , ф ), v2k e C ” (D) .

Hence, we have:

v ₂ k +1 ( р , ф ) = ^A v ^{2 k ( р} , ф+ ^ 1^k + ^^рф = v 2 k ( р , ф ). 4 р - ¹

And, for v _{2 k} ( р , ф ), we have:

л, (_Лт\- ^A v 2 k -1 ⁽Р, ф ) ⁺ h2 к ^{Р ф v} 2 к ^{( р} , ^{ф ) -}           /----т

Р - ¹

if h 2 _к ( ф ) = - A v ₂ к -1 (1, ф ), k е N , then v_2к е C ( D ) .

Note that v _{2 к + 1} е C ® ( D ) , v _{2 к} е C ( D ) .

Let us determine the functions w_k ( т , ф ). Rewrite relation (9) as follows:

V¹ k^ Cw wk.    2 ^wk , 2 4 5² wk    Г 1 X Ю^ 1 к x з_ ² h 2 к +1,3 ^{( ф )}     4 h 2 к +1,1 ^{( рт} , ^{ф )}

L р ^тт ⁺ ср д ⁺ cp ^т ^{- T}^^wk ^- L р  ^h2к ⁽Ф) ⁺ р —гч^{- +} р —h=— .

к-0   V дт        дт        дф         J  к-0    к              \T             VTJ

Hence, taking into account (5), we have:

52 W lw0 = —у- - Tww0 - h0, wо(О,ф) = 0, lim w0(т,ф) - 0,(11)

дт2

Iw 1 = 0, w 1(0,ф) = Ф1(ф) - v0(1,ф), lim w1(т,ф) - 0,(12)

T^® cW  h\ 3(ф)

lw2--c —0 +—’— , w2(0,ф) = 0, lim w2(т,ф) -0,(13)

дт   4?

W--c^w-, w3(0,ф) = 0, lim Wз(т,ф) -0,(14)

дт dw-   752 wn  ^и(тр,ф)

lw4--c —2-c2---^ + ——-;=— , w4(0,ф) = 0, lim w4(t,ф) -0,(15)

дт     дф2     VtT dw,   + д2 w.

lw5 --c —3 -c2---у, w5(0,ф) = 0, lim w5(т,ф) - 0,(16)

дт     дф2

dW/| lw6--c —-4-c2---2-, w6(0, ф) = -v 1(1,ф), lim W6(т,ф)-0,(17)

дт     дф2

d w_j - _{2    9} д ² w_j - ₄

lWj - -c        c2----^ , wj(0, ф) = 0, lim w=(т, ф) - 0, j - 7,8,9,(18)

дт       дф^

lw5к+1 --cdw5к-1 -c2 д w5к-3 , w5к+1(0,ф) = -Vk(1,ф), lim W5к+1(т,ф)-0,(19)

дт       дф2

lW10к --c^w-1 к-2 -c2 д w102'-4 + h2к(ф), w 10к(0,ф) = 0, lim wwк(т,ф) -0,(20)

дт        дф^

dw10i    2д W10k-2   h2к+1,3(ф)            х л г         / х л /Л1Х lW10 к+2 -- c -4т02 - c „ 2' 2 +----7=^- , w 10 к+2(0,ф) = 0, lim w10 к+2(т,ф) - 0 , дт        дф2       Тт3

CwlOt-i-2    2 д W10t   h2к+1,1(тр,ф)        /л х л г          / х л /ллх lw-0к+4--c —г212-c ----0к +----h=----, w 10к+4(0,ф) = 0, lim wwк+4(т,ф) -0.(22)

дт      дф^      т

Let us prove the following statement.

Lemma. The problem z "(t) - xZz(t) - —c^, t е (0, ®), к - 0,1,3, z(0) - z0, lim z(t) - 0 , 4tk has a unique solution (here с, z0 are constants).

Proof. We know that the corresponding linear homogeneous differential equation of the second or der z "(t) - x/fz(t) - 0 has two independent solutions:

z-(t) - xIl2/5 (4t5/4 /5), z2(t) - x/tK2/5 (4t5/4 /5) , where 12/5(5),K2/5(5) are the modified Bessel functions.

Solutions to the homogeneous equation have the following property:

Г ^{1  4} ,5/4 ⁴

z 1 ( t ) = O t ⁸ e ⁵ I

_Г 1

’ ^z 2 ⁽ t ) = ^O t ⁸

I

-  — ⁴ 1 ^5/4 '

e ⁵

, t ^да , z ₁(0) = 0, z ₂(0) ^ 0, W ( z 1 , z ₂) = —, 0 ^ c 1 = const. c

7                                                        ¹

The solution to the inhomogeneous equation with the corresponding boundary conditions can be represented as

z ⁽ t ) = zz (0 ) ) ⁺ c ^C 1 ( z 2 ⁽ t^ O s k ^{/2 z} 1 ^{( 5} ) ds ^{+ z} 1 ⁽ t ) I s k ^/2 z 2 ⁽ s ) ds ) .

The asymptotic behavior of the solutions z 1 ( t ) and z ₂ ( t ) to the homogeneous equation implies that z ( t ) = O ( t ^{— k + 1)/2} ) when t ^ ro, and z ( t ) = O ( t ^{2 — k /2} ) , k = 0,1,3 , when t ^ 0. This completes the proof of Lemma.

By virtue of Lemma, there exist unique solutions to boundary value problems (11)–(22). It follows from the properties of z 1 ( t ) and z ₂( t ) that w _{10 k} ( t ) = O (1/ t ^1/2), w _{10 k +2}( t ) = O (1/ t ² ), w _{10 k +4}( t ) = O (1/ t ), t ^ ro, and the remaining solutions w n ( t ) are exponentially small when t ^ ro.

Let us consider boundary value problem (10), (6). Since a = X t /( a — 1), 0 <  a <  1, then the following decomposition takes place:

Гл -----  1       11 Г 1

V1 — a = 1 +>---

= ^ 1 j !2 1 2

—

4 Г 1

¹ | ... | 2^— j ⁺ 1 1 ^{( — a )}} .

Therefore, homogeneous differential equation (10) can be written in the following form: 2^

—41 — Va — 1z0 = 0,  (t,ф)eD1, zo(O, ф) = ф2(ф)-vo(a,ф), lim z0(t,ф) = 0, д t2                                                                         t д2 z д tг

[    д z 0 д ² z

- zk = Gk  zo ,----,----- k k I 0  дt  дф

' 0

. 2 ’

....’ z k —1

^д z k —1 ^д 2 z k —1

’   д t  ’ д ф ²

,t , ( t , ф ) e D i ,

and the boundary conditions can be represented as z2n(0,ф) = -vn(a,ф), z2n-1(0,ф) = 0, lim zk(t,ф) = 0, t ^x where the functions Gk z^

a z o ^д 2 z 0

⁰’ д t ’ д ф ² ”

₇     ^д z k —1 ^д 2 ^zk —1

...’ k ^— 1 ’   д t  ’  д ф ²

,t depend linearly on the previous solu-

tions and on the derivatives of these solutions, i.e. on z k _— 1

^{д z}k —1   ^{д 2 z}k —1

’   д t  ’ дф ²

and on the variable t .

These problems have unique solutions, which decrease exponentially when t ^ да :

z 0 ( t , ф ) Цф 2 ( ф ) — V 0 ( a , ф ) ) e "^a^¹t , z k ( t , ф ) = e -^{4 а — Г !} P_k ( t , ф ), P k e C ” ( D) .

We have determined all terms of formal asymptotic solution (4). Let us estimate the remainder of this expansion.

Let          u ⁽ P ’ ф ’ £ ) = V 2„ +1 ⁽ P ’ ф ’ £ ) ⁺ Z 4 _n +2 ⁽ t ’ ф ’ ^{X ) +} W 0 n +6 ^{( т} ’ ф ’ M ) ⁺ R 2 n +1 ⁽ P ’ ф ’ ^e ) ,         ^where

2 n +1                                       4 n +2                                         । 10 n +6

V2 n+1 ( Р’ф’£ )=E £kvk ( Р’ф) , Z4 n+2 ( t ’ ф’ X)=E Xzk ( t ’ ф) , W10 n+6 (T’ ф’ M ) = - E Mkwk (Т,ф) , k=0                                 k=0

R 2 n ₊ i ( p , ф , £ ) is the remainder of the series.

Then we obtain the following problem for the residual function R _n( р , ф , £ ):

£^R2n+1(Р’ф,£) — VP — 1R2n+1(Р’ф’£) = O(£2n+2)’ £ ^0,     (P’ф) e D,(23)

R2n+1(1,ф,£) = O(e4/^), R2n+1(а,ф,£) = O(£2m+2), £ ^0, фе[0,2n] .(24)

It is impossible to apply the maximum principle directly, since ^ p — 1 >  0 for 1 <  p <  а . Therefore, first of all, we replace R _{2 n+1}( р , ф , £ ) = ( а - p 2 /2) r _{2 n+1}( р , ф , £ ).

Then problem (23)–(24) takes the form:

^s ^ ^r2n +1 -   4^2 ^ r n¹ - ^{f      +} „ 4 ^S г ^ ^r 2 n +1 = O ( ^{s 2} ” ^{+ 2} ) , ^p , Ф ) ^e D ,

• 2 a - p2  dp   (        2 a - p2 J        v     ’

Г 2n +1 (1, Ф , s ) = O ( e ^{- 1/} ) , r n +i ( a, Ф , e ) = O( e ² m ^{+ 2}), e ^ 0, ^ е [0,2 п ].

For this problem, the maximum principle [3] can be applied. As a result, we obtain an asymptotic estimation: r_{2n + 1} ( p , ф , s ) = O( s ^{2 m + 1}), s ^ 0, ( p , ф ) e D .

Therefore, R 2n +1 ( p , Ф , s ) = O ( s ^{2 m + 1}), s ^ 0, ( p , ф ) e D .

This completes the proof of Theorem.