Harmonic interpolating wavelets in Neumann boundary value problem in a circle Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 91-100

DOI: 10.15826/umj.2019.1.009

HARMONIC INTERPOLATING WAVELETS IN NEUMANN BOUNDARY VALUE PROBLEM IN A CIRCLE

Dmitry A. Yamkovoi

Krasovskii Institute of Mathematics and Mechanics,

Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, Russia, 620990 [email protected]

Abstract: The Neumann boundary value problem (BVP) in a unit circle is discussed. For the solution of the Neumann BVP, we built a method employing series representation of given 2n-periodic continuous boundary function by interpolating wavelets consisting of trigonometric polynomials. It is convenient to use the method due to the fact that such series is easy to extend to harmonic polynomials inside a circle. Moreover, coefficients of the series have an easy-to-calculate form. The representation by the interpolating wavelets is constructed by using an interpolation projection to subspaces of a multiresolution analysis with basis 2n-periodic scaling functions (more exactly, their binary rational compressions and shifts). That functions were developed by Subbotin and Chernykh on the basis of Meyer-type wavelets. We will use three kinds of such functions, where two out of the three generates systems, which are orthogonal and simultaneous interpolating on uniform grids of the corresponding scale and the last one generates only interpolating on the same uniform grids system. As a result, using the interpolation property of wavelets mentioned above, we obtain the exact representation of the solution for the Neumann BVP by series of that wavelets and numerical bound of the approximation of solution by partial sum of such series.

Keywords: Wavelets, Interpolating wavelets, Harmonic functions, Neumann boundary value problem.

Introduction

Subbotin and Chernykh [1] constructed real 2n-periodic orthogonal wavelets and applied them to represent and analyze solutions of Dirichlet, Neumann, and Poisson boundary value problems for harmonic and biharmonic functions. In [2] the Dirichlet BVP in a unit circle was solved by means of interpolating-orthogonal periodic wavelets from [3]. In the present paper, we propose to use the same wavelets for solving the Neumann BVP in a unit circle. Moreover, our main interest is the exact representation of the solution for the Neumann BVP by series of wavelet bases and behavior of partial sums of such series. For the sake of convenience, we give the reader an adequate background for further study and partially repeat sections with interpolating and interpolating-orthogonal 2n-periodic wavelet construction from [1, 3].

1. Preliminaries

Consideration of autocorrelation functions for orthonormal scaling functions instead of orthonormal scaling functions is commonly used construction technique for interpolating wavelets in R. It is equivalent to replacement of scaling <p(x) function by function, which Fourier transform coincides with |(^(w)|2.

Let e be a fixed number from (0,1/3] and let (p£(w) be a Fourier transform of Meyer-type (see [4, 5]) function:

& (w)=0, M > (1+ e)/2;

&(w) = 1, M< (1 - e)/2;

02(u) + 02(w - 1) = 1, (1 - e)/2 < w < (1 + e)/2.

We also require that the function ^(w) is even and smooth on R with the symmetry center of its graph on the interval ((1 — e)/2, (1 + e)/2) at the point w = 1/2. Define functions ps(x) (s = 1,2) as in [1] and function p3(x) as in [2] such that:

= ^

(1 + (p£(w) — ps(w — 1) — ps(w + 1)) + i(sign w

|w| < (1 + e)/2, 0,

|w|> (1 + e)/2,

lp2 (w) = P (w) + i(sign w)^(w), P(w) = <p£ (w) ( <p£(w — 1) + <p£(w + 1)) ,

LP3(w) = ^2(w).

Here @(w) is a smooth even function on R vanishing together with its derivative at the points w = (—1 ±e)/2 and w = i1 ±e)/2, with the support {((-1 — e)/2, (—1+e)/2) U ((1 — e)/2, (1+ e)/2) } and even on intervals ± ((1 — e)/2, (1 + e)/2) with respect to their centers w = ±1/2. Functions ps(x)(s = 1, 2, 3) generates interpolating in C(R) systems {ps(2jx — k) : k € Z} (j € Z) on the grids {1/2j : l € Z} (j € Z). For s = 1,2 these systems are also orthogonal in L2(R). Unless otherwise stipulated, throughout the paper s = 1,2,3.

The 1-periodization process of the function ps(2jx)

Peips(2jx) = ^ ps(2j(x + =: $2'0(2nx), j € Z (1.1)

^eZ

converges uniformly on the interval [—1/2,1/2] (see [1]). Calculating the coefficients av in the expansion of the function $i'0(2nx) by the trigonometric system {e2nivx : v € Z}, we get

$s'°(2nx) = ^ ave2nivx, j € Z. (1.2)

veZ

Using (1.1), we find all coefficients av (v € Z)

1 1

a = I \ ,n i 2j (x + ,,))e-2niradx = ^ I ,n (2j

J (2j (x + ^))e-2nivxdx = ^ J Ps(2j (x + ^))e-2nivxdx

0 ^eZ jueZ 0

M+1

= [substitution: x + ^ = t] ^ J Ps(2j¿)e-2niv(i-^ dt = = J tp8(2H)e~2wU/tdt = ¿J2H) =

R

Substituting the coefficients av (v € Z) in (1.2), we obtain:

(2vrx) = 2"^ £ fa^y™*, j(=z,

veA^nZ

where A^ = 2j ((—1 — e)/2, (1 + e)/2). Replacing the variable x by x/(2n), we obtain 2n-periodic wavelet systems

veAinZ

which are interpolating on the grids {xlj := 2ttI/2:> : I = 0,2-? — 1} for s = 1,2,3 and orthogonal in L2(R) for s = 1, 2.

It is easy to see, that for n € Z

<1'^' 2 "[.,■) = E = «I'pi.r).

v eA^nz

So the sequence of spaces (1.3) has only 2J distinct linearly independent terms. Hence, we can

assume in the following discussion that k = 0,23 — 1. _

Define system of spaces {Vi := sp&n{&sh(x) : I; = 0,2^ -1} : j £ Z}. As follows from Ai n Z = {0} for j < 0 and <ps(0) = 1, we see that

*°'°(x) = E & (v )eivx = 1

veAgnz

and

^(x)=2~3 E 2 j < 0,

veA^nz

i.e., for all integers such that j < 0 and for all k € Z relation $S'fc(x) = $S'°(x) = const holds and thus we can consider the system of spaces {Vj} only for j € N U {0}. Further, for j € N U {0} define

spaces Wi as direct complement of vj to with the interpolation system {^ik (x) : k = 0,23 — 1} on the grid '■ I = 0,2-? — 1}, which is interpolating basis of 27r-periodic continuous functions.

Show that the ^ik(x) = H+h2k+1(x) holds for all j € N U {0} and for all k = 0,23 _ i. Since Vj C Vsj+1 (j € N U {0}), we see that

2J+1-1

•I'pi.r) E b.,M+hn(x), j € NU {0}, k = 0,23-1. (1.4)

Jn-

n=0

Using interpolating condition of basis (x) : k=0,2-î+1 — 1} on the grid {xj+1 : 1=0,2j+1 - 1}

and assuming x := 2irl/2:>+1 in (1.4), we find the coefficients bn (n = 0, 2-?+1 — 1):

2j+1_1

i.k( 2n1 A ^i+1.n( 2n1 A

(*£) = E E M„,„ ¡ = T,

n=0 n=0

so

2nn \

n

= n = 0.2Î+1-1

In view of bn obtained, the sum on the right side of the expression (1.4) may be written as two sums over even and odd indices

2j+1 -1 „ 2j -1

: .hi

n=0 n=0

2 j_1

+ EIj) -i-r1'2"-'(•'•) = .i-f(l^)^2k(x)+

2j_1

n=0

2j _1

+

n=0

2j+1_1

As a result, we have

2j -1

&>k{x) = $i+l>2k(x) + £ $i>k ( ^) ^+1'2ra+1 (a;), j'GNU {0}, k = 0, 2* - 1

n=0

i.e.,

2j -1

k

W -

' ra=0

^+1'2k(x) = ^'k(x) - £ 1) j jgNu {0}, k = 0, 23 ~ 1,

evj+1 evj

and it implies that

j = { £ Cj+1,k^s+1'k(x): Cj+1,k € R}

k=0

2j-1 2j-1 2j-1

= {'52 Cj+1, 2k j1' 2k (x) + £ Cj+1-2k+1 2k+V)} = { £ Cj+1- 2k ^,k(x)-k=0 k=0 k=0 2j-1 2j-1 , . 2j-1

Cj+1, 2k+1

2j+1-1

, 2k+1

k=0 n=0 k=0

2j-1 2j-1 2j-1

{E - E E ^H'i2^1^

k=0 n=0 k=0

2j-1 2j-1 2j-1 n=0 ' " k=0 n=0

2J -1 2 -1 2 -1 + E cj+1- 2n+1^+1' 2n+1(x^ = {¿2 cj+1-2k ^'k(x) + £ dj,ra$s+1' ^V) :

2 j_1

d3,n = " E CJ+l,2fc^'fc(27F(^+|1)) + CJ+l;2n+l} = V? © W/'.

k=0

In view of definitions of spaces Vsj and W, for all j € N U {0} and for all k = 0,2j — 1 relation

^s'k(x) = $s+1,2k+1(x)

holds.

Denote the interpolation projection of a function f € C2n (the space of continuous 2n-periodic functions) onto the Vsj by

2j-1 2 k

Ss,2i{x]f) = E/(^r)'!^'3 GNU{0}. (1.5)

k=02

Since (J°=0 = C2n, for f € C2n we have

Ss,2j (x; f) ^ f (x), (1.6)

R

2j-1 2j-1

f (x) = f (0) + EE cj,k^s'k(x) = f (0) + ££ Cj-k ^s+1'2k+1 (x) • (1.7)

j=0 k=0 j=0 k=0

Find all coefficients cj<k, (je N U {0}, k = 0,2-?' - 1) from (1.7). Because of Ss.2j(x]f) € Vi, Ss 2J+1 (x; f ) € V/+1 and definition of spaces we have Ss 2j+i (x; f ) — Ss 2j (x; f ) € W, i.e.,

2j-1 2j-1 (Ss,2j+1 (x; f ) — Ss,2j (x; f )) _ 2i+i = £ j$Ss+1>2k+1(x2Hi1) = £ j= Cj,1,

x ^j^1 k=Q k=Q

where j € N U {0} and l = 0,2j — 1. Using definition (1.5), we rewrite Ss 2j+i(x; f) and take

x ._ x21+1 >Xj .- j | 1

2j+1-1 2j+1-1

J+1 k=Q k=Q

Consequently,

9,/ = ^,2J+i(.tj2/+1; /) - ^(.T2^1; /) = /(-'f.,') - ¿^(x2^1; /), j e N u {o}, l = o,- i.

With (1.3), (1.7) and preceding expression the following relation holds for a function f € C2n

2j-1 2j-1

\k/

j=Q k=Q j=Q k=Q

2J -1 2 -1 f (x) = f (0) + EE Cj,k j (x) = f (0) + ££ (f jl1) — Ss,2j jl1 ; f )) X

2j -1

x j^x) = f (0) + ££ (f (x^ki1) — Ss,2j j1 ; f))2-(j!1) x (1.8)

j=Q k=Q

£ ?

X > <A, —TT e

V eA^+1nz

V \riv(x-'2-K('2k+l)/'2j+1) s I 2i+1

The definition of W imply Vsj = Vs0 ® (®j=01Wj). Then Ss-2j(x; f) is the partial sum of order 2j for (1.7) and from (1.6) series (1.7) converges uniformly. Thus for J € Z

J-1 2j-1

Ss,2J(x;f) = f(0) + ££ (f(xl+i1) — Ss,2j(x^1;f^$ss+1,2k+1(x) (1.9)

j=0 k=0

and as J ^ to

Ss,2J(x;f) ^ f(x).

R

2. Application to the solution of the Neumann BVP in a circle

Setting of the Neumann BVP in the unit circle K1 (see, for example, [6]):

AU(r,x) = ^ + + -2 ^ =0, U € C^m n C^(Kl),

dr2 r dr r2 dx2 (2 1)

dU v ' '

— (l,x) = gi(x) € C-2TT,

where reix (0 < r< 1, 0 < x< 2n) are points of the unit circle K1 centered at the origin of the polar coordinate system. It has been well known that necessary condition of solvability of the Neumann problem is

2n

/ g1(x)dx = 0, (2.2)

Q

and the problem have a unique solution up to an additive constant. Define harmonic in the unit circle polynomials $S,h(r, x):

<1'^' i r. x) := 2~j & (J^yWe^*-2*^, je N U {0}, k = 0, ï? - 1

v eA^nz

and consider series

2j _1

U (1,0) + EE (u (1, ■) - Ss . 2j (1, U (1, •))) j+^j ,2h+1(1, x).

j=0 h=0

Since U(r, x) is a harmonic in the unit circle function with continuous boundary value U(1,x), it follows that the above series converges uniformly on the boundary of K1 by taking into account (1.8) and (1.9) (where for f (x) we take U(1,x)). Because of maximum principle for harmonic functions, we obtain the following representation for U(r,x) in form of uniformly convergent in K\ series

2j _1

U(r,x) = U(1,0) + EE (u(1, ■) - Ss .2j (1, ■; U(1, O))^1 j ,2h+1(r,x) = 7=0 k=0 2j _1

= U (1,0)+EE (u (1, ■) - Ss . 2j (1, ■; U (1, •))) (x2++1)2_(j+1) x (2.3)

s

j=0 h=0

v eA^+1nz

Using (1.8), we have the following representation for function g1(x) € C2n in form of uniformly

convergent in K\ series

2j -1

siW^iioi + EE^')-^^!))«1)^11 E Vs^yi'-ww^.

j=° k=° VeAi+1nz

We may extend terms of the series into the interior of the unit circle to harmonic polynomials cj,k(g1)$S'k(r, x) and, consequently, we may extend the series into the interior of the unit circle to harmonic in K\ and in continuous K\ function.

2j -1

g1(r,x) := g1 (0) + EE (g1 (■) - S s , 2j (■; g1 ^(x2++1)2-(j+1)x

j=° k=° (2.4)

E (Ps\——)r'-'e

veA^+1nz

Because of series in (2.3) converges uniformly, we can perform a term-by-term differentiation with respect to r and multiplication by r and as result we get

dU j-1

j=0 h=0

veA^+1nz

As is easy to see that this function is harmonic in K1. In view of setting of the Neumann BVP, we

dU , dU

have -Q-{r,x)\r=l = gi(x), this implies that for 0 < r < 1 the equality r-^-(r,x) = g\(r,x) holds

as equality of two harmonic functions which are equal at the boundary of K1. Hence

dU 2j -1

T—(r,x)=gi( 0) + £ £ (gi(-) - >'„.,, x

j=0 k=0

veA^+1nZ

In consequence of (2.2), we also have

2n

J g1(r, x)dx = 0. (2.5)

dU

Indeed if we expand function gi(r,x) = r-^-(r,x) in a series by system {r\n\e%nx- : n € Z}

(for instance, with the use of Poisson kernel), then we get for 0 < r < 1

2n 2n

9i(r,x) = ± J9l(l,t)Pr(x-t)dt = ± I Y^ai^e^-^dt. 0 0 neZ

Interchanging of integration and summation and using (2.2), we arrive at

2n

neZ\{0} 0

resulting in (2.5).

Thus, using (2.5) and taking into account <ps(0) = 1, we obtain

2j -1

51(0) + £ £ (W) — Ss,2j(■; g1^(xJ2++1)2-(j+1) = 0, j=0 k=0

and numerical series on the left side of the equality converges. Consequently, the following equality holds

2j -1

91 (r,x) = £ £ (,„(•) ^ (•:'/,' 1 £

j=0 k=0 veA|+1nZ\{0}

Therefore, by setting

2n

-O'+D £ ^J^JH^-a^fc+i)/^1)

veA^+1nZ\{0}

0

0

we obtain

r

J=0 fc=Q

2j -1

j=Q k=Q veA|+1nz\{0}

where the series converges uniformly in K\. Setting

r +1 ,2k+1 ,%/ x) vl'f' '(/-.:= / ^-(/ --'V =

0

= 2"^ £ j GNU {0}, = 0,2^-1,

^ — i

f/(r,x) = U( 1,0) EE („,(•) - Ss^(-]gl))(xfpl1)^-1(r,x), re- € 77,. (2.6)

and calculating the U(r, x) from the preceding equality, we formulate the following theorem.

Theorem 1. Under conditions of setting of the Neumann BVP (2.1) we obtain for s = 1, 2, 3

2j — 1

EE

j=0 fc=0

Series in (2.6) converges uniformly in K\ and Î7(l,0) is a constant.

Proof follows from preceding equations. □

Also we obtain the error for approximation of solution U(r, x) of the problem (2.1) by partial sums of series (2.6) denoted by Ss,2j(r, x; U, 1 ). Denote by (f)C2n the best approximation

of a function f in C2n by trigonometric polynomials of order N—J = |_2J—1(1 — e)J.

Theorem 2. Under conditions of setting of the Neumann BVP (2.1) for s = 1,2,3 and J € Z+ := {j € Z : j > 0} the function Ss 2j (r, x; U, approximates the solution U(r, x) of problem (2.1) with accuracy guaranteed by the inequality

\\U(r,x) - Ss2j(r,X] U, 'I'-1) Hc(A'i) < + 11^11)^(51)^,

Estimates for norm of the operator Ss ,2j (interpolation projection onto the subspace Vjs) can be found in Theorem from [2].

Proof. For convenience introduce the following notation:

'"'/./■''//I < (</iH >',.2 (•:'/,m-'f:,1)- i € N U {0}, /,• 0.2' 1. Using Euler's formula, we can represent (2.6) in the form

2j — 1

U(r, x) = U(0,0) + E £ Cj,fe(5fi)2 E &1 ^^ 1 — cos | z/(x -

2n(2k + 1).

2j+1/ v V v 2j+1

j=0 k=0 veA^+1nN

Harmonic Interpolating Wavelets in Neumann Boundary Value Problem in a Circle 99 and partial sum Ss-2j (r, x; U, ) in the form

J -1 2j-1

Ss,2J(r, x; U, ) = U(0,0) + £ £ Cj,k(g1)2-j x

j=0 k=0

2n(2k + 1),

£ ps

veAi+1nN

cos v x —

2j+1 v 2j+1

Note that the following representations hold

2n

1

(r,x) =

0 M=1 ^ j=0 k=0

2n/ / J -1 2j-1

n 0 M=1 ^ j=0 k=0

(2.7)

It follows from

2n

1

n J ^ ^

0 ^=1

0 ^=1 veA^+1nZ+

veA|+1nN

v

where N+ = |"2j' (1 + e)] and the second equality holds in view of

2n

1 Jcos(Ai(x-0)cos - =

7T

0

= cos - >)) A^1 n Z+.

Let Ss 2j(r, x;g1) be a partial sum of series in (2.4), then

2n

|U(r,x)-Ss.2j(r,x-,U,*-l)\ = |iy ^ _ S,2nr.

n J ^ ^ 0 ^=1

<

2n

1 f|r cos(^(x — £))

n J I ' ^ 0 ^=1 2n 2n

1 ( f^A1/2f 1 COs(^(x — {))

de <

* £( j (j |Ecos , =

0 0 ^=1

-V2tt(tt^2—) \\gi{x)) -Sa .2j{r,x] gi) | |C(2jr) < -^=(1 + ||SSi2j||)£jv£iJ(sri)c2,r, n M=1 ^ V3

where the first equality follows from (2.7), the second equality follows from Parseval's identity, the second inequality follows from Holder's inequality and the last inequality follows from Theorem in [2]. As the final result we have

||U(r,x)-Ss<2j(r,x-,U,*-1)\\C{Kl) < ^(1 + 115^11)^(51)^.

□

3. Conclusion

Theorem 1 gives the solution (2.6) (up to an additive constant) of the problem (2.1) in form of uniformly convergent in K\ series of harmonic interpolating 2-/r-periodic wavelets. In this case, coefficients of series in (2.6) have an easy-to-calculate form in preference to calculating coefficients (integrals) in case of implementing orthogonal 2n-periodic wavelets. This useful fact simplify the numerical implementation of the suggested method.

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