Composite Structures 126 (2015) 227–232

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Convergence theorem for the Haar wavelet based discretization method

http://dx.doi.org/10.1016/j.compstruct.2015.02.0500263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: juri.majak@ttu.ee (J. Majak).

J. Majak a,⇑, B.S. Shvartsman b, M. Kirs a, M. Pohlak a, H. Herranen a

a Dept. of Machinery, Tallinn University of Technology, 19086 Tallinn, Estoniab Estonian Entrepreneurship University of Applied Sciences, 11415 Tallinn, Estonia

a r t i c l e i n f o a b s t r a c t

Article history:Available online 25 February 2015

Keywords:Haar wavelet methodAccuracy issuesConvergence theoremNumerical evaluation of the order ofconvergenceExtrapolation

The accuracy issues of Haar wavelet method are studied. The order of convergence as well as error boundof the Haar wavelet method is derived for general nth order ODE. The Richardson extrapolation method isutilized for improving the accuracy of the solution. A number of model problems are examined. Thenumerically estimated order of convergence has been found in agreement with convergence theoremresults in the case of all model problems considered.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, the Haar wavelets are most widely used wavelets forsolving differential and integro-differential equations, outperformingLegendre, Daubechie, etc. wavelets (Elsevier scientific publicationstatistics). Prevalent attention on Haar wavelet discretizationmethods (HWDM) can be explained by their simplicity. The Haarwavelets are generated from pairs of piecewise constant functionsand can be simply integrated. Furthermore, the Haar functions areorthogonal and form a good transform basis.

Obviously, the Haar functions are not differentiable due to dis-continuities in breaking points. As pointed out in [1] there are twomain possibilities to overcome latter shortcomings. First, the quad-ratic waves can be regularized (‘‘smoofed’’) with interpolatingsplines, etc. [2,3]. Secondly, an approach proposed by Chen andHsiao in [4,5], according to which the highest order derivativeincluded in the differential equation is expanded into the seriesof Haar functions, can be applied. Latter approach is applied suc-cessfully for solving differential and integro-differential equationsin most research papers covering HWDM [1,4–28]. Following thepioneering works Chen and Hsiao in [4,5] Lepik developed theHWDM for solving wide class of differential, fractional differentialand integro-differential equations covering problems from elasto-statics, mathematical physics, nonlinear oscillations, evolutionequations [1,6–10]. The results are summarized in monograph[11]. It is pointed out by Lepik in [1,11] that the HWDM is conve-nient for solving boundary value problems, since the boundary

conditions can be satisfied automatically (simple analyticalapproach).

Composite structures are examined by use of wavelets first in[12,2]. In [12] the free vibration analysis of the multilayer compos-ite plate is performed by adapting HWDM. The static analysis ofsandwich plates using a layerwise theory and Daubechies waveletsis presented in [2]. The delamination of the composite beam isstudied in [13]. During last year Xiang et al. adapted HWDM forfree vibration analysis of functionally graded composite structures[14–18]. In [14–18] a general approach for handling boundaryconditions has been proposed. In all above listed studies the Haarwavelet direct method is applied. The weak form based HWDMhas been developed in [19], where the complexity analysis of theHWDM has been performed. Recent studies in area of waveletbased discretization methods cover solving fractional partialdifferential equations by use of Haar, Legendre and Chebyshevwavelets [20–25]. In [26–28] the Haar wavelets are utilized forsolving nuclear reactor dynamics equations. The neutron pointkinetics equation with sinusoidal and pulse reactivity is studiedin [26]. In [27,28] are solved neutron particle transport equations.In [29–31] the HWDM is employed with success for solvingnonlinear integral and integro-differential equations.

Most of papers overviewed above found that the implementationof the HWDM is simple. Also, the HWDM is characterized mostcommonly with terms ,,simple’’, ‘‘easy’’ and effective‘‘ (see [1,14–18,25–28] and others). The review paper [32] concludes that theHWDM is efficient and powerful in solving wide class of linear andnonlinear reaction–diffusion equations.

However, no convergence rate proof found in literature for thismethod. It is shown in several papers [33–35] that in the case of

228 J. Majak et al. / Composite Structures 126 (2015) 227–232

function approximation with direct expansion into Haar wavelets,the convergence rate is one. This result hold good for functionapproximation in integral equations, but does not hold good forHWDM developed for differential equations above, since in thesemethods instead of the solution its higher order derivative isexpanded into wavelets.

The aim of the current study is to clarify the accuracy issues ofthe HWDM, based on approach introduced by Chen and Hsiao in[4,5], and featured for solving general nth order ordinal differentialequations (ODE). This question is open from 1997 up to now.Answer to this question allows to give scientifically foundedestimate to HWDM, also to make comparisons with other methods.

2. Haar wavelet family

In the following the Haar wavelet family is defined by usingnotation introduced by Lepik [1]. Let us assume that the integra-tion domain ½A;B� is divided into 2M equal subintervals each oflength Dx ¼ ðB� AÞ=ð2MÞ. The maximal level of resolution J isdefined as M ¼ 2J . The Haar wavelet family hiðxÞ is defined as agroup of square waves with magnitude �1 in some intervals andzero elsewhere

hiðxÞ ¼1 for x 2 n1ðiÞ; n2ðiÞ½ Þ;�1 for x 2 n2ðiÞ; n3ðiÞ½ Þ;0 elsewhere;

8><>: ð1Þ

where

n1ðiÞ ¼ Aþ2klDx; n2ðiÞ ¼ Aþð2kþ1ÞlDx;n3ðiÞ ¼ Aþ2ðkþ1ÞlDx; l¼M=m; Dx¼ ðB�AÞ=ð2MÞ: ð2Þ

In Eqs. (1) and (2) j ¼ 0;1; . . . ; J and k ¼ 0;1; . . . ;m� 1 stand fordilatation and translations parameters, respectively. The index i iscalculated as i ¼ mþ kþ 1. Each Haar function contains one square

wave, except scaling function h1ðxÞ � 1. The parameter m ¼ 2 j

(M ¼ 2J) corresponds to a maximum number of square waves canbe sequentially deployed in interval ½A;B� and the parameter kindicates the location of the particular square wave. Since thescaling function h1ðxÞ � 1 does not include any waves herem ¼ 0; n1 ¼ A; n2 ¼ n3 ¼ B. The Haar functions are orthogonal toeach other and form a good transform basis

Z 1

0hiðxÞhlðxÞdt ¼ 2�j i ¼ l ¼ 2 j þ k;

0 i – l:

(ð3Þ

Any function f ðxÞ that is square integrable and finite in the interval½A; B� can be expanded into a Haar wavelets as

f ðxÞ ¼X1i¼1

aihiðxÞ: ð4Þ

The Haar coefficients

ai ¼ 2 jZ B

Af ðxÞhiðxÞdx; i ¼ 1; . . . ;2 j þ kþ 1 ð5Þ

can be determined from minimum condition of integral squareerror as

Z B

AE2

Mdx!min; EMj j ¼ f ðxÞ � f MðxÞj j; f MðxÞ ¼X2M

i¼0

aihiðxÞ: ð6Þ

In Eq. (5) f ðxÞ and f MðxÞ stand for the exact and approximatesolutions, respectively. The integrals of the Haar functions (1) oforder n can be calculated analytically as [1]

pn;iðxÞ ¼

0 for x 2 A; n1ðiÞ½ Þ;x�n1ðiÞð Þn

n!for x 2 n1ðiÞ; n2ðiÞ½ Þ;

x�n1ðiÞð Þn�2 x�n2ðiÞð Þnn!

for x 2 n2ðiÞ; n3ðiÞ½ Þ;x�n1ðiÞð Þn�2 x�n2ðiÞð Þnþ x�n3ðiÞð Þn

n!for x 2 n3ðiÞ;B½ Þ:

8>>>>><>>>>>:

ð7Þ

Note that the integrals of the Haar functions are continuousfunctions in interval ½A;B�. Also, the first integrals of the Haarfunctions are triangular functions (a ¼ 1Þ.

3. Convergence analysis of Haar wavelet discretization method

Let us consider nth order ordinal differential equation (ODE) ingeneral form

Gðx;u;u0;u00; . . . uðn�1Þ;uðnÞÞ ¼ 0; ð8Þ

where prime stand for derivative with respect to x. According tomost commonly used approach introduced in [4,5] instead ofsolution of the differential equation its higher order derivative isexpanded into Haar wavelets

f ðxÞ ¼ dnuðxÞdxn ¼

X1i¼1

aihiðxÞ: ð9Þ

Using notation introduced in previous section, the sum in Eq. (9)can be rewritten as

f ðxÞ ¼ a1h1 þX1j¼0

X2 j�1

k¼0

a2 jþkþ1h2 jþkþ1ðxÞ: ð10Þ

In Eqs. (9) and (10) i ¼ mþ kþ 1; j ¼ 0;1; . . . ; J; k ¼ 0;1;

. . . ;m� 1;m ¼ 2 j (M ¼ 2JÞ. By integrating relation (9) n times oneobtains the solution of the differential equation (8) as

uðxÞ ¼ a1ðB� AÞn

n!þX1j¼0

X2 j�1

k¼0

a2 jþkþ1pn;2 jþkþ1ðxÞ þ BTðxÞ: ð11Þ

In Eq. (11) BTðxÞ and pn;2 jþkþ1ðxÞ stand for boundary term and nthorder integrals of the Haar functions are determined by formula(7), respectively.

Without loss of generality it can be assumed in the followingthat A ¼ 0; B ¼ 1 since the differential equations can be convertedinto non-dimensional form by use of transform s ¼ ðx� AÞ=ðB� AÞi.e. x ¼ Aþ ðB� AÞs (see [19]).

Theorem 1. Let us assume that f ðxÞ ¼ dnuðxÞdxn 2 L2ðRÞ is a continuous

function on ½0;1� and its first derivative is bounded

8x 2 ½0;1� 9g :df ðxÞ

dx

�������� 6 g; n P 2 ðboundary value problemsÞ:

ð12Þ

Then the Haar wavelet method, based on approach proposed in[4,5], will be convergent i.e. jEM j vanishes as J goes to infinity. the con-vergence is of order two

EMk k2 ¼ O1

2Jþ1

� �2" #

: ð13Þ

Proof. It implies from Eqs. (5), (6) and (10) The error at the Jthlevel resolution can be written as

EMj j ¼ uðxÞ � uMðxÞj j ¼X1

j¼Jþ1

X2 j�1

k¼0

a2 jþkþ1pn;2 jþkþ1ðxÞ�����

�����: ð14Þ

J. Majak et al. / Composite Structures 126 (2015) 227–232 229

Expanding quadrate of the L2-norm of error function, oneobtains

EMk k22 ¼

Z 1

0

X1j¼Jþ1

X2 j�1

k¼0

a2 jþkþ1pn;2 jþkþ1ðxÞ !2

dx

¼X1

j¼Jþ1

X2 j�1

k¼0

X1r¼Jþ1

X2r�1

s¼0

a2 jþkþ1a2rþsþ1

Z 1

0pn;2 jþkþ1ðxÞpn;2rþsþ1ðxÞdx:

ð15Þ

In order to estimate complex expression (15) it is reasonable first toderive estimates for its components: the coefficients ai and the inte-grals of the Haar wavelets pn;iðxÞ. By use of and formulas (5) and (1)

the coefficients ai ði ¼ 2 j þ kþ 1Þ can be evaluated as

ai ¼ 2 jZ 1

0f ðxÞhiðxÞdx ¼ 2 j

Z n2

n1

f ðxÞdx�Z n3

n2

f ðxÞdx� �

¼ 2 j n2 � n1ð Þf ðf1Þ � n3 � n2ð Þf ðf2Þ½ �: ð16Þ

In Eq. (16) f1 2 ðn1; n2Þ and f2 2 ðn2; n3Þ. It follows from Eq. (2) that

n2 � n1 ¼ n3 � n2 ¼ 1=ð2mÞ ¼ 1=ð2jþ1Þ and the expression for coeffi-cients ai in Eq. (16) can be reduced to

ai ¼12

f ðf1Þ � f ðf2Þ½ � ¼ 12

f1 � f2ð Þ dfdxðfÞ; f 2 f1; f2ð Þ: ð17Þ

It implies from Eq. (17) and assumption (12) of the theorem that

ai 6 g1

2jþ1 : ð18Þ

Note that there are no significant differences in estimation of coef-ficients in cases where function itself of its nth order derivative isexpanded into Haar wavelets. The formula (18) is in accordancewith the results obtained in [36].

However, the function estimation is principally different, sincethe integrals of the Haar wavelets pn;iðxÞ are not orthogonal i.e.all terms in Eq. (15) should be considered. Before direct estimation

of the integralR 1

0 pn;2 jþkþ1ðxÞpn;2rþsþ1ðxÞdx it is reasonable to deriveestimates on the integrals of the Haar wavelets pn;iðxÞ. It is assumedin the following that n P 2 (boundary value problems). Particularcase n ¼ 1 needs little different approach but leads to the sameorder of convergence (studied by authors in [38]). Let us proceedfrom formula (7) and derive upper bound of the function pn;iðxÞin each subinterval. First note that the function pn;iðxÞ � 0 forx 2 ½0; n1ðiÞ�.

In the interval x 2 ½n1ðiÞ; n2ðiÞ� the function pn;iðxÞ is monotoni-cally increasing (power law). Thus, the upper bound value of thefunction pn;iðxÞ is reached at x ¼ n2ðiÞ as follows (see Eqs. (2) and(7))

pn;iðxÞ ¼ pn;2 jþkþ1 6n2ðiÞ � n1ðiÞ½ �n

n!¼ 1

n!

1

2jþ1

� �n

; x

2 n1ðiÞ; n2ðiÞ½ �: ð19Þ

In the interval x 2 ½n2ðiÞ; n3ðiÞ� the function pn;iðxÞ is monotoni-cally increasing if

x 6 n2 þ n3 � n2ð Þ 1

21=n�1 � 1

� �: ð20Þ

The inequality (20) can be derived from formulas (2) and (7) and

condition dpn;iðxÞdx > 0. Since the right hand size of the inequality

(20) is greater or equal to n3 for considered values of parameternðn P 2Þ it can be stated that the function pn;iðxÞ has a maximumat x ¼ n3ðiÞ in the interval x 2 ½n2ðiÞ; n3ðiÞ�. The maximum value of

the function pn;iðxÞ can be obtained by substituting x ¼ n3ðiÞ inEq. (7) as

pn;iðxÞ ¼ pn;2 jþkþ1 62n � 2

n!

1

2jþ1

� �n

; x 2 n2ðiÞ; n3ðiÞ½ �: ð21Þ

In the interval x 2 ½n3ðiÞ;1� the function pn;iðxÞ can be expandedas (see formula (7))

pn;iðxÞ ¼1n!

Xn

k¼2

n

k

� �x� n2ð Þn�k 1

2jþ1

� �k

þ �1

2jþ1

� �k" #

: ð22Þ

Obviously, all terms in sum (22) corresponding to the odd values ofthe parameter k are equal to zero and the function pn;iðxÞ has amaximum at x ¼ 1. Thus, denoting n1 ¼ floorðn=2Þ, considering that

k!ðn� kÞ! P ðn1!Þ2 and applying geometric progression formulas oneobtains estimate on pn;iðxÞ as

pn;iðxÞ ¼ pn;2 jþkþ1 683

1

n1!ð Þ21

2jþ1

� �2

; x 2 n3ðiÞ;1½ �: ð23Þ

The function pn;iðxÞ is monotonically increasing in interval ½0;1�,since it is monotonically increasing in each subinterval of ½0;1� asshown above. Furthermore, the upper bound of the function pn;iðxÞin interval ½0;1� is determined by inequality equation (23).

Next the quadrate of the L2-norm of error function can be esti-mated. Inserting Eq. (18) in Eq. (15) one obtains

EMk k22 6 g2

X1j¼Jþ1

X2 j�1

k¼0

X1r¼Jþ1

X2r�1

s¼0

1

2jþ1

1

2rþ1

Z 1

n1

pn;2 jþkþ1ðxÞpn;2rþsþ1ðxÞdx:

ð24Þ

Introducing the following notation

Cn ¼83

1

n1!ð Þ2; ð25Þ

one can express the estimates on upper bounds of the integrals ofHaar wavelets pn;2 jþkþ1ðxÞ and pn;2rþsþ1ðxÞ as

pn;2 jþkþ1ðxÞ 6 Cn1

2jþ1

� �2

; pn;2rþsþ1ðxÞ 6 Cn1

2rþ1

� �2

: ð26Þ

The coefficient Cn depends on order of the differential equationonly, not on level of the resolution.

Inserting Eq. (26) in Eq. (24) yields

EMk k22

���j¼s6 g2C2

n

X1j¼Jþ1

X1r¼Jþ1

1

2jþ1

� �3 12rþ1

� �3

2 j2rð1� n1Þ

61

36g2C2

n1

2Jþ1

� �4

: ð27Þ

Based on Eq. (27) it can be stated that the convergence of the Haarwavelet method considered is of order two, since the integrationdomain is divided into 2M ¼ 2Jþ1 equal subintervals i.e.

EMk k2 ¼ O1

2Jþ1

� �2" #

; ð28Þ

The error bound can be expressed as

EMk k2 6gCn

61

2Jþ1

� �2

¼ 49

gðfloorðn=2Þ!Þ2

12Jþ1

� �2

: ð29Þ

The theorem is proved. h

230 J. Majak et al. / Composite Structures 126 (2015) 227–232

4. Numerical validation of the rate of convergence andextrapolation of results

In the following the analytical formulas are provided for numeri-cal estimation of the order of convergence and extrapolation of theresults.

4.1. Theoretical background

Let us consider the asymptotic error expansion in powers of thestep size h as

FðhÞ � Fð0Þ ¼ ahk þ OðhlÞ; 0 < k < l: ð30Þ

Here FðhÞ denote the value obtained by any numerical method withstep size h; Fð0Þ is an unknown exact value, a is unknown constantindependent on h and k is theoretical order of accuracy of thenumerical method. Such expansions have been proven for a widerange of finite difference and finite element solutions [36].

Denote two numerical solutions found on nested gridshi�1; hi ¼ hi�1=2 as follows Fi�1 ¼ Fðhi�1Þ; Fi ¼ FðhiÞ. Applying (30)for these solutions one can write the following equality

Fi�1 � Fð0Þ ¼ ahki�1 þ Oðhl

i�1Þ;Fi � Fð0Þ ¼ ahk

i þ OðhliÞ:

ð31Þ

By taking a linear combination of these two solutions, one canobtain the error estimate as

Fð0Þ � Fi ¼Fi � Fi�1

2k � 1þ Oðhl

iÞ ð32Þ

or another approximation of the value F(0) as

Ri ¼ Fi þFi � Fi�1

2k � 1¼ Fð0Þ þ Oðhl

iÞ: ð33Þ

This formula is simple Richardson extrapolation formula[36,37]. In other words, the approximate solutions Ri have errorwith higher order in relation to h than Fi. Therefore, if numericalsolutions of the problem for two grids and the theoretical orderof accuracy k of the numerical method are known, a simple linearextrapolation formula (33) eliminates the leading term from errorexpansion equation (30) and leads to reasonably accurate results[36,37]. The Richardson extrapolation is an efficient method forerror estimation and increase the accuracy of finite differenceand finite element solutions of different problems of mathematicalphysics.

On the other hand, if exact solution Fð0Þ is known, Eq. (31) givesimple method for estimating the order of convergence ofnumerical method as

Fi�1 � Fð0ÞFi � Fð0Þ ¼ 2k þ Oðhl�k

i Þ; k � kEi

¼ logFi�1 � Fð0ÞFi � Fð0Þ

� ��logð2Þ: ð34Þ

In practice, the order of convergence of numerical method canbe estimated even when the exact solution F(0) is unknown. In lat-ter case, the theoretical order of accuracy can be estimated usingthree solutions on a sequence of nested grids Fi�2; Fi�1; Fi;

hi�2=hi�1 ¼ hi�1=hi ¼ 2. The following ratio can be obtained fromthree equations similar to Eq. (34):

ki ¼Fi�2 � Fi�1

Fi�1 � Fi¼ 2k þ Oðhl�k

i Þ: ð35Þ

Further, from Eq. (35) the theoretical order of accuracy k can beeasily estimated [37]:

k � ki ¼ logðkiÞ= logð2Þ: ð36Þ

Here ki is a value of observed order of accuracy and Eq. (36) givesexperimental method for determining or verifying the value oftheoretical order of accuracy k of the numerical method.Obviously, Eq. (36) can be used only for ki > 0, i.e. three successivevalues Fi�2; Fi�1; Fi must be monotonic. The proximity of obtainedvalues of the ratio ki or observed order of accuracy ki respectively

to theoretical values 2k or k is a confirmation of asymptotic error

expansion (30) with leading term ahk.Moreover, the following formula can be used to estimate the

order of convergence of the improved values Ri :

l � li ¼ logRi�2 � Ri�1

Ri�1 � Ri

� ��logð2Þ: ð37Þ

This formula follows from Eq. (33) and can be obtained similarly toEq. (36). In the next sections the order of convergence of the HWDMis evaluated numerically in the case of three model problems.

4.2. Free transverse vibrations of the orthotropic rectangular plates ofvariable thickness

In the current section the order of convergence of the HWDMtreated by author in [12] is examined. Assuming that the principaldirections of orthotropy coincide with natural co-ordinate systemone can represent the equation of motion governing natural vibra-tion of a thin orthotropic rectangular plate as

Dx@4w@x4 þ Dy

@4w@y4 þ 2T

@4w@x2@y2 þ 2

@T@x

@3w@x@y2 þ 2

@T@y

@3w@y@x2

þ 2@Dx

@x@3w@x3 þ 2

@Dy

@y@3w@y3 þ 2

@2Dx

@x2

@2w@x2 þ 2

@2Dy

@y2

@2w@y2

þ @2D@y2

@2w@x2 þ

@2D@x2

@2w@y2 þ 4

@2Dxy

@x@y@2w@y@x

¼ �qc@2w@t2 � kw; ð38Þ

where

Dx ¼ E�xc3=12; Dy ¼ E�yc

3=12; Dxy ¼ Gxyc3=12; D ¼ E�c3=12;

T ¼ Dþ 2Dxy; E�x ¼Ex

1� mxmy; E�y ¼

Ey

1� mxmy; E� ¼ myE�x ¼ mxE�y:

ð39Þ

In Eqs. (38) and (39) m is a Poisson’s ratio, D and E stand for flexuralrigidity and modulus of elasticity, respectively. The transversedeflection of the plate and variable plate thickness are denoted byw ¼ wðt; x; yÞ and c ¼ cðxÞ, respectively. The variables q and k standfor the mass density and the modulus of a Winkler type foundation,respectively. It is assumed that the edges of the plate aresimply supported along y ¼ 0; y ¼ b and the other two edges(x ¼ 0; x ¼ aÞ are clamped or simply supported. According to theLévi approach the time-harmonic-dependent solution can beexpanded as

wðt; x; yÞ ¼ wnðxÞ sinðnpy=bÞeixt : ð40Þ

In Eq. (40) x is the harmonic frequency, n – positive number andi ¼

ffiffiffiffiffiffiffi�1p

. The system (38)–(40) can be written in terms of non-di-mensional variables as

WIVn þ

6�c2�c0�c3 W 000

n þ3ð�c�c00 þ 2�c02Þ

�c2 � 2ak2� �

W 00n � ga2 6�c2�c0

�c3 W 0n

þE�yE�x

a4 � 3ð�c�c00 þ 2�c02Þ�c2 a2mx �X2 c2

0�c2 þ 12K=�c3

� �Wn ¼ 0; ð41Þ

where prime denotes derivative with respect to s and

Table 1Numerical results for X. Simply supported plate (all sides).

j Fund. freq. Extrap_res ki kEi

li

3 48.6554484 48.651658 48.650394 2.0075 48.650716 48.650402 2.009 2.0026 48.650481 48.650402 2.002 2.000 4.0017 48.650422 48.650402 2.001 2.000 4.0008 48.650407 48.650402 2.000 2.000 4.000

Table 3Numerical results for second order eigenvalue problem considered in [38].

j Fund. freq. Extrap.res. ki kEi

li

4 12.59505 12.4172 12.3579 1.9606 12.3711 12.3557 1.947 1.9987 12.3595 12.3556 1.991 2.019 4.4598 12.3566 12.3556 2.000 2.078 -

Table 4Numerical results for fourth order eigenvalue problem considered in [38].

j Fund. freq. Extrap.res. ki kEi

li

4 16.66665 16.5364 16.4930 2.0106 16.5041 16.4933 2.011 2.0077 16.4960 16.4933 1.996 2.041 3.322

J. Majak et al. / Composite Structures 126 (2015) 227–232 231

s ¼ xa; Wn ¼

wn

c; �c ¼ c

a; k ¼ npa

b; g ¼ E� þ 2Gxy

E�x;

K ¼ kaE�y; X2 ¼ 12qa2x2

E�yc20

: ð42Þ

In Eqs. (41) and (42) c0 stands for the thickness of the plate at s ¼ 0.According to the HWDM considered the higher order derivativeincluded in ODE (41) is expanded into Haar wavelets as

d4Wn

ds4 ¼ aT H; ð43Þ

where aT is unknown coefficient vector. The numerical results forfundamental frequency of the orthotropic rectangular plates ofvariable thickness are given in Table 1 (all sides simply supported)and Table 2 (simply supported along y ¼ 0 ; y ¼ b, clamped alongx ¼ 0; x ¼ a).

In Tables 1 and 2 the j is the value of resolution (see section 2,

m ¼ 2 jÞ. The second and third columns contain the values of thefundamental frequency and its extrapolation performed by

formula (33), respectively. The orders of convergence ki and kEi

given in columns four and five of Tables 1 and 2 are computedby formulas (36) and (34), respectively. Obviously, both orders ofthe convergence have limit value two. Latter results are in accor-dance with convergence theorem proved above. It can be seen fromTables 1 and 2 that the order of convergence computed with use ofthe exact solution converges faster to two (column five) in the caseof both boundary conditions considered. The order of convergenceof the extrapolated results given in last column of the Tables 1 and2 is equal to four (see Eq. (37)). Thus, the fundamental frequencieswith improved accuracy have higher order of convergence equal tofour.

4.3. Second, fourth and sixth order eigenvalue problems considered byShi and Cao [38]

In the current section the order of convergence of the HWDMintroduced by Shi and Cao in [38] is examined. Let us consider firstthe following second-order eigenvalue problem

�y00 þ cosðxÞþ2cosð2xÞþ3cosð3xÞð Þy¼ ky; yð0;kÞ¼ yðp;kÞ¼ 0: ð44Þ

The orders of convergence calculated on results given in [38] arepresented in Table 3.

The structure of Table 3 coincide with that of Tables 1 and 2 i.e.the columns contain the same parameters. In can be seen fromTable 3 that the computed rates of convergence have values near

Table 2Numerical results for X. Simply supported-clamped plate.

j Fund. freq. Extrap.res. ki kEi

li

3 61.1728054 61.172392 61.172254 1.5575 61.172236 61.172184 1.407 1.9036 61.172194 61.172180 1.877 1.976 3.9207 61.172183 61.172179 1.970 1.994 3.9808 61.172180 61.172179 1.993 1.999 3.995

two. In [38] the results are given with four decimal places. Latterfact has obviously impact on improved solution (column three)and on its rate of convergence (column six). In this reason the onlyvalue in column six differ from theoretically expected value four.The last value in column six is not present due to division by zero(the last two values obtained by extrapolation formula (33)coincide in column four).

Next the following fourth order eigenvalue problem isconsidered

yðIVÞ ¼ ky; 0 6 x 6 1; yð0Þ ¼ y0ð0Þ ¼ yð1Þ ¼ y00ð1Þ ¼ 0: ð45Þ

The orders of convergence calculated on results given in [38](see Table 4 in [38]) are presented in Table 4.

Similarly to second order eigenvalue problem (Table 3) it can beconfirmed that the computed orders of convergence have valuesnear two in both cases with and without use of the exact solution(columns 4 and 5, respectively). The extrapolated solution and itsorder of convergence are more affected by low number (four) ofdecimal placed considered in [38].

Finally, the following sixth order eigenvalue problem isconsidered

yðIVÞ ¼ ky; 0 6 x 6 1; yð0Þ ¼ y0ð0Þ ¼ yð1Þ ¼ y00ð1Þ ¼ 0: ð46Þ

The orders of convergence calculated on results given in [38] (seeTable 1 in [38]) are presented in Table 5.

The results in Table 5 are similar to ones given in Tables 3 and 4and confirm that the order of convergence of the HWDM is equal totwo. In Table 5 the rate of convergence of the extrapolated resultscoincide unexpectedly well with theoretical value four despite tolow number of decimal places presented in solution [38].

4.4. Free vibration analysis of functionally graded (FG) cylindricalshells based on the shear deformation theory [17]

In the current section the order of convergence of the HWDMtreated in [17] is examined. The governing equations and boundaryconditions are omitted herein for conciseness sake (see formulas

Table 5Numerical results for sixth order eigenvalue problem considered in [38].

j Fund. freq. Extrap.res. ki kEi

li

4 2.80685 2.8272 2.8340 2.0836 2.8319 2.8335 2.118 1.9777 2.8331 2.8335 1.970 2.000 4.000

Table 6Numerical results for FG cylindrical shells considered in [17].

j Fund. freq. Extrap.res. ki kEi

li

2 12.3083 8.763 7.5814 7.637 7.262 1.6555 7.33 7.228 1.655 1.875 3.2336 7.251 7.225 1.875 1.958 3.5037 7.231 7.224 1.958 1.982 3.1708 7.226 7.224 1.982 2.000

232 J. Majak et al. / Composite Structures 126 (2015) 227–232

(22) and (24) in [17]). The orders of convergence computed basedon results given in [17] (see Table 2 in [17]) are presented inTable 6.

Again, the order of convergence two of the HWDM can beconfirmed based on results given in Table 6. Obviously, the lownumber decimal places for initial solution given in Table 6 hasconsiderable impact on columns three and six (extrapolated resultsand their order of convergence).

5. Conclusion

The accuracy issues of the HWDM, open from year 1997, are clari-fied. The convergence theorem is proved for general nth order ODE(assuming n P 2, boundary value problems). As result it is shownthat the order of convergence is two and the error bound has beenderived. The numerical validation of the results of convergencetheorem has been performed by utilizing a number of case studies.The accuracy of the HWDM considered can be improved by adoptingRichardson extrapolation method [37]. The particular case wherethe order of differential equation is equal to one correspond to initialvalue problem, which are less actual for composite structures andare omitted herein for conciseness sake. Latter problem is studiedby authors and the same order of convergence is obtained.

In the case of all model problems considered the theoreticalvalue equal to two of the order of convergence has been confirmed.The theoretical value equal to four of the order of convergence ofthe extrapolated results has been observed in the case of firstmodel problem. In the case of second and third model problem,where the initial solution has less number of decimal places (theseresults are taken from literature) the order of convergence of theextrapolated results differ from four as it can be expected.

The convergence theorem proved for ODE obviously hold goodalso for integro-differential equations. However, the integralequations where the derivatives are not present need futureinvestigation. Extension of the results for partial differentialequations has been also foreseen in future study.

Acknowledgment

This research was supported by the EU structural funds project,‘‘Smart Composites – Design and Manufacturing’’ 3.2.1101.12-0012, targeted financing project SF0140035s12.

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