APPLICATION OF THE HAAR AND B-SPLINE WAVELETS TO APPROXIMATE SOLUTION OF THE BOUNDARY PROBLEMS

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1 ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2011 Seria: MATEMATYKA STOSOWANA z. 1 Nr kol Anna KORCZAK Institute of Mathematics Silesian University of Technology APPLICATION OF THE HAAR AND B-SPLINE WAVELETS TO APPROXIMATE SOLUTION OF THE BOUNDARY PROBLEMS Summary. This paper presents an application of the wavelet theory to function approximation and to approximate solutions of chosen differentialequations.thereareconsideredthefunctionsofclassl 2 (R)andthe first and the second order linear differential equations in interval[0, 1] with appropriate boundary conditions. Haar wavelet and linear B-spline wavelet were applied to approximate these functions and solving these problem. ZASTOSOWANIE FALEK HAARA I B-SPLAJNOWEJ W PRZYBLIŻONYM ROZWIĄZYWANIU ZAGADNIEŃ BRZEGOWYCH Streszczenie. W artykule przedstawiono zastosowanie teorii falek w aproksymacji funkcji oraz w aproksymacji rozwiązań wybranych równań różniczkowych.rozważanesąrównieżfunkcjeklasyl 2 (R)orazrównania różniczkowe liniowe rzędów pierwszego i drugiego, określone na przedziale [0, 1], z odpowiednimi warunkami brzegowymi. Do aproksymacji wyżej wymienionych funkcji oraz równań zostały zastosowane falka Haara oraz falka B-splajnowa liniowa Mathematics Subject Classification: 80M99, 65M99. Wpłynęło do Redakcji(received): r.

2 150 A. Korczak 1. Introduction Wavelettheoryisarespectivelynewfieldofmathematics.Ithasbeenveryintensively developed since eighties and is promising in usage to various applications. Wavelets find an application in many different problems like analyzing acoustic emission pulses[1], technical diagnostics[4], local images filtering[6], finite elements method[3]. Their popularity is caused by the combination of theoretical and applied mathematics represented by this approach. WaveletisakindoffunctionΨ(x) L 2 (R)suchthattheset B Ψ ={2 j/2 Ψ(2 j x k);j Z,k Z} (1) isabaseinspacel 2 (R)[2].FamillyB Ψ iscalledthewaveletbase. Previousdefinitionimpliesthatanyfunctionf(x) L 2 (R)canbeshowedas linearcombinationofbasefunctionsψ jk =2 j/2 Ψ(2 j x k)(ifψisawavelet). Thatmeansthatfunctionfcanbewrittenasthefollowingseries f(x)= f jk Ψ jk (x), (2) j Zk Z wheref jk arethefouriercoefficientsdefinedasfollows f jk = f(x),ψ jk (x) = f(x)ψ jk (x)dx. 2. Haar wavelet ThesimplestexampleofwaveletistheHaarwaveletwhichisdefinedasfollows 1, x [0, 1 2 ), H(x)= 1, x [ 1 2,1), 0, x/ [0,1). Function H(x) is connected with the set B H ={2 j/2 H(2 j k);j Z,k Z}. (3) FunctionsH jk (x) B H (forfixedjandk)havetheform 2 j/2, x [k2 j,(k+ 1 2 )2 j ), H jk (x)= 2 j/2, x [(k+ 1 2 )2 j,(k+1)2 j ), 0, x/ [k2 j,(k+1)2 j ). (4)

3 Application of the Haar and B-spline wavelets ThesupportoffunctionH jk (x)istheinterval supph jk (x)= [ k2 j,(k+1)2 j]. (5) BasicpropertyoftheHaarwaveletistheorthogonalityofbaseH jk H j1k 1,H j2k 2 =0 for j 1 j 2 k 1 k 2. (6) AnyfunctionH jk isobtainedbytranslatingandscalingofthehaarwavelet Haar wavelet approximation Function f(x) belonging to the appropriate function space can be approximatedbyusingthehaarwaveletandthebaseb H associatedwithit. Weapproximatefunctionfonlyininterval[ 2 m,2 m ],outofitwetake value0.moreover,wedefinethevaluer Z,whichisthemaximumapproximation (resolution) level of function f. Forfixedmandrtheapproximationoffunctionfisasfollows f(x)=f m (x)+ r j=m 2 j m 1 k= 2 j m f jk H jk (x). (7) where,forj<m,wehave f m (x)= 2 m 2 m 0 f(x)dx, x [ 2 m,0), 0 2 m f(x)dx, x [0,2 m ). 2 m Example 2.1 In this example, we will approximate function f(x), defined as follows { sinx+0.5, x [0,1), f(x)= 0, x/ [0,1). Functionf(x)istheelementofspaceL 2 (R)andcanbeapproximatedbyusing the Haar wavelet. Forthefollowingapproximationlevels:r=1,2,3,4,Figure1illustratesthe comparisons of function f(x) and its approximations. Whereas, the error of approximationsforlevels:r=1,2,3,4,aredisplayedinfigure2.basingonthe

4 152 A. Korczak r=1 r= r= r= Fig. 1. Comparison of function f(x) and its approximations Rys. 1. Porównanie przebiegu funkcji f(x) i jej aproksymacji presented diagrams we can say that the increase of approximation level causes the double decrease of the approximation error. Calculatingtheapproximationerrorinpointsx i = i 10 fori=0,1,2,...,10 the efficiency of function approximation can be compared for different wavelets. Errorsinpointsx i forthehaarwaveletareshownintable1. WaveletbaseB H givesanopportunitytoapproximatethesolutionsoflinear differential equations. We consider differential equation of the form forx [0,1],withtheinitialcondition y +f(x)y=g(x), (8) y(0)=y 0, wheretheunknownfunctiony=y(x)andthegivenfunctionsf(x)ig(x)belong totheappropriateclassin[0,1].byusingwaveletbaseb H wearelookingforthe solution having the following form ỹ r (x)=a+ r 2 j 1 j=0k=0 a jk H jk (x), (9)

5 Application of the Haar and B-spline wavelets r=1 r= r= r= Fig. 2. Error of the function approximation Rys. 2. Błąd aproksymacji funkcji Functionỹ r dependsonparametersaanda jk,whichareunknownandneedto befound.thisfunctionisconstantineachofintervals [ i2 r 1,(i+1)2 r 1), i=0,...,2 r+1 1[2]. Example 2.2 ByusingthewaveletbaseB H wewillfindtheapproximatesolutionofthe following differential equation of the first order y (x)+y(x)=x 2 sinx, x [0,1], (10) with initial condition y(0)=1. Exact solution of this equation is the following y(x)= 1 2 e x ( 3+e x cosx 2e x xcosx+e x x 2 cosx+e x sinx e x x 2 sinx).(11)

6 154 A. Korczak Table 1 Function approximation error x i r=1 r=2 r=3 r= By using described algorithm of finding approximate solution we obtain the approximate solution in points x i =(i+ 1 2 )2 r 1 fori=0,...,2 r+1 1. Forthefollowingapproximationlevels:r=1,2,3,4,Figure3illustratescomparison of the exact solution of considered problem and its approximations. Whereas, the approximation errors for these levels are illustrated in Figure 4. Basing on the presented diagrams we can say that the increase of approximation level causes the decrease of approximation error. InthenextexampleofwaveletbaseB H applicationthreeboundaryproblems for linear differential equation of the second order will be considered. Let us consider the equation y (x)+f(x)y (x)+g(x)y(x)=h(x), x [0,1], (12) with the following boundary conditions: { y(0)=y 0, y (0)=y 0, (13)

7 Application of the Haar and B-spline wavelets r=1 r= r= r= Fig. 3. Comparison of the exact and approximate solution of differential equation(10) Rys. 3. Porównanie przebiegu rozwiązania dokładnego równania różniczkowego i jego aproksymacji(10) { { y(0)=y 0, y(1)=y 1, y (0)=y 0, y(1)=y 1. (14) (15) Theunknownfunctiony=y(x)andthegivenfunctionsf(x),g(x)ih(x)belong to the appropriate class. Solutionofthisproblemissoughtasthesum(9).Similarlylikeincaseofthe firstorderequationtheunknownelementsareaanda jk. Example 2.3 ByusingthewaveletbaseB H wewillfindtheapproximatesolutionofthe following linear differential equation of the second order y (x)+y(x)= cosxe x, x [0,1], (16)

8 156 A. Korczak r= r= r= r= Fig. 4. Error of approximate solution of differential equation(10) Rys. 4. Błąd aproksymacji rozwiązania równania różniczkowego(10) with the boundary conditions of the second type y(0)=0 and y(1)=0. Exact solution of this equation is the following: y(x) = 1 10 (2cosx 2ex cosxcos2x+5esinx 5e x sinx+ecos2sinx 2ctg1sinx+2ecos2ctg1sinx+2esin2sinx ectg1sin2sinx +e x cosxsin2x 2e x sinxsin2x). By applying the proposed approach we obtain the approximated solution in points x i =(i+ 1 2 )2 r 1 fori=0,...,2 r+1 1.Forthefollowingapproximationlevels:r=1,2,3,4,Figure 5 shows the comparison of the exact and approximate solution of the considered

9 Application of the Haar and B-spline wavelets r=1 0 r= r= r= Fig. 5. Comparison solution of differential equation(16) and its approximation Rys. 5. Porównanie przebiegu rozwiązania równania różniczkowego(16) i jego aproksymacji problem. Whereas, approximation errors for levels: r = 1, 2, 3, 4, are illustrated in Figure 6. According to the presented diagrams we can say that with the increase of approximation level the approximation error decreases. 3. B-spline wavelet B-spline wavelet is often defined by means of the scaling functions[7]. Scaling function of the m-order B-spline wavelet is defined as the following convolution: φ m (t)=(φ m 1 φ 1 (t))= φ m 1 (t x)φ 1 (x)dx= 1 0 φ m 1 (t x)dx. (17) Moreover, the m-order B-spline wavelet can be defined as the linear combination of scaling functions: Ψ m (x)= 3m 2 k=0 q k φ m (2x k), (18)

10 158 A. Korczak r=1 r= r= r= Fig. 6. Error of approximate solution of differential equation(16) Rys. 6. Błąd aproksymacji rozwiązania równania różniczkowego(16) where q k =( 1) k 2 1 m m l=0 ( ) m φ 2m (k+1 l). (19) l ItisworthtonoticethatthefirstorderB-splinewaveletistheHaarwavelet. Scaling function of the linear B-spline wavelet is defined by the formula x, x [0,1), φ 2 (x)= 2 x, x [1,2), (20) 0, x/ [0,2). Furtherscalingfunctionform=2isdenotedasφ(x).Then,thetwo-scalerelation for this function is the following x j k, x [k2 j,(k+1)2 j ), φ j,k (x)= 2 (x j k), x [(k+1)2 j,(k+2)2 j ), (21) 0, x/ [k2 j,(k+2)2 j ), wherex j =2 j x k.

11 Application of the Haar and B-spline wavelets m=1 2.0 m= m= m= Fig.7. Scalingfunctionsform=1,2,3,4 Rys.7. Funkcjeskalującedlam=1,2,3,4 Functionφ j,k (x)isdefinedasthescaledandtransformedfunctionφ(x)by usingparametersjandk.functionsφ j,k (x)arehelpfulinformingthefunctions approximation. Linear B-spline wavelet is defined by the formula: x 6, x [0,1 2 ), Ψ 2 (x)= 1 6 ( 7x+4), x [1 2,1), 1 6 (16x 19), x [1,3 2 ), 1 6 ( 16+29), x [3 2,2), 1 6 (7x 17), x [2,5 2 ), 1 6 ( x+3), x [5 2,3), 0, x/ [0,3). (22)

12 160 A. Korczak m= m= m= m= Fig.8. B-splinewaveletsoforder1,2,3,4 Rys.8. FalkiB-splajnowerzędu1,2,3,4 Ψ 2 (x)waveletisassociatedwiththewaveletbase[5](x j =2 j x k): ψ j,k (x)= x j 2 j/2 6, x [k2 j,(k+ 1 2 )2 j ), ( 7x j +4) 2j/2 6, x [(k+1 2 )2 j,(k+1)2 j ), (16x j 19) 2j/2 6, x [(k+1)2 j,(k+ 3 2 )2 j ), ( 16x j +29) 2j/2 6, x [(k+3 2 )2 j,(k+2)2 j ), (7x j 17) 2j/2 6, x [(k+2)2 j,(k+ 5 2 )2 j ), ( x j +3) 2j/2 6, x [(k+5 2 )2 j,(k+3)2 j ), 0, x/ [k2 j,(k+3)2 j ]. (23) Similarly like in case of the Haar wavelet, parameter j is responsible for scaling andparameterkisresponsiblefortranslatingthefunctionψ jk.

13 Application of the Haar and B-spline wavelets Linear B-spline wavelet approximation[5] Foranyfixedpositiveintegerr N,thefunctionf(x)definedover[0,1]may be represented by the B-spline scaling functions in the following way where s k = 1 0 f(x)= 2 r+1 1 k= 1 s k φ r+1,k (x), (24) f(x) φ r+1,k (x)dx, k= 1,0,...,2 r+1 1. (25) Functions φ r+1,k (x)arethedualfunctionsforφ r+1,k (x)givenbytherelation 1 0 Φ r+1 Φ r+1 dx=i, (26) whereφ r+1 =[φ r+1, 1,φ r+1,0,...,φ r+1,2 r+1 1] T andiistheidentitymatrixof dimension.(2 r+1 +1) (2 r+1 +1) Let From(26)and(27)weget P r+1 = 1 0 Φ r+1 Φ T r+1dx. (27) Φ r+1 =(P r+1 ) 1 Φ r+1. (28) Example 3.1 In this example we approximate the function f(x) defined as follows { sinx+0.5, x [0,1), f(x)= 0, x/ [0,1). For the first approximation level the comparison of function f(x) and its approximationisshowninfigure9.

14 162 A. Korczak Fig.9. Comparisonoffunctionf(x)anditsapproximationforr=1 Rys.9. Porównanieprzebiegufunkcjiijejaproksymacjidlar=1 r= r= r= r= Fig. 10. Error of the function approximation Rys. 10. Błąd aproksymacji funkcji Diagramoff(x)anditsapproximationarepracticallythesame,evenforfirst approximation level which confirms the good precision of approximation. Error analysis for the successive levels gives the opportunity to compare efficiency of approximation. Approximation errors for levels: r = 1, 2, 3, 4, are illustrated in Figure10.Basingonthepresenteddiagramswecansaythattheincreaseof

15 Application of the Haar and B-spline wavelets Error of the approximate solution Table 2 x i r=1 r=2 r=3 r= approximation level causes the decrease of approximation error. Errors in points x i fortheb-splinewaveletareshownintable B-spline wavelet approximation[5] Let us consider the linear differential equation of the first order y (x)+f(x)y=g(x), (29) with the initial condition y(0)=y 0, (30) wheref(x),g(x)arethegivenfunctionsofclassl 2 [0,1],y 0 isthegivenreal numberandy(x)istheunknownfunction.thedifferentiationofthevectorsφ r+1 andψcanbeexpressedinfollowingway Φ r+1 =D ΦΦ r+1, Ψ =D Ψ Ψ, (31) whered Φ andd Ψ denotethe(2 r+1 +1) (2 r+1 +1)operationalmatricesof derivative for B-spline scaling functions and wavelet.

16 164 A. Korczak Approximation of the function f(x) can be expressed y(x)=c T Ψ(x), (32) wherecisanunknownvectorofdimension2 r+1 +1.From(32)weobtain y (x)=c T Ψ (x)=c T D Ψ Ψ(x). (33) Using(29)and(33)wehave C T D Ψ Ψ+f(x)C T Ψ=g(x). (34) From(30)and(32)weget C T Ψ(0)=y 0. (35) Tofindthesolutiony(x)firstwecollocateequality(34)inx i = 2i 1 2 r+1 2,for i=1,2,...,2 r+1 1.Theresultingequationwithinitialcondition(35)generates 2 r+1 +1linearequations.SolutionoftheseequationsisvectorCwhichcanbeset into(32) and gives the approximation(29). Example 3.2 ByusingwaveletbaseΨ j,k,wewillfindtheapproximatesolutionofthefollowing differential equation y (x)+y(x)=x 2 sinx, x [0,1] (36) with the initial condition y(0)=1. Diagramoff(x)anditsapproximationarepracticallythesameevenforfirst approximation level(figure 11) what means the good precision of approximation. Error analysis at the successive levels gives opportunity to compare the efficiency ofapproximations.approximationerrorsforlevels:r=1,2,3,4,aredisplayedin Figure12.Basingonthepresenteddiagramswecansaythatwiththeincreaseof approximation level the approximation error decreases. We consider now the second order linear differential equation with the boundary condition y (x)+f(x)y +g(x)y(x)=h(x), (37) y(0)=y 0, y(1)=y 1, (38)

17 Application of the Haar and B-spline wavelets Fig. 11. Comparison of the solution of differential equation and its approximation for r=1 Rys. 11. Porównanie przebiegu rozwiązania równania różniczkowego i jego aproksymacji dlar=1 r=1 r= r= r= Fig. 12. Error of approximate solution of the differential equation Rys. 12. Błąd aproksymacji rozwiązania równania różniczkowego wheref(x),g(x),h(x)arethegivenfunctionsofclassl 2 [0,1],y 0 andy 1 arethe given real numbers and y(x) is the unknown function.

18 166 A. Korczak Tofindtheapproximationoffunctiony(x)weproceedasincaseofthefirst order linear differential equation. We substitute into equation(37) the following functions y(x)=c T Ψ(x), y (x)=c T Ψ (x)=c T D Ψ Ψ(x), (39) y (x)=c T DΨ 2Ψ(x). We obtain: From(38)and(39)wehave: C T D 2 ΨΨ+f(x)C T D Ψ Ψ+g(x)C T Ψ=h(x). (40) C T Ψ(0)=y 0, C T Ψ(1)=y 1. (41) Tofindthesolutiony(x)firstwecollocateequation(40)inpointsx i = 2i 1 2 r+1 2, fori=1,2,...,2 r+1 1.Theresultingequationwithboundaryconditions(38) generates2 r+1 +1linearequations.SolutionoftheseequationsisvectorCwhich canbesetinto(39)andgivestheapproximation(37). Example 3.3 ByusingwaveletbaseΨ j,k,wewillfindtheapproximatesolutionofthefollowing differential equation y (x)+y(x)= cosxe x, x [0,1] (42) with the boundary conditions and y(0)=0 y(1)=0. Diagramoff(x)anditsapproximationarepracticallythesameevenforfirstapproximation level(figure 13) which confirms the good precision of approximation. Error analysis at the successive levels gives opportunity to compare the efficiency of approximation. Approximation errors for levels: r = 1, 2, 3, 4, are illustrated infigure14.basingonpresenteddiagramswecansaythatwiththeincreaseof approximation level the approximation error decreases.

19 Application of the Haar and B-spline wavelets Fig. 13. Comparison of the exact solution of differential equation and its approximation forr=1 Rys. 13. Porównanie przebiegu dokładnego rozwiązania równania różniczkowego i jego aproksymacjidlar=1 r=1 r= r= r= Fig. 14. Error of approximate solution of the differential equation Rys. 14. Błąd aproksymacji rozwiązania równania różniczkowego 4. Conclusions This paper presents the comparison of Haar s wavelet and linear B-spline wavelet with regard to the efficiency in function approximations and in approximate

20 168 A. Korczak solutions of the first and second order differential equations. Presented examples prove that linear B-spline wavelet gives better results in functions approximation and in finding the approximate solutions of differential equations than Haar wavelet. Approximation errors calculated for each level of B-spline wavelet are much smaller than for Haar wavelet. Moreover, the examples show that errors for the Haar wavelet approximation counted for particular levels of approximation decrease much faster than approximation errors generated by using the B-spline wavelet. References 1. Boczar T.: Application of signal processing elements for the characteristics of acoustic emission pulses generated by partial discharges. Phys. Chem. Solid State 8(2007), Grzymkowski R., Zielonka A.: Application of the wavelets theory in the boundary problems. WPKJS, Gliwice 2004(in Polish). 3. Jiawei X., Xuefeng Ch., Zhengjia H., Yinghong Zh.: A new wavelet-based thin plate element using B-spline wavelet on the interval. Comput. Mech. 41(2008), Katunin A., Korczak A.: The possibility of application of B-spline family wavelets in diagnostic signal processing. Acta Mech. Autom. 3(2009), Lakestani M., Razzaghi M., Dehghan M.: Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Math. Probl. Eng (2006), Samavati F., Mahdavi-Amiri N.: A filtered B-spline models of scanned digital images. J. Sci. 10(2000), Ueda M., Lodha S.: Wavelets: an elementary introduction and examples. University of California at Santa Cruz, Santa Cruz Omówienie W niniejszym artykule dokonano porównania falki Haara i falki B-splajnowej liniowej pod względem efektywności aproksymacji funkcji oraz rozwiązań równań różniczkowych rzędów pierwszego i drugiego. Falka Haara jest najstarszą falką, zdefiniowaną już w 1910 roku, dobrze poznaną i obecnie powszechnie stosowaną w wielu dziedzinach. Falka Haara należy do grupy falek ortogonalnych i ze względu

21 Application of the Haar and B-spline wavelets na jej prostą postać funkcyjną jej użycie nie wymaga skomplikowanych algorytmów. Natomiast falka B-splajnowa liniowa nieco bardziej skomplikowana jest falką biortogonalną, co daje szersze możliwości zastosowania jej w wielu dziedzinach nauki. Analizując przykłady zawarte w tym artykule można zauważyć, że falka B-splajnowa liniowa daje dużo lepsze rezultaty w aproksymacji rozpatrywanych funkcji oraz rozwiązań równań różniczkowych. Na każdym z poziomów rozdzielczości błędy aproksymacji, liczone dla falki B-splajnowej liniowej, były mniejsze, aniżeli liczone dla falki Haara. W wybranych przykładach można również zaobserwować, że w przypadku falki Haara błąd liczony na poszczególnych poziomach aproksymacji spada dużo szybciej niż w przypadku falki B-splajnowej. Należy podkreślić, że teoria falek B-splajnowych jest teorią stosunkowo nową, intensywnie rozwijaną przez liczne ośrodki naukowe na całym świecie. Obecnie ze względu na swoje ciekawe własności i zastosowania jest tematem wielu prac naukowych.