Haar wavelet collocation method to solve problems arising in induction motor

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1 ISSN , England, UK Journal of Informaton and Computng Scence Vol., No., 07, pp Haar wavelet collocaton method to solve problems arsng n nducton motor A. Padmanabha Reddy *, C. Sateesha, S. H. Manula Department of Studes n Mathematcs, V. S. K. Unversty, Ballar, 58304, Karnataka, Inda, E-mal: paddu7_math@redffmal.com. MOB: (Receved September 8, 06, accepted February 07, 07) Abstract. In ths paper, Haar wavelet collocaton method has been employed to solve problems occur n the mathematcal modelng of nducton motor. To approxmate the soluton algorthm based on Haar wavelet s consdered. The order of convergence s estmated for dscussed problems. The accuracy ssues of the solutons are demonstrated by comparng wth other numercal technques exstng n the lterature. Keywords: Haar wavelets; nducton motor; collocaton method; convergence analyss.. Introducton Alfred Haar [] ntroduced the noton of wavelets whch are called Haar wavelets. These wavelets placed a crucal role for the numercal soluton of dfferental or ntegral equatons. At present there are two approaches to applyng the Haar wavelet for ntegratng ordnary dfferental equatons (ODE). In case of the frst method for ntegratng ODE concept of operatonal matrx s ntroduced by Chen and Hsao [, 3]. Another approach s called drect method due to Lepk [4]. In ths approach Haar functons are ntegrated drectly. The drect method s easly applcable for calculatng ntegrals of arbtrary order but the operatonal matrx method has been used manly for frst order ntegrals. Haar wavelets conssts of pecewse constant functons and are therefore the smplest orthogonal wavelets wth compact support. Here the fact that Haar wavelets are not contnuous and hence dervatves do not exsts at the brakng ponts. So that t s not possble to apply the Haar wavelets for solvng ODEs drectly. The man advantage of the method s that t can be used drectly wthout usng restrctve assumptons. The ffth-order dfferental equatons arse n modelng of nducton motor wth two rotor crcuts. Ths model contans two stator state varables, two rotor state varables and one shaft speed. Normally, two more varables must be added to account for the effects of a second rotor crcut representng deep bars, a startng cage or rotor dstrbuted parameters. To avod the computatonal burden of addtonal state varables when addtonal rotor crcuts are requred model s often lmted to the ffth order and rotor mpedance s algebracally altered as functon of rotor speed. Ths s done under the assumpton that the frequency of rotor currents depends on rotor speed. Ths approach s effcent for the steady state response wth snusodal voltage [5]. The exstence and unqueness solutons of ffth order boundary value problems (BVPs) are dscussed by Agarwal [6]. Over the years some researchers have worked on nducton motor problems by usng dfferent methods for numercal solutons. Reddy et al. [7] have demonstrated the superorty of the HWCM for the soluton of seventh order ODEs of nducton motor wth two rotor crcuts. Sddq et al. [8] estmated the soluton for lnear specal case ffth-order two-pont boundary value problems by non-polynomal sextc splne method (NPSS). Faraeyan and Jallan [9] have found the numercal soluton by ffth order BVPs n off step ponts (OSPM). Mehd Golom et al. [0] solved ffth order dfferental equatons by He s Varatonal teraton method (VIM). Ths paper deals wth ffth order ODEs arsng n modelng of nducton motor; these problems have the followng general representaton: (5) () () (3) (4) y ( x) f ( x, y, y, y, y, y ), x ( c, d), () subect to the followng condtons: Case I: Intal value problem: () () (3) (4) y( c), y ( c), y ( c), y ( c), y ( c). () Case II: Boundary value problems of Type : Publshed by World Academc Press, World Academc Unon

2 Journal of Informaton and Computng Scence, Vol. (07) No., pp y( c), y ( c), y ( c), y( d), y ( d). (3) () () () Case III: Boundary value problem of Type : () (3) () y( c) 3, y ( c) 3, y ( c) 3, y( d) 3, y ( d) 3. (4) Where ' s, ' s, ' s, ' s, ' s, c and d are real constants for Ths artcle s organzed as, n secton notatons of Haar wavelets and ther ntegrals are ntroduced. In secton 3, numercal algorthm based on wavelets s ntroduced. In secton 4 convergence analyss s presented. In secton 5 we reported our numercal results wth comparson. In the fnal secton concluson of our work has been dscussed.. Haar wavelets and ther ntegrals,,3. L [ c, d] In ths secton, we obtan orthogonal bass for the subspaces of called Haar wavelet famly. For ths notatons ntroduced n Ref. [4] are used. The nterval [, ] s dvded nto ( d c) length t J, where s called maxmal level of resoluton. We have coarser resoluton values 0,,,..., J and translaton parameter k 0,,,...,. Wth these two parameters Haar wavelet n Haar famly s defned as, for t [ ( ), ( )), h ( t), for t [ ( ), 3( )), (5) 0, otherwse, J cd J subntervals of equal here m k, ( ) c kt, ( ) c (k ) t and 3 ( ) ( ) J. Above equatons are vald for. and are called father and mother wavelets n Haar wavelet famly and are gven by, for t [ c, d), h () t 0, otherwse, (6) h() t h () t c d where, p. Any functon whch s havng fnte energy on[ cd, ],.e. sum of Haar wavelets:, for t c, p, h ( t), for t p, d, (7) 0, otherwse, th f L [ c, d] can be decomposed as nfnte f ( x) a h ( x), (8) where a ' s are called Haar coeffcents. If s ether pecewse constant or wsh to approxmate by pecewse constant on each subnterval then the above nfnte seres wll be termnated at a fnte number of terms. Snce, we have explct expresson for each member of Haar famly (5-7). We can ntegrate as many tmes depend upon the context. The followng notatons are used for tmes of ntegraton of members n the famly defned on [ cd, ): f P, ( t)... h( x) dx, t t t c c c (9) JIC emal for subscrpton: publshng@wau.org.uk

3 98 for For, (9) becomes, we have A. Padmanabha Reddy et al.: Haar wavelet collocaton method to solve problems arsng n nducton motor d E, P, ( t) dt. (0) c P,( t ) ( ),! t c () 0, f t [ c, ( )), P ( ( )) t, f t [ ( ), ( )),! () t ( t ( )) ( t ( )), f t [ ( ), ( )),! ( ( )) t ( t ( )) ( t 3 ( )), f t [ 3 ( ), d ).!, 3 () 3. Method of soluton Haar wavelet collocaton method: The proposed method s as follows [4, 7]. Approxmate hghest dervatve n terms of Haar wavelets J (5) y ( x) a h ( x). (3) (4) (3) () () Decompose y ( x), y ( x), y ( x), y ( x) and n terms of ntegrated Haar functons and replace these n to the gven lnear dfferental equaton. ( x Dscrtze equaton obtaned n above at collocaton ponts: l xl),,,... J xl l, J where xn c nt, n 0,,,...,. Resultng nto J J lnear algebrac system. Calculate the wavelet coeffcents and obtan the approxmate soluton for problem of nducton motor. In ths paper, problems orgnatng n modelng of nducton motor defned over [0,] are consdered. The proposed method s further smplfed wth the help of partcular ntal or boundary condtons. For IVPs: c d Integrate equaton (3) from 0 to x fve tmes we obtaned the approxmate soluton. a ' s c 0 and BVPs : 0,. yx () 3 4 () x () x (3) x (4) y( x) y(0) xy (0) y (0) y (0) y (0) ap5, ( x). (4)! 3! 4! Usng equaton () we can fnd the soluton for ntal value problem. Approxmate soluton for BVPs (3) (4) can be estmated by fndng the y (0), y (0) values usng boundary condtons of type and () (4) y (0), y (0) values usng boundary condtons of type, these unknowns are expressed as follows Type : Type : Where, (3) y (0) a ( 4E 6 E ), (5) J J 5, 4, (4) y (0) a (7E 46 E ), (6) J 5, 4, M () y (0) a (4 E E ), (7) , 4, 6 (4) y (0) 4 4 a (4E E ). (8) M , 4, JIC emal for contrbuton: edtor@c.org.uk

4 Journal of Informaton and Computng Scence, Vol. (07) No., pp (9) E P ( x) dx and E P ( x) dx. 4, 4, 5, 5, Convergence analyss of Haar wavelet dscretzaton method (HWDM) The accuracy ssue of the HWDM was open from year 997. Ths ssue s clarfed by J. Maak et al. [] n 05. Followng results are due to notatons ntroduced n Ref. [].The general form of ffth order ODE s () () (3) (4) (5) f x, y, y, y, y, y, y 0. (0) Expand ffth order dervatve nto Haar wavelets as 5 d y( x) ( ) 5 ah x () dx a h a h ( x). () k k 0 k0 In equatons () and () k, k 0,,...,. Integratng equaton () fve tmes, we obtan the soluton of DE (0) as Here P ( ) x 5, k Let us assume that 6 ( ) : d y x 6 dx a ( ) ( ) ( ). k 5, k 5! 0 k0 y x a P x B x (3) can be calculated wth ad of equaton () and 5 d y( x) dx 5 J Bx () s a boundary term. s a contnuous and ts next dervatve s bounded on [0,] L ( R) a Let y J ( x) ( ) ( ) a P x B x k 5, k be the approxmaton to unknown 5! 0 k0 wavelets. The absolute error at the th J Norm of the error n Hlbert space resoluton s denoted as E J and s gven by k 5, k J k0 y,.e. by Haar E J y( x) y J ( x) a P ( x). (4) L ( R )[] s defned as J k 5, k 0 J k0 E a P ( x) dx r r r k s 5, k 5, s J k0 rj s0 0 a a P ( x) P ( x) dx, (5) J. Maak et al. [, ] have shown that a, for J k and P 5, ( x) are monotoncally ncreasng on[0,). Equaton (5) can be estmated as r 4 4 E J. (6) r 4 J k 0 rj s0 6 6 Above equaton can be smplfed as JIC emal for subscrpton: publshng@wau.org.uk

5 00 A. Padmanabha Reddy et al.: Haar wavelet collocaton method to solve problems arsng n nducton motor factrzaton and, m,. r m J rj 4 E J 36 J 0 J. (7) Therefore, E J O. J (8) From equaton (8), we can conclude that the convergence s of order two. 5. Numercal studes We consdered some problems of nducton motor whose exact solutons are known. The approxmate soluton for each problem s devsed by the HWCM. All computatons are carred out by MATLAB software. Example : Consder the ntal value problem [0], (5) y ( x) 5( x )sn( x) 5( x x 5)cos( x) xy( x), x [0,], (9) wth ntal condtons: () () (3) (4) y(0) 5, y (0) 5, y (0) 5, y (0) 5, y (0) 5. (30) Usng the method of soluton solvng equaton (9), we get the Haar coeffcents [ a, a,..., a J ] [.8,.9,...,.4] for J=4.. By the help of Haar wavelet soluton y(x) s found. Obtaned soluton s represented wth analytc soluton y( x) 5( x)cos( x) n Table. Example : Consder the nducton motor problem [5], y (5) ( x) sn( x) y( x) cos( x)( sn( x)) sn( x)(sn( x) ), x [0,], (3) wth boundary condtons: () () () y(0), y() cos() sn(), y (0), y () cos() sn(), y (0). (3) Worked out the equaton (3) of Type, we found the Haar coeffcents [ a, a,..., a J ] [0.38, 0.33,..., 0.08] for J=5. Deduced the soluton usng a ', s equatons (4-6) and s represented wth the analytc soluton y( x) cos( x) sn( x) n Table. Example 3: Consder the nducton motor problem [5], x cos( ) x ( sn( )) 5 x y x y x e x e x e sn( x ), x [0,], (33) wth boundary condtons: () () () y(0), y() e( sn()), y (0) 0, y () e(cos() sn() ), y (0). (34) Haar coeffcent vector of Type BVP for J=4 s [ a, a,..., a J ] [0.8, 3.4,..., 0.9]. Approxmate x soluton y(x) s found usng equatons (4-6) and compared wth the exact soluton y( x) e ( sn( x)) n Table 3. Example 4: Consder the boundary value problem [9], (5) ( ) ( ) (5 0 ) x y x y x x e, x [0,], (35) wth boundary condtons: () () (3) y(0) 0, y() 0, y (0), y () e, y (0) 0. (36) [ a, a,..., a J ] [ 35.4,.73,...,5.6] are Haar coeffcents of Type BVP for J=6. These fndngs, equatons (4), (7) & (8) are approxmated the soluton. The comparson of obtaned soluton and analytc soluton x( x) e x are tabulated n Table Results and dscusson Absolute errors obtaned by HWCM are compared to other exstng technques, exhbt the results that as the level of resoluton ncreases absolute error curves of HWCM are goes closer to x-axs (Where the a s ' JIC emal for contrbuton: edtor@c.org.uk

6 Journal of Informaton and Computng Scence, Vol. (07) No., pp absolute errors are zero). The comparson of approxmate soluton, exact soluton at collocatons ponts wth J=3,4,5 & 6 for Examples,, 3 & 4 have been demonstrated n Fgures, 3, 5, 7 respectvely. Here n each fgure approxmate soluton concded wth the exact soluton, ths assures the exactness of HWCM results. Absolute errors obtaned by the Examples,, 3 & 4 are drawn n Fgures, 4, 6& 8 for varous resolutons. In all these fgures as the level of resoluton ncreased absolute errors tended towards x axs. These graphs concluded that decreasng of absolute errors show the accuracy of the HWCM solutons. To check the effcency of the method quanttatvely 6 Tables are desgned. In Tables -4 approxmate soluton, exact soluton, absolute errors are at the grd ponts 0., 0.,, 0.9 wth J=4, 5, 4 & 6 for Example -4 are lsted. Each tables explored the comparson of exact and Haar soluton. By ths observaton concluded that approxmate solutons are much closed to analytc solutons. In Table 5 Maxmum absolute errors obtaned by Examples, & 3 for dfferent values of J are compared to non polynomal splne method. Ths table ensures that maxmum absolute errors of HWCM are very less compared to NPSM. Accuracy of solutons obtaned for Example 4 s examned n Table 6. HWCM has gven least maxmum absolute errors for J=7, 8, 9 compared to OSPM and NPSS methods. Tables -6 depct as accuracy of the wavelet solutons are less satsfactory for small value of J; n case ncreases occur n the resoluton better accuracy can be acheved. Fg.. Comparson of exact and Haar soluton of Example for J=4 Fg.. Absolute errors for varous resolutons of Example JIC emal for subscrpton: publshng@wau.org.uk

7 0 A. Padmanabha Reddy et al.: Haar wavelet collocaton method to solve problems arsng n nducton motor Fg.3. Comparson of Exact and Haar Soluton for Example wth J=3 Fg.4. Absolute errors by HWCM wth J=4, 5, 6&7 for Example Fg.5. Comparson of Exact and Haar Soluton for Example 3 wth J=4 JIC emal for contrbuton: edtor@c.org.uk

8 Journal of Informaton and Computng Scence, Vol. (07) No., pp Fg.6. Absolute errors by HWCM wth J=4, 5, 6&7 for Example 3 Fg.7. Comparson of Exact and Haar Soluton for Example 4 wth J=5 Fg.8. Absolute errors by HWCM wth J=4, 5, 6&7 for Example 4 JIC emal for subscrpton: publshng@wau.org.uk

9 04 A. Padmanabha Reddy et al.: Haar wavelet collocaton method to solve problems arsng n nducton motor Table. Comparson of exact and Haar soluton of Example X Exact Soluton Approxmate Soluton Absolute error.509e e e E E E E E E-06 Table. Comparson of exact and Haar Soluton of Example X Exact Soluton Approxmate Soluton Absolute error E-.4506E E-0.08E E E E E-0.553E-0 Table 3. Comparson of exact and Haar Soluton of Example 3 X Exact Soluton Approxmate Soluton Absolute error.0804e-09.3e E E E E E E E-08 Table 4. Comparson of exact and Haar Soluton of Example 4 X Exact Soluton Approxmate Soluton Absolute error E E E E E E E E E-08 JIC emal for contrbuton: edtor@c.org.uk

10 Journal of Informaton and Computng Scence, Vol. (07) No., pp Table 5. Comparson of Maxmum absolute errors Example No. HWCM Non-Polynomal Splne(NPS)[5] h=/0 h=/0 9.46E E.63E E E 4.79E E--06 Table 6. Maxmum absolute errors of HWCM compared wth OSPM and [8] Example No. 0 OSPM[9] HWCM NPSS [8] 40 J=7,8, E-09.80E- 3.7E-.85E E-06.99E-06.9E E- 3.47E-.46E E E Concluson The present method has been tested on ntal and boundary value problems (Type and Type) of nducton motor. The order of convergence for ffth order ODEs s found. The approxmate solutons obtaned by HWCM are n good agreement wth exact solutons. We concluded from graphs and tables that the numercal results obtaned by HWCM are better than other exstng methods. Analyzng the numercal studes we observed that the proposed method gves more precse results by ncreasng level of resoluton. So that the method s effcent and reasonable for nducton motor problems. 8. Acknowledgment Author A. Padmanabha Reddy s grateful to Vson Group on Scence and Technology, Govt. of Karnataka, Inda, for fnancal assstance under the scheme Seed Money to Young Scentsts for Research (SMYSR-FY-05-6/GRD-497). 9. References A. Haar, Zur theorc der orthogonalen Funktonsysteme, Math. Annal. 69: 33-37(90). C. F. Chen and C. H. Hsao, Haar wavelet method for solvng lumped and dstrbuted -parameter systems, IEEE Proc. Control Theory Appl. 44: 87 94(997). C. F. Chen and C. H. Hsao, Wavelet approach to optmzng dynamc systems, IEEE Proc. Control Theory. Appl. 46: 3 9(990). U. Lepk and H. Hen, Haar wavelets wth applcatons, Sprnger. (04). S. S. Sddq and M. Sadaf, Applcaton of non-polynomal splne to the soluton of ffth-order boundary value problems n nducton motor, J. Egyp. Math. Soc. 3: 0-6(05). R. P. Agarwal, Boundary value problems for hgher order dfferental equatons, Worl.Sc. Sngapore. 8: (986). A. Padmanabha Reddy, S. H. Manula, C. Sateesha and N. M. Buurke Haar wavelet approach for the soluton of seventh order ordnary dfferental equatons, Math. Mod. Eng. Prob. 3: 08-4(06). Sddq S. S. Akram G. and Salman A. Malk, Nonpolynomal sextc splne method for the soluton along wth convergence of lnear specal case ffth-order two-pont boundary value problems, Appl. Math. Comput. 90: 53-54(007). K. Faraeyan and R. Jallan, Numercal soluton of ffth-order boundary-value problems n off step ponts, Math. JIC emal for subscrpton: publshng@wau.org.uk

11 06 Sc. 4: 43-46(00). A. Padmanabha Reddy et al.: Haar wavelet collocaton method to solve problems arsng n nducton motor P. Mehd Gholam, G. Behzad and G. Mohammad, He s varatonal teraton method for solvng dfferental equatons of ffth order, Gen. Math. Notes. : 53-58(00). J. Maak B. S., Shvartsman, M. Krs, M. Pohlak and H. Herranen, Convergence theorem for the Haar wavelet based dscretzaton method, Compos. Struct. 6: 7-3(05). J. Maak, B. Shvartsman, K. Karust, M. Mkola, A. Haavaoe and M. Pohlak, On the accuarcy of the Haar wavelet dscretzaton method, Compos. Part B. 80: 3-37(05). JIC emal for contrbuton: edtor@c.org.uk