A NEW NUMERICAL APPROACH FOR SOLVING HIGH-ORDER LINEAR AND NON-LINEAR DIFFERANTIAL EQUATIONS

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1 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S67 A NEW NUMERICAL APPROACH FOR SOLVING HIGH-ORDER LINEAR AND NON-LINEAR DIFFERANIAL EQUAIONS by Ayd SECER a* ad Selv ALUN b a Departmet of Mathematcal Egeerg, Yldz echcal Uversty, Istabul, urey b Yldz echcal Uversty, Istabul, urey Orgal scetfc paper Itroducto I ths paper, the Legedre wavelet operatoal matrx method has bee troduced for solvg hgh-order lear ad o-lear mult-pot: tal ad boudary value problems. It has bee suggested that the techque s rest upo practcal applcato of the operatoal matrx ad ts dervatves. he dfferetal equato s preseted that t s coverted to a system of algebrac equatos va the propertes of Legedre wavelet together wth the operatoal matrx method. As a result of ths study, the scheme has bee tested o fve lear ad o-lear problems. he results have demostrated that ths method s a very effectve ad advatageous tool solvg such problems. Key words: shfted Legedre polyomals, Legedre wavelet, operatoal matrx, hgh-order dfferetal equatos Hgh-order dfferetal equatos have substatal atteto, because of ther fascatg mathematcal structures ad propertes ad they play a mportat role the thermal scece ad mechacal egeerg. Flud-flow, heat trasfer, ad other related physcal pheomea of terest are gaed by prcples of coservato ad are symbolzed terms of dfferetal equatos deotg these prcples. For stace, fourth-order dfferetal equatos are used the umercal aalyss of vscoelastc ad elastc flows, the free vbrato aalyss of beam structures, deformato of beams ad plate deflecto theory []. A fourth order aalogue of t s the Orr-Sommerfeld equato expla to great correctess the cross-stream behavor of chael flud-flow. Moreover, sxth-order dfferetal equatos arse the free vbrato aalyss of rg structures ad astrophyscs []. Some related applcatos of hgh-order dfferetal equatos ca be foud [-8]. I [], Noor ad Mohyud-D preseted the varato terato method for solvg fourth-order boudary value problems ad [4, 7] they llustrated the homotopy perturbato method for solvg ffth-order ad sxth-order boudary value problems, respectvely. Also [8], a Legedre Petrov-Galer method was demostrated for the soluto of the fourth-order boudary value problems. I [5, 6], the umercal soluto of ffth-order boudary value problems was preseted by usg a ew cubc B-sple method ad a sxth-degree B-sple approxmato, respectvely. I [], El-Gamel et al. appled sc-galer method for solvg sxth-order boudary value problems. * Correspodg author, e-mal: aydsecer@gmal.com

2 S68 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 he orthogoal fuctos ad polyomal seres are very mportat feld scece ad egeerg. Bloc-pulse fuctos, se-cose fuctos, Legedre, Laguerre, ad Chebyshev polyomals are the most commoly utlzed amog these fuctos. What maes these fuctos mportat s that they permt the udertag problem to be reduced to system of algebrac equatos ad the approxmato of aalytc fuctos. he problem s solved by trucatg seres of orthogoal bass fuctos ad utlzg operatoal matrx ad ts dervatves. he operatoal matrx of dervatves [D] s gve: d ψ () t = Dψ () t () where ψ( t) = [ ψ, ψ,..., ψ N ] ad ψ ( =,,..., N) are orthogoal bass fuctos, orthogoal o a certa terval [ ab., ] he matrx [D] ca be uquely detfed o the bass of the specfc orthogoal fuctos [9-]. May papers whch are related to the applcato of operatoal matrx of dervatve ca be foud the lterature [9-]. Wavelet fuctos have attracted cosderable atteto recet years ad have made a sgfcat cotrbuto to both appled ad theoretcal vestgatos. he most basc feature of wavelets s to be able to aalyze accordg to scale. Especally, wavelets are commoly utlzed sgal aalyss, tme-frequecy aalyss, electromagetc feld computatos, ad may other areas. Daubeches, Bel, Meyer, ad Mallat played a very mportat role uderstadg the cotet of wavelets. here has bee a sgfcat crease the umber of studes o wavelets due to ther cotrbutos. Moreover, may dfferet types of wavelet fuctos have bee preseted over the past few years. Daubeches, Haar, Legedre, Shao, Hermta, Chebyshev wavelets are some of them. We prefer to use Legedre wavelets ths paper due to ther orthogoalty ad smplcty. May applcatos of Legedre wavelets ca be vewed from [3-9]. I ths paper, frst of all the Legedre wavelets have bee troduced. he operatoal matrx of dervatve s the utlzed to compute the coeffcets of Legedre wavelets by usg shfted Legedre polyomals. he operatoal matrx of Legedre wavelet s geeralzed order to solve hgh-order lear ad o-lear mult-pot tal ad boudary codtos. Prelmares ad otatos I ths secto, we gve some otatos, deftos, ad prelmary facts to use forthcomg sectos of ths paper. Wavelets ad Legedre wavelets May researchers have utlzed wavelets for solvg mathematcal ad egeerg problems ad they have bee gag more promece recetly. Moreover, researchers prefer to use Legedre wavelets due to ther clarty ad orthoormalty. Wavelets establsh a famly of fuctos formulated from dlato parameter a ad the traslato parameter b chage cotuously, we have the followg famly of cotuous wavelets: ( ) / t b ψ,,, ab t = a ψ ab R a a If these parameters a ad b are restrat to dscrete values as = a, = b, the: ψ t ψ t / ( ) = ( ) forms a orthogoal bass. We use multresoluto of aalyss (MRA) for structure wavelets.

3 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S69 Defto. Let { V j } j Z of subset of L ( R ) be the creasg sequece ad φ be the scalg fucto. If t satsfes the followg codtos, we call { V j } j Z wth scalg fucto φ MRA [7]: () j V j s dese L ( R ), () jv j = {}, (3) f( t) V ( j j f t) V, ad (4) { ϕ ( t )} Zs a orthogoal bass for V. Legedre wavelets ψ () t m = ψ ( mt,,,) have four parameters: parameter, ca be presumed ay postve teger, m the order of shfted Legedre polyomals, ad t the ormalzed tme. hey are defed o the terval [,] by: + + m+ Pm ( t ), t ψ () = m t (), otherwse where =,,..., ; =,,...,( / m M ). he coeffcet [( m + )/] s for orthoormalty. Let Pm () t deote the shfted Legedre polyomals of order m, the defe the Pm () t by: ad the orthogoalty codto s: Fucto approxmato m m P ( t) = ( ) m = m+ ( m+ )! t ( m )! (!), for P() tp()d t t= m +, for m= m Suppose that a fucto f() t s defed over [,]. he f() t may be expaded the terms of Legedre wavelet: f() t c ψ () t (5) = m m = m= where cm = [ f( t), ψ m ( t)] whch (.,.) deotes the er product. Let the fte seres eq. (5) s trucated, the t ca be wrtte: where C ad ψ () t are matrces [5]: M f() t c ψ () t = C ψ () t (6) m = m= =,,,,...,,,...,,,...,,,..., M M,,, M C c c c c c c c =,,,,...,,,...,,,...,,,..., M M,,, M ψ ψ ψ ψ ψ ψ ψ ψ he operatoal matrx of dervatve Mohammad ad Hosse [5] derved Legedre wavelets operatoal matrx hs paper. I ths secto, the theorem ad corollary are just metoed: m (3) (4) (7)

4 S7 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 heorem. Let ψ () t be the Legedre wavelets vector gve secto Wavelets ad Legedre wavelets, the we get: dψ ( t) ( t) = Dψ (8) where [D] s the ( M + ) operatoal matrx of dervatve defed: F O O O F O D = O O F th whch F s a ( M + )( M + ) matrx ad ts (,) rs elemet s defed: ( )( ) ( ) + r s, r =,..., M +, s =,..., r ad ( r + s) odd F rs, = (), otherwse Corollary. If we use eq. (8) the we have operatoal matrx for th dervatve: (9) d ψ () t = D ψ () t () where [D] s the th power of matrx [D]. By usg the property of the product of two Legedre wavelets vector fuctos, we have: e ψψ = ψ E () where e s a gve vector ad E s a ( M + ) ( M + ) matrx depedet o vector e [5]. Solvg hgh-order lear dfferetal equatos I ths secto, the Legedre wavelet operatoal matrx method s mplemeted to solve th order lear dfferetal equatos. Cosder the followg equato: d u d u du ( ) ( )... ( ) ( ) ( ), h t + h t + + h t + h t u= gt t < t< t (3) wth these tal codtos: or wth these boudary codtos: u( t) = u, ( t) = u, ( t) = u,..., ( t) = u (4) ( ), ( ), ( ),..., ( ) u t = u t = u t = u t = u ( ) ( ) ( ) ( ) u t = u, t = u, t = u,..., t = u =,,..., / f eve

5 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S7 ( ), ( ), ( ),..., ( ) u t = u t = u t = u t = u du d u d u ( ) ( ) ( ) ( ) u t = u, t = u, t = u,..., t = u =,,..., ( + )/ f odd Frst we presume that the uow fucto ut () s approxmated ad gve: ut () = C ψ () t (5) where C s a uow vector ad ψ () t s the vector gve eq. (). By utlzg eq. () we obta: du ( t ) = C Dψ ( t ) d C t ( ) = D ψ ( ) d u t C t ( ) = D ψ ( ) We ca also approxmate h( t), h( t),..., h ( t) ad gt ( ) as: ( ) = ψ ( ) ( ) = ψ ( ) h t H t h t H t ( ) = ψ ( ) ( ) = ψ ( ) h t H t g t G t where vectors H, H,..., H ad G are gve by eq. (6). Substtutg eqs. (6) ad (7) eq. (3) we obta: R t H t C D t H t C D t () = ψ() ψ() + ψ() ψ() +... (6) (7) + H ψ() t C Dψ() t + H ψ() t C ψ() t G ψ () t = (8) If we use the product operato matrx of Legedre wavelets (6), the we obta: R t = + H t t D C H t t D C +... ( ) ψ ( ) ψ ( )( ) ψ ( ) ψ ( )( ) ( ) ( )( ) ( ) ( ) ( ) + H ψ t ψ t D C + H = ψ t ψ t C ψ t G ( ) ( ) ( ) ( )... ( ) ( ) ( ) ( ) = ψ t H D C + ψ t H D C + + ψ t H D C + ψ t H C ψ t G where H, H,..., H are the product operato matrces ad ca be calculated by usg eq. (). We obta ( M + ) lear equatos by computg: ( ) ( ) d,,..., ( ) ψ trt t = j = M + (9) j

6 S7 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 Also by substtutg tal codtos (4) eqs. (5) ad (6) we obta: ( ) ψ ( ) u t = C t = u du ( t ) = C Dψ ( t ) = u d u ( t ) = C D ψ ( t ) = u d u ( ) D t C = ψ ( t ) = u We obta ( M + ) set of lear equatos by usg eqs. (9) ad (). hese lear equatos ca be solved for uow coeffcets of the vector C. Accordgly ut () whch s gve eq. (3) ca be computed. Solvg hgh-order o-lear dfferetal equatos I ths secto, the Legedre wavelet operatoal matrx method s mplemeted for solvg th order o-lear dfferetal equatos. Cosder the followg equato: wth the tal codtos: or boudary codtos: d u du d u ( t) = H tu, ( t), ( t),..., ( t), t t t < < du d u d u ( ), ( ), ( ),..., ( ) () () u t = u t = u t = u t = u () ( ), ( ), ( ),..., ( ) u t = u t = u t = u t = u ( ), ( ), ( ),..., ( ) u t = u t = u t = u t = u ( ), ( ), ( ),..., ( ) u t = u t = u t = u t = u du d u d u u( t) = u, ( t) = u, ( t) = u,..., ( t ) = u =,,..., / f eve =,,...,( + )/ f odd Frst we presume that the uow fucto ut () s approxmated ad gve: ut () = C ψ () t (3) where C s a uow vector ad ψ () t s the vector whch gve eq. (). By utlsg eq. () we obta: C D ψ() t = H tc, ψ(), t C D ψ(),..., t C D ψ() t (4) Also by substtutg tal codtos () eqs. (5) ad (6) we obta:

7 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S73 u( t ) = C ψ ( t ) = u d u ( t ) = C D ψ ( t ) = u d u ( t ) = C D ψ ( t ) = u (5) d u ( t ) = C D ψ ( t ) = u o obta the soluto ut (), we frst compute eq. (4) at ( M + ) pots. For a better result, we utlze the frst ( M + ) roots of shfted Legedre P () t. If we use ( M + ) these equatos collectvely wth eq. (5), the we obta ( M + ) o-lear equatos. hese o-lear equatos ca be solved by usg Newto s teratve method. Accordgly ut () gve eq. () ca be computed. Illustratve examples I ths secto, we have preseted some examples we have wated to demostrate the performace of the proposed techque solvg hgh-order lear ad o-lear dfferetal equatos. It s show that the Legedre wavelet operatoal matrx method yeld better results. Example. We frst cosder the followg fourth-order lear tal value problem: 4 3 d u d u d u du t 4 ( t) cost + s t + cost + 6u = 4e s t, < t < 4 3 subject to these tal codtos: 3 d d d () = u, (), (), () 3 u t t = the exact soluto of the prevous system s: ut ( ) = e t s t Usg these parameters: M = 6, =. he approxmate ad exact solutos have bee dsplayed tab.. Example. Cosder the followg ffth-order lear tal value problem [5]: d u d u d u d u du 5 ( t ) 4 3 ( t t ) ( t 4t) 4 3 tu= 4e t cost t + 4t + 6t 4t +, < t< subject to these tal codtos: d d d d ( ) = u, ( ), ( ) 6, 3 ( ) 4, 4 ( ) u t t t = the exact soluto of the prevous system s: t ut ( ) = e s t+ t

8 S74 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 able. Comparso betwee exact ad umercal solutos for Example x Exact M = 6, = soluto y(x) Approxmate soluto Absolute error Usg these parameters: M = 6, =. he approxmate ad exact solutos have bee dsplayed tab.. Example 3. Cosder the followg sxth-order lear boudary value problem [3]: 6 3 d u d u d u u + + u = e 6 3 ( 5t + 78t 4 ), < t < subject to these boudary codtos: du d u du d u u() =, () =, ()=, u() =, () =, ()= e e e able. Comparso betwee exact ad umercal solutos for Example x Exact soluto y(x) M = 6, = Approxmate soluto Absolute error the exact soluto of the prevous system s: ut t 3 () = e t hese parameters M = 8, = have bee used. he approxmate ad exact solutos have bee dsplayed tab. 3.

9 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S75 able 3. Comparso betwee exact ad umercal solutos for Example 3 x Exact soluto y(x) M = 8, = Approxmate soluto Absolute error Example 4. Cosder the followg ffth-order o-lear boudary value problem [6]: 5 d u 5u e =, < t < ( + t) subject to these boudary codtos: d d d () =, u() = u l, (), ().5, () u t t = the exact soluto of the prevous system s: ( ) = l( + t ) u t Usg these parameters: M = 5, =. he approxmate ad exact solutos have bee dsplayed tab. 4. Example 5. Cosder the followg sxth-order o-lear boudary value problem [7]: d u t u t t 6 () = e t (), < < 6 able 4. Comparso betwee exact ad umercal solutos for Example 4 x Exact soluto y(x) M = 6, = Approxmate soluto Absolute error

10 S76 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 subject to these boudary codtos: du d u u() =, () =, ()= du d u e e e u() =, () =, ()= the exact soluto of the prevous system s: ut () = e t hese parameters M = 8, = are used. he approxmate ad exact solutos have bee dsplayed tab. 5. Cocluso able 5. Comparso betwee exact ad umercal solutos for Example 5 x Exact soluto y(x) M = 6, = Approxmate soluto Absolute error I ths study, we have solved hgher order dfferetal equatos by utlzg Legedre wavelet operatoal matrx method. he udertag problem has bee reduced to solve a system of algebrac equatos. We have also developed a very effcet algorthm to solve th -order tal ad boudary value problems wth the proposed method o the MAPLE computer algebra system. All of the prevous umercal results ad graphcal represetatos have bee prepared by usg MAPLE. hese results show that the Legedre wavelet operatoal matrx method s a effcet mathematcal tool for fdg the umercal solutos of dfferetal equato. Refereces [] Noor, M. A., Mohyud-D, S.., A Effcet Method for Solvg Fourth-Order Boudary Value Problems, Computers ad Mathematcs wth Applcatos, 54 (7), 7-8, pp. - [] El-Gamel, M., et al., Sc-Galer Method for Solvg Lear Sxth-Order Boudary Value Problems, Mathematcs of Computato, 73 (3), 73, pp [3] Wag, Y., et al., A Note o the Numercal Soluto of Hgh-Order Dfferetal Equatos, Joural of Computatoal ad Appled Mathematcs, 59 (3),, pp [4] Noor, M. A., Mohyud-D, S.., A Effcet Algorthm for Solvg Ffth-Order Boudary Value Problems, Mathematcal ad Computer Modellg, 45 (7), 7-8, pp [5] Feg-Gog, L., A New Cubc B-Sple Method for Lear Ffth Order Boudary Value Problems, Joural of Appled Mathematcs ad Computg, 36 (), -, pp. -6

11 Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S77 [6] Caglar, H. N., Caglar, S. H., he Numercal Soluto of Ffth-Order Boudary Value Problems wth Sxth- Degree B-Sple Fuctos, Appled Mathematcs Letters, (999), 5, pp. 5-3 [7] Noor, M. A., Mohyud-D, S.., Homotopy Perturbato Method for Solvg Sxth-Order Boudary Value Problems, Computers ad Mathematcs wth Applcatos, 55 (8),, pp [8] She,.., et al., A Legedre Petrov-Galer Method for Fourth-Order Dfferetal Equatos, Computers ad Mathematcs wth Applcatos, 6 (),, pp. 8-6 [9] Sade, I., Bohar, M., Optmal Cotrol of a Parabolc Dstrbuted Parameter System va Orthogoal Polyomals, Optmal Cotrol Appl. Methods, 9 (998), 3, pp. 5-3 [] Razzagh, M., Arabshab, A., Optmal Cotrol of Lear Dstrbuted Parameter Systems Va Polyomal Seres, Iteratoal Joural of Systems Scece, (989), 7, pp [] Chag, R., Yag, S., Solutos of wo-pot-boudary-value-problems by Geeralzed Orthogoal Polyomals ad Applcatos to Cotrol of Lumped ad Dstrbuted Parameter Systems, Iteratoal Joural Cotrol, 43 (986), 6, pp [] Balaj, S., A New Approach for Solvg Duffg Equatos Ivolvg Both Itegral ad No-Itegral Forcg erms, A Shams Egeerg Joural, 5 (4), 3, pp [3] Saadatmad, A., Dehgha, M., A New Operatoal Matrx for Solvg Fractoal-Order Dfferetal Equatos, Computers ad Mathematcs wth Applcatos, 59 (), 3, pp [4] Mohammad, F., et al., A New Operatoal Matrx for Legedre Wavelets ad Its Applcatos for Solvg Fractoal Order Boudary Value Problems, Iteratoal Joural of Systems Scece, 6 (), 3, pp [5] Mohammad, F., Hosse, M. M., A New Legedre Wavelet Operatoal Matrx of Dervatve ad Its Applcatos Solvg the Sgular Ordary Dfferetal Equatos, Joural of he Fral Isttute, 348 (), 8, pp [6] Mshra, V., Saba, Wavelet Galer Solutos of Ordary Dfferetal Equatos, Iteratoal Joural of Math. Aalyss, 5 (), 9, pp [7] Khellat, F., Yousef, S. A., he Lear Legedre Mother Wavelets Operatoal Matrx of Itegrato ad Its Applcato, Joural of he Fral Isttute, 343 (6),, pp. 8-9 [8] Mohammad, F., et al., Legedre Wavelet Galer Method for Solvg Ordary Dfferetal Equatos wth No-Aalytc Soluto, Iteratoal Joural of Systems Scece, 4 (), 4, pp [9] Razzagh, M., Yousef, S., Legedre Wavelets Operatoal Matrx of Itegrato, Iteratoal Joural of Systems Scece, 3 (), 4, pp Paper submtted: Jue, 7 Paper revsed: November 7, 7 Paper accepted: November, 7 8 Socety of hermal Egeers of Serba Publshed by the Vča Isttute of Nuclear Sceces, Belgrade, Serba. hs s a ope access artcle dstrbuted uder the CC BY-NC-ND 4. terms ad codtos