wavelet collocation method for solving integro-differential equation.

Transcription

1 IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): Vol. 5, Issue 3 (arch. 5), V3 PP -7 wavelet collocatio method for solvig itegro-differetial equatio. Asmaa Abdalelah Abdalrehma Uiversity of Techology Applied Sciece Departmet Abstract: - Wavelet collocatio method for umerical solutio th order Volterra itegro diferetial equatios (VIDE) by epadig the ukow fuctios, as series i terms of chebyshev wavelets secod kid with ukow coefficiets. The aim of this paper is to state ad prove the uiform covergece theorem ad accuracy estimatio for series above. Fially, some illustrative eamples are give to demostrate the validity ad applicability of the proposed method. Keywords: chebyshev wavelets secod kid; itegro-differetial equatio; operatioal matri of itegratios; uiform covergece; accuracy estimatio. I. INTRODUCTION Basic wavelet theory is a atural topic. By ame, wavelets date back oly to the 98s. o the boudary betwee mathematics ad egieerig, wavelet theory shows studets that mathematics research is still thrivig, with importat applicatios i areas such as image compressio ad the umerical solutio of differetial equatios [], itegral equatio [],ad itegro differetial equatios [3,8]. The author believes that the essetials of wavelet theory are sufficietly elemetary to be taught successfully to advaced udergraduates [4]. Orthgoal fuctios ad polyomials series have received cosiderable attetio i dealig with various problems wavelets permit the accurate represetatio of a fuctios ad operators. Special attetio has bee give to applicatio of the Legedre wavelets [5], Harr wavelets [6] ad Sie-Cosie wavelets [7]. The solutio of itegro-differetial equatios have a major role i the fields of sciece ad igieerig whe aphysical system is modeled uder the differetial sese, it fially gives a differetial equatio, a itegral equatio or a itegro-differetial equatios mostly appear i the last equatio [8]. I this paper the operatioal matri of itegratio for Hermite wavelets is derived ad used it for obtaiig approimate solutio of the followig th order VIDE. u () ()g()+ k, t u (s) t dt () where s k(,t) ad g() are kow fuctios, ad u() is a ukow fuctio. II.SOE PROPERTIES OF SECOND CHEBYSHEV WAVELETS Wavelets costitute a family of fuctios costructed from dilatio ad traslatio of a sigle fuctio t called the mother wavelet.whe the dilatio parameter a ad the traslatio parameter b vary cotiuously we have the followig family of cotiuous wavelets as [9]. a,b t a tb, a, b R, a. a The secod chebyshev wavelets,m t (k,, m, t) ivolve four argumets,,,, k is assumed ay positive iteger, m is the degree of the secod chebyshev polyomials ad t is the ormalized time. They are defied o the iterval,) as,m t k U m k t +, k t < () oterwise where U m t U m t m,,, (3) here U m t are the secod chebyshev polyomialsof degree m with respect to the weight fuctio w t t o the iterval [-,] ad satisfy the followig recursive formula U t, U t t, U m + t tu m t U m t, m,, The set of chebyshev wavelets are a orthoirmal set with respect to the weight fuctio w t w ( k t + ). Iteratioal orgaizatio of Scietific Research P a g e

2 III.FUNCTION APPROXIAT A fuctio f(t) defied over [, ) may be epaded as wavelet collocatio method for solvig itegro-differetial equatio. f t m C m m (t) (4) were C m (f t, m t ) I which.,. deoted the ier product i L w [,). If the ifiite series equatio (.34) is trucated, the it ca be writte f t k m C m m t C T (t) (5) C C, C,, C (), C,, C (),, C,, C T (t) (t), (t),, (t), (t),, for Chebyshev wavelets of Secod kid. A fuctio f t defied over, may be epaded as: f t m f m m t (t), (6) (t), (t) T Covergece Aalysis where f m f t, m t (7) I eq.(5)..,. deotes the ier product with weight fuctio w t. If the ifiite series i eq.(4) is tracated the eq.(4) ca be writte as: f t f t f m m m F T m where F ad m are k matrices give by F f, f,, f, f,, f,, f k,, f T (8) (t) t, t,, t, t,, t, t,, t T Theorem(): (Covergece Aalysis theorm) Assume that a fuctio f(t) L w,, w t with t L, ca be epaded as ifiite series of secod kid chebyshev wavelets, the the series coverges uiformly to f(t). Proof: sice f m f t, m (t) the f m f(t) m (t)w t dt k+ f(t)u m k t + w k t + dt If we make use of the substitutio k t + cos θ i (5), yields cos θ + f m k f k si θ si m + θ By usig the itegratio by parts, The eq.() becomes cos θ + si mθ si m + θ f m k f k m m + cos θ + + 3k f k si θ cos θ + f m m 3k f k si θ si mθ m + 3k f After performig itegratio by parts agai yields, si mθ m (9) () si m + θ m + cos θ + k si θ si m + θ () Iteratioal orgaizatio of Scietific Research P a g e

3 Cosider: f m m 3 5k wavelet collocatio method for solvig itegro-differetial equatio. cos θ + cos m θ cos mθ cos mθ cos m + θ k m m + cos θ + cos m θ cos m + θ m + 3 5k k m + cos m + θ cos m + 4 θ m + 3 () cos θ + cos m θ cos mθ cos mθ cos m + θ k m m + cosθ + cos m θ cos mθ cos mθ cos m + θ k m m + cosθ + m cos m + θ m cos m θ + m + cos m θ k m m + m cos m + θ + 4m cos mθ + m + cos m < L m m + L m m + m + 4m + m + L m m + 3m + Thus, we get cos θ + cos m θ cos mθ cos mθ cos m + θ k m m + < L 3m + (3) m m + Similarly, cos θ + cos m θ cos m + θ cos m + θ cos m + 4 θ k m + m + 3 cos θ + k m + cos m + 4 θ (m + ) cos m + θ + m + 3 cos m + θ + m + 3 cos m θ (m + ) m + 3 cos θ + k Thus we get < L Usig eqs.(3)ad(4), oe ca get m cos m + 3 θ m cos m + θ + m + θ cos m θ m m + m + cos m + 4 θ + m + 4 cos m + θ + m + 3 cos m θ m + m + 3 L m + m + 3 6m + 4m + 6 L m + m + 3 3m + m + 3 cos θ + cos m θ cos m + θ k m + < L m + m + 3 3m + m + 3 cos m + θ cos m + 4 θ m + 3 (4) Iteratioal orgaizatio of Scietific Research 3 P a g e

4 5k f m < 3 m + wavelet collocatio method for solvig itegro-differetial equatio. f m < 5k3 L 3m + m m m + L 3m + m + 3 m + m + m + 3 m+ 5k L m m Fially sice k,the f m < L + 5 m Accuracy Estimatio of m (): If the fuctio f() is epaded iterms of fourth kid chebyshev wavelets, f m f m m () (5) It is ot possible to perform computatio a ifiite umber of terms, therfore we must trucate the series i (5). I place of (5), we take f k m f m m () (6) so that f f + + m f m m () or f f r() (7) where r() is the residual fuctio r + m f m m () (8) we must select coefficiets i eqs.(7) ad (8) such that the orm of the residual fuctio r() is less tha some covergece criterio,that is for all greater tha some value. f f w d < Theorem () Let f() be a cotiuaus fuctio defied o [,), ad f () < L, the we have the followig accuracy estimatio c k, < L + m (9) + 5 m 4 where Proof:- c k, r w d Sice r w d + r w d f m w d m + m f m m () w d or f m w d + m f k + m U m k + k + d Let t k +, f U m t t d we have, the + < m U m t t d m f + L m m. 8 m 4 Iteratioal orgaizatio of Scietific Research 4 P a g e

5 wavelet collocatio method for solvig itegro-differetial equatio. wavelet collocatio method for VIDE with mth order: I this sectio the itroduced wavelets collocatio will be applied to solve VIDE with mth order, ( u ) i g i + K i,j, t u (s) i t dt, s () s With the followig coditios u i a is i,,, l s,,,, Afuctio u i which is defied o the iterval, ca be epaded ito the secod chebyshev wavelet series u i i c i i (t) () Where c i are the wavelet coefficiets. Itegrate eq.() m times,yields m j u i c i i t dt + j a j! m j () Usig the followig formula i t dt t ( )! i (t)dt therefore eq.() becomes u c i t i (t)dt + i j (3) This leads to Let K, t i u c i L i + j I similar way, we ca get t! j j! a j ()! ad L i K, t i t dt s j j! a j i,,, u (s) i c i L s i + j a j! sj (4) Substitutig eqs () ad (4) i (), yield or i c i i (t) g i + K i,j, t c i L s i + i c i i (t) A i g i + where A i K, t L i s t dt B j K, t t j dt s j i j a j! sj dt s a sj B j! j () (5) j (6) i,,,, j,,,,-s- (7) Net the iterval, is devided i to l ad itroduce the collocatio poits l k k, k,,,l eq() is satisfied oly at the collocatio poits we get asystem of liear equatios l s a sj i c i [ i () A i ] g i + j j! B j () (8) The matri form of this system s a sj is C FG+ j B j () where F (), Gg() j!.desig of the matri A:- Whe chebyshev wavelets secod kid are itegrated m times, the followig itegral must be evaluated. L i K, t i t dt, i,,,, L i t k! () Therefore the matri A i ca be costructed as follows s Sice A i K, t L i t dt i,,,, l k < l k Iteratioal orgaizatio of Scietific Research 5 P a g e

6 wavelet collocatio method for solvig itegro-differetial equatio. A i s K, t L i t dt i K i, t L i s t dt i > IV. WAVELETS ETHOD FOR VIDE WITH NTH ORDER For solvig VIDE with mth order the matri L i i sectio above will be followed to get s a sj i c i [ i ( L A L )] g L + j B j ( L ) L a, b But A i L L L j! K L, t L i s t dt where i,, B j ( L ) K L, t t s dt s where L i t as i eq(7),(8) that is A i L A L, F i L i L A i L F L Numerical Results: I this sectio VIDE is cosidered ad solved by the itroduced method. parameters k ad are cosidered to be ad 3 respectively. Eample : Cosider the followig VIDE: U e e (t) U t dt Iitial coditios U(), U (). The eact solutio U e e +. Table shows the umerical results for this eample with k, 3 with error -3 ad k, 4, with error -4 are compared with eact solutio graphically i fig. Table :some umerical results for eample Eact solutio Approimat solutio k,3 Approimat solutio k, Fig :Approimate solutio for eample Eample : Cosider the followig VIDE : U (5) si + cos + ( t)u (3) t dt Iitial coditios U(), U (), U"(), U 3 (). The eact solutio U cos. Table shows the umerical results for this eample with k, 3 with error -3 ad k, 4, with error -4 are compared with eact solutio graphically i fig,. Iteratioal orgaizatio of Scietific Research 6 P a g e

7 wavelet collocatio method for solvig itegro-differetial equatio. Table :some umerical results for eample Eact solutio Approimat solutio k,3 Approimat solutio k, Fig :Approimate solutio for eample V. CONCLUSION This work proposes a powerful techique for solvig VIDE secod kid usig wavelet i collocatio method compariso of the approimate solutios ad the eact solutios shows that the proposed method is more faster algorithms tha ordiary oes. The covergece ad accuracy estimatio of this method was eamied for several umerical eamples. REFERENCES [] Asmaa A. A, 4, Numerical, Solutio of Optimal Cotrol Problems Usig New Third kid Chebyshev Wavelets Operatioal atri of Itegratio, Eg&Tech joural,vol 3,part(B),: [] Tao. X. ad Yua. L.. Numerical Solutio of Fredholm Itegral Equatio of Secod kid by Geeral Legedre Wavelets, It. J. I. Comp ad Cot. 8(): [3] A.Barzkar,.K.Oshagh,, Numerical solutio of the oliear Fredholm itegro-differetial equatio of secod kid usig chebysheve wavelets, World Applied Scices Joural (WASJ),Vol.8, N(): [4] E.Johasso, 5, Wavelet Theory ad some of its Applicatios. [5] Jafari. H ad Hosseizadeh. H.. Numerical Solutio of System of Liear Itegral Equatios by usig Legedre Wavelets, It. J. Ope Problems Compt. ath., 3(5): [6] Shihab. S. N. ad ohammed. A.. A Efficiet Algorithm for th Order Itegro- Differetial Equatios Usig New Haar Wavelets atri Desigatio, Iteratioal Joural of Emergig & Techologies i Computatioal ad Applied Scieces (IJETCAS). (9): [7].Razzaghi & S.Yousefi.. Si-Cosie wavelets operatioal matri of itegratio ad it s applicatios i the calculus of variatios. Vol 33. No : [8] A.Arikoglu & I.Ozkol. 8. Solutio of itegral itegro-differetial equatio systems by usig differetial trasform method. Vol 56,Issue [9] Arsalai.. & Vali.. A., Numerical Solutio of Noliear Problems With ovig Boudary Coditios by Usig Chebyshev Wavelets, Applied athematical Scieces,Vol.5(): Iteratioal orgaizatio of Scietific Research 7 P a g e