Research Article The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations

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1 ISRN Applied Mthemtics Volume 214, Article ID 21399, 7 pges Reserch Article The Applictios of Crdil Trigoometric Splies i Solvig Nolier Itegrl Equtios Ji Xie, 1 Xioy Liu, 2 d Liig Xu 1 1 Deprtmet of Mthemtics d Physics, Hefei Uiversity, Hefei 2361, Chi 2 Deprtmet of Mthemtics d Physics, Uiversity of L Vere, L Vere, CA 9175, USA Correspodece should e ddressed to Xioy Liu; liu@lvere.edu Received 3 Decemer 213; Accepted 15 Jury 214; Pulished 4 Mrch 214 Acdemic Editors: Y. M. Cheg d L. You Copyright 214 Ji Xie et l. This is ope ccess rticle distriuted uder the Cretive Commos Attriutio Licese, which permits urestricted use, distriutio, d reproductio i y medium, provided the origil work is properly cited. The crdil trigoometric splies o smll compct supports re employed to solve itegrl equtios. The ukow fuctio is epressed s lier comitio of crdil trigoometric splies fuctios. The simple system of equtios o the coefficiets is deducted. Whe solvig the Volterr itegrl equtios, the system is trigulr, so it is reltively stright forwrd to solve the olier system of the coefficiets d good pproimtio of the origil solutio is otied. The sufficiet coditio for the eistece of the solutio is discussed d the covergece rte is ivestigted. 1. Itroductio Trigoometric splies were itroduced y Schoeerg i [1]. Uivrite trigoometric splies re piecewise trigoometric polyomils of the form k cos (α k + k si (α k (where = (1 (where α = < α 1 < α 2 < < α re rel umers i ech itervl d they re ture etesios of polyomil splies. Needless to sy, trigoometric splies hve their ow dvtges. A umer of ppers hve ppered to study the properties of the trigoometric splies d trigoometric B- splies (cf. [2 4] sice the. I my previous ppers (cf. [5 7], low degree orthoorml splie d crdil splie fuctios with smll compct supports were costructed. The method c e eteded to costruct higher degree orthoorml or crdil splies. Ulike i the ook (cf. [1], y the crdil splies we me the specific splies stisfyig crdil iterpoltio coditios, which mes tht the crdil fuctio hs the vlueoetoeiterpoltiopoitdvluezerotllother iterpoltio poits. Crdil splies re ot oly useful i iterpoltio prolems, ut they re lso useful i deductio of umericl itegrtio formuls [6] d i solvig itegrl equtios. Itegrl equtios pper i my fields, icludig dymic systems, mthemticl pplictios i ecoomics, commuictio theory, optimiztio d optiml cotrol systems, iology d popultio growth, cotiuum d qutum mechics, kietic theory of gses, electricity d mgetism, potetil theory, d geophysics. My differetil equtios with oudry vlue c e reformulted s itegrl equtios. There re lso some prolems tht c e epressed oly i terms of itegrl equtios. I this pper we focus o the Volterr itegrl equtios of the secod kid: y ( =g( +λ K (, t f(y(tdt, (,, (2 where λ is comple umer, the kerel K(, t, f(y, d g( re kow fuctios, d y( is ukow fuctio to e determied. Thispperhssisectios.ISectio 2, uivrite trigoometric crdil splie o smll compct support is costructed d properties re studied. I Sectio 3, the pplictios of trigoometric crdil splies o solvig the Volterr itegrl equtios re eplored. The ukow fuctio is epressed s lier comitio of trigoometric

2 2 ISRN Applied Mthemtics crdil splie fuctios. The simple system of olier equtios o the coefficiets is deducted. It is reltively simple to solve the lier system sice the system is trigulr, d good pproimtio of the origil solutio is otied. The sufficiet coditio for the eistece is discussed d the covergece rte is ivestigted. I Sectio 4, the pplictios of trigoometric crdil splies o solvig the systems of Volterr itegrl equtios re eplored. I Sectio 5,umericl emples re give o solvig the olier Volterr itegrl equtios d system of olier Volterr itegrl equtios. Sectio 6 cotisthecoclusioremrks. h y 1 h Figure 1: The grph of T 1,h (. 2. A Crdil Trigoometric Splie with Smll Support To costruct crdil trigoometric splies, let μ h ( = { 1, h { 2 < h 2, {, elsewhere. ThisisthezerodegreepolyomilortrigoometricBsplie. Let T,h ( = μ h (. A cotiuous uivrite crdil trigoometric splie with smll support is T 1,h ( = I 1T ( I 1 T ( = 1 2 si (h/2 h/2 Eplicitly, h/2 μ h (+t cos tdt. 1 2 si (h/2 (si (h 2 si ( h 2, for h, { T 1,h ( = 1 2 si (h/2 (si (h 2 +si ( + h 2, for h, { {, for >hor < h. The grph of T 1,h ( is Figure 1. Propositio 1. If y( C 1 [, ], y ( eists d is ouded o the fiite itervl [, ] (where <, for y [,]d y iteger,suchthth := ( / < 1;let TLy ( = j= (3 (4 (5 y(+jht 1,h ( jh; (6 the y( TLy(( 6h 2 M [,] y (. If y( C 1 (,, y ( eists o (, d oth y ( d y ( re ouded, for y (, d y chose h<1,let TL(y ( = y(jht 1,h ( jh ; (7 j= the y( TLy(( 6h 2 M (, y (. 3. Numericl Method Solvig Itegrl Equtios To solve the Volterr itegrl equtios (2 iitervl (,, weleth = ( /, i = + ih, i =,1,...,. Furthermore, let y ( = f(y( = c k T 1,h ( k, f(c k T 1,h ( k, K (, t = K( i, j T 1,h ( i T 1,h (t j, i= j= g ( = pluggig i (2, we get c k T 1,h ( k = i= j= g( k T 1,h ( k ; λ T 1,h ( i K( i, j T 1,h (t j g( k T 1,h ( k. f(c k T 1,h (t k dt Lettig = k,werrivefork =,1,2,3,4,...,, (where α = T 1,1(T 1,1 (d, β = T 1,1(T 1,1 ( 1d, t g( k =c k λ( α 2 f(c +βf(c 1 hk (, k k 1 (8 (9 λ(βf(c i 1 +αf(c i +βf(c i+1 hk ( i, k λ(βf(c k 1+ α 2 f(c k hk ( k, k, (S1

3 ISRN Applied Mthemtics 3 which is trigulr system of +1olier equtios o ukows c,c 1,...,c. Notice tht the coefficiet mtri for the system is trigulr mtri, which mes tht we solve c i = f(c i +W i,wherew i is umer ot depedig o c i,fori =,1,...,. For the covergece rte of solutio of the Volterr itegrl equtios (2, we hve the followig Propositio 2. Propositio 2. Give tht y(, g( C[, ], y (, d g ( eist d re ouded i [, ], K(, y C[, ] [, ], ( 2 / s y t K(, y (s+t = 2eistdreoudedi[, ] [, ].Furthermore,K(, y stisfies the coditio λ K(,y(y( u(d <LMm y ( u(, [,] (1 where LM < 1. Let e iteger, h = ( /, let i = +ih, y i =y( i, i =,1,2,...,,dc,c 1,...,c stisfies the lier system (S1 = λ +λ K (, t f (t dt λ K (, t f (t dt K (, t f (t dt λ K (, t f (t dt +g( g ( [,] = λ (K (, t K (, tf(t dt +λ K (, t (f (t f (tdt+(g( g ( [,] 2 48 λ m { s+t=2 s y t K(,y } [,] [,] f ( [,] ( h2 the y ( = c i T 1,h ( kh ; (11 Plug i Therefore +LM f ( f ( [,] + g ( g ( [,]. (14 f ( [,] 1 1 LM g ( [,]. (15 y ( y( [,] =O(h2, (12 where y( is the ect solutio of (2. Proof. Let y ( = f(y(t = c k T 1,h ( k, f(c k T 1,h ( k, K (, t = K( i, j T 1,h ( i T 1,h (t j, i= j= g ( = g( k T 1,h ( k, (13 where the coefficiets re the solutios of ove system (S1. The f ( f ( [,] = λ K (, t f (t dt + g ( λ K (, t f (t dt g ( [,] f ( f ( [,] 1 λ ( (48 1 LM 1 LM 2 m { s+t=2 s y t K(,y } [,] [,] g ( [,] +7 g ( [,] h2. (16 4. Numericl Method Solvig Systems of Itegrl Equtios The system of Volterr itegrl equtios is criticl to my physicl, iologicl, d egieerig models. For istce, forsomehettrsferprolemsiphysics,thehetequtios re usully replced y system of Volterr itegrl equtios [8]. My well-kow models for eurl etworks i iomthemtics, ucler rector dymics prolems, d thermoelsticity prolems re lso sed o system of Volterr itegrl equtios ([9, 1]. Our method could e eteded to solve the system of Volterr itegrl equtios. Give m y s ( =g s ( + λ p K p,s (, t f p,s (y p (tdt, p= (17 (,, s=1,2,...,m,

4 4 ISRN Applied Mthemtics i itervl (,, weleth = ( /, i =+ih, i=,1,...,. Furthermore, let (s,p=1,2,...,m y p ( = f p,s (y p ( = c k,p T 1,h ( k, f p,s (c k,p T 1,h ( k, K p,s (, t = K p,s ( i, j T 1,h ( i T 1,h (t j, i= j= g s ( = pluggig i (17, we get c k,s T 1,h ( k = g s ( k T 1,h ( k ; m λ p p= i= j= T 1,h ( i K p,s ( i, j T 1,h (t j f p,s (c k,p T 1,h (t k dt g s ( k T 1,h ( k. (18 (19 Let = k, we rrive for k =,1,2,3,4,...,,(where α= T 1,1(T 1,1 (d, β= T 1,1(T 1,1 ( 1d, t g s ( k m =c k,s λ p ( α 2 f p,s (c,s +βf p,s (c 1,s p= m k 1 λ p p= m hk p,s (, k (βf p,s (c i 1,s +αf p,s (c i,s +βf p,s (c i+1,s hk p,s ( i, k λ p (βf p,s (c k 1,s+ α 2 f p,s (c k,s hk p,s ( k, k, p= (S2 which is trigulr system of ( + 1m olier equtios o ukows {c,s,c 1,s,...,c,s }. Notice tht the coefficiet mtri for the system is trigulr mtri, which mes tht we solve c i,s =f p,s (c i,s +W i,s,wherew i is umer, for i=,1,...,. For the covergece rte of solutio of the Volterr itegrl equtios (2, we hve the followig Propositio 3. Propositio 3. Give tht y p (, g p ( C[, ], y p (,d g p ( eists d is ouded i [, ], K p,s(, y C[, ] [, ], ( 2 / s y t K p,s (, y (s + t = 2 eist d re ouded i [, ] [, ].Furthermore,K p,s (, y stisfies the coditio m p= λ p K p,s (, y (y p ( u p (d m <LMm [,] p= λ p y p ( u p (, (2 where LM < 1. Let e iteger, h = ( /, let i = +ih, y i =y( i, i =,1,2,...,, c,p,c 1,p,...,c,p stisfies the lier system (S1 the y p ( = c i,p T 1,h ( kh; (21 y p ( y p ( [,] =O(h2, (22 where {y p (} is the ect solutio of ( Numericl Emples Emple 1. Give tht y( = g( + K(, tf(y(tdt, where k(, t = t( t/( t +.1 2, f(y = y 2, g( = l cos. Let (, = (, 1, = 1, h =.1, i = ih. y( = 1 c kt 1,h ( k, y 2 (t = 1 c2 k T 1,h( k, K(, t = t( t/( t = 1 i= 1 j= (t( i j /( j T 1,h ( i T 1,h (t j, g( = 1 g( kt 1,h ( k. Pluggig ito the itegrl equtio, we rrive t c =g(, g( k =c k ( α 2 c2 +βc2 1 hk(, k (βc 2 i 1 +αc2 i +βc 2 i+1 hk( i, k (βc 2 + α 2 c2 k hk( k, k, (k=1,2,...,1. (23 The solutio is [c, c 1, c 2, c 3, c 4, c 5, c 6, c 7, c 8, c 9, c 1 ] = [ , , , , , , , , , ]. Compred with the ect solutio [ , , , , , , , , , 1.1]. The error E <.3 = 3h 3. Emple 2. u( = t (1/4 si(2 (1/2+ 1/(1+ 1/2 u 2 (tdt, 1/2 1/2, with the ect solutio u( = t.

5 ISRN Applied Mthemtics 5 We choose =1, h =.1, d k = (1/2+kh,d let u ( = 1 c kt 1,h ( k plug ito (S1.Thesystemhs the form g( k =c k ( α 2 c =g(, 1 β (c (c1 2 +1h β ( (ci α (c 2 i +1 + β ( α 1 (c (ck 2 +1h, (k=1,2,...,1. β (c 2 i+1 +1h (24 The solutio is [c, c 1, c 2, c 3, c 4, c 5, c 6, c 7, c 8, c 9, c 1 ] = [ , , , , , , , , , , ]. Comprig with the ect solutios [u(, u( 1, u( 2, u( 3, u( 4, u( 5, u( 6, u( 7, u( 8, u( 9, u( 1 ] = [ , , , , ,, , , , , ], the error E <.2. Notice tht our ccurcy is much etter th the oe i the pper [11] lthough they choose h =.1 i the pper. Emple 3. The equtio of percoltio i [12] y ( = e A( t (1+( t l A (y (t 1/p dt, (25 where A>1physicl prmeter, costt p>1; ccordig to pper [13], we c oti uique oegtive otrivil solutio. Let (, = (, 1, = 1, h =.1, i = ih. y( = 1 c kt 1,h ( k, K(, t = e A( t (1+( t l A = 1 i= 1 j= ea( i j (1 + ( i j l AT 1,h ( i T 1,h (t j, g( =. LetA = 2, p = 2; pluggig ito the itegrl equtio, we rrive t c =, =c k ( α 2 c1/p +βc 1/p 1 hk(, k (βc 1/p i 1 +αc1/p i +βc 1/p i+1 hk( i, k (βc 1/p + α 2 c1/p k hk( k, k, (k=1,2,...,1. (26 The result we got is [c,c 1,c 2,c 3,c 4,c 5,c 6,c 7,c 8,c 9,c 1 ]= [,.326,.1373,.35223,.71853,.12921, , ,.51886,.75148, ]. We do ot hve the ect solutio. Nevertheless, compre with y( k = k e A(k t (1 + ( k tl A(y(t 1/p dt,[y(, y( 1, y( 2, y( 3, y( 4, y( 5, y( 6, y( 7, y( 8, y( 9, y( 1 ] = [,.311,.13632,.3539,.71478, , ,.3362,.58289, , ]. For the sme itegrl equtio, let A = 2, p = 3; we rrive t [c, c 1, c 2, c 3, c 4, c 5, c 6, c 7, c 8, c 9, c 1 ] = [.1293,.51423,.11623, , , ,.81744, , , ]. Compre with y( k = k e A(k t (1 + ( k t l A(y(t 1/p dt, [y(, y( 1, y( 2, y( 3, y( 4, y( 5, y( 6, y( 7, y( 8, y( 9, y( 1 ] = [.1284,.51156, , ,.3497, , , , , ]. Emple 4. Cosider the equtio y( (1/ 6 si(y(t cos( tdt = (1/6 cos +rcsi (1/6 = g(. The ect solutio is y=rcsi. Applyig the method o the itervl [, 1], leth = 1/1, = 1, i = ih, K(, t = cos( t = 11 i= 1 T 1,h( i 11 j= 1 cos( i t j T 1,h (t j, g( = (1/6 cos +rcsi (1/6 = 11 i= 1 g( it 1,h ( i,dy( = +1 k= 1 c kt 1,h ( k ; pluggig ito the itegrl equtio (2, we rrive t c =g(, g( k =c k λ( α 2 si (c +βsi (c 1 hk (, k λ(βsi (c i 1 +αsi (c i +β si (c i+1 hk ( i, k λ(βsi (c + α 2 si (c k hk ( k, k, (k=1,2,...,1. (27 The solutio is [c,c 1,c 2,c 3,c 4,c 5,c 6,c 7,c 8,c 9,c 1 ] = [ , , , , , , , , , ]. Compred with the ect solutio: [y(, y( 1, y( 2, y( 3, y( 4, y( 5, y( 6, y( 7, y( 8, y( 9, y( 1 ] = [ , , , , , , , , , ]. The error E <.7. Emple 5. Cosider the equtio y( (1/8 (1/ 1 2 +t 2 e y(t dt = g( =l (1/8 2+(1/8 2 +1;the ect solutio is y( = l. Applyig the method o the itervl [1, 2], leth = 1/1, = 1, i = 1 + ih, K(, t = 1/ 2 +t 2 = 11 i= 1 T 1,h( i 11 j= 1 K( i, j T 1,h (t j, g( = l (1/8 2 + (1/ = 11 i= 1 g( it 1,h ( i,d

6 6 ISRN Applied Mthemtics y( = +1 k= 1 c kt 1,h ( k ; pluggig ito the itegrl equtio, we rrive t g ( = si = g( k T 1,h ( k, c =g(, g( k =c k λ( α 2 ec +βe c 1 hk(, k λ(βe c i 1 +αe c i +βe c i+1 hk( i, k λ(βe c + α 2 ec k hk( k, k, (k=1,2,...,1. (28 h ( = cos 1 2 si2 = pluggig ito the system, we get c,1 =, h( k T 1,h ( k ; g( k =c k,1 ( α 2 (c2,1 +c2,2 +β(c2 1,1 +c2 1,2 h (β (c 2 i 1,1 +c2 i 1,2 +α(c2 i,1 +c2 i,2 (3 The solutio is [c, c 1, c 2, c 3, c 4, c 5, c 6, c 7, c 8, c 9, c 1 ] = [ , , , , , , , , , ]. Compred with the ect solutio [y(, y( 1, y( 2, y( 3, y( 4, y( 5, y( 6, y( 7, y( 8, y( 9, y( 1 ]= [ , , , , , , , , , ]. The error E <.1. Emple 6. Cosider system of Volterr itegrl equtios: u ( = si + (u 2 (t + V 2 (tdt, V ( = cos 1 2 si2 + (u (t V (t dt. (29 The ect solutios re u( = si, V( = cos. I itervl (, 1, weleth = 1/1, i = ih, i =,1,...,1. Furthermore, let u ( = V ( = (u( 2 = (V( 2 = c k,1 T 1,h ( k, c k,2 T 1,h ( k, (c k,1 2 T 1,h ( k, (c k,2 2 T 1,h ( k, (u ( V ( = (c k,1 c k,2 T 1,h ( k, K (, t =1= T 1,h ( i T 1,h (t j, i= j= +β(c 2 i+1,1 +c2 i+1,2 h (β(c 2,1 +c2,2 +α 2 (c2 k,1 +c2 k,2 h, c,2 =1, h( k =c k,2 ( α 2 (c,1c,2 +β(c 1,1 c 1,2 h (k=1,2,...,1, (β (c i 1,1 c i 1,2 +α(c i,1 c i,2 +β(c i+1,1 c i+1,2 h (β(c,1 c,2 + α 2 (c k,1c k,2 h, (k=1,2,...,1, (31 which is erly trigulr system of ( olier equtios o ukows {[c,1,c 1,1,...,c,1 ], [c,2,c 1,2,...,c,2 ]}; we eed to solve two olier equtios: c i,1 = f 1 (c i,1,c i,2 +W i,1, c i,2 = f 2 (c i,1,c i,2 +W i,2,wherew i,1,w i,1 re umers, ech time for i=,1,...,. Solutios re {[c,1,c 1,1,...,c,1 ], [c,2,c 1,2,...,c,2 ]} = {[ , , , , , , , , , ], [ , , , , , , , , , ]}. Compred with the ect solutio: {[, , , , , , , , , , ], [1, , , , , , , , , , ]}. The error < Coclusios The proposed method is simple d effective procedure for solvig olier Volterr itegrl equtios, s well s

7 ISRN Applied Mthemtics 7 system of olier Volterr itegrl equtios. The methods c e dpted esily to the Volterr itegrl equtios of the first kid, which hve the form g( = A K(, ty(tdt. The covergece rte could e higher if we use more complicted orthoorml or crdil splies. Nevertheless, the resultig system of coefficiets will e more complicted olier systems, which could tke more time d effort to solve. [13] D. Wei, Uiqueess of solutios for clss of o-lier volterr itegrl equtios without cotiuity, Applied Mthemtics d Mechics,vol.18,o.12,pp ,1997. Coflict of Iterests The uthors declre tht there is o coflict of iterests regrdig the pulictio of this pper. Ackowledgmets The work ws fuded y the Nturl Sciece Foudtio ofahuiproviceofchiudergrto.12885ma15 d the Mjor Project of the Nture Sciece Foudtio of the Eductio Deprtmet, Ahui Provice, uder Grt o. KJ214ZD3. Refereces [1] I. J. Schoeerg, O trigoometric splie iterpoltio, Jourl of Mthemtics d Mechics,vol.13,pp ,1964. [2] T.Lyche,L.L.Schumker,dS.Stley, Qusi-iterpolts sed o trigoometric splies, Jourl of Approimtio Theory, vol. 95, o. 2, pp , [3]T.LychedR.Wither, Astlerecurrecereltiofor trigoometric B-splies, Jourl of Approimtio Theory, vol. 25,o.3,pp ,1979. [4] A. Shrm d J. Tzimlrio, A clss of crdil trigoometric splies, SIAM Jourl o Mthemticl Alysis, vol. 7, o. 6, pp ,1976. [5] X. Liu, Bivrite crdil splie fuctios for digitl sigl processig, i Treds i Approimtio Theory, K. Kopotum, T. Lyche,dM.Nemtu,Eds.,pp ,VderiltUiversity, Nshville, Te, USA, 21. [6] X. Liu, Uivrite d ivrite orthoorml splies d crdil splies o compct supports, Jourl of Computtiol d Applied Mthemtics, vol. 195, o. 1-2, pp , 26. [7] X. Liu, Iterpoltio y crdil trigoometric splies, Itertiol Jourl of Pure d Applied Mthemtics,vol.4,o.1, pp , 27. [8] R. Kress, Lier Itegrl Equtios,vol.82ofApplied Mthemticl Scieces, Spriger, Berli, Germy, [9] B. L. Moiseiwitsch, Itegrl Equtios, Dover,NewYork,NY, USA, 25. [1] A. D. Polyi, Hdook of Itegrl Equtios, CRCPress, Boc Rto, Fl, USA, [11] A. Vhidi Kmyd, M. Mehriezhd, d J. Seri-Ndjfi, A umericl pproch for solvig lier d olier volterr itegrl equtios with cotrolled error, IAENG Itertiol Jourl of Applied Mthemtics,vol.4,o.2,pp.69 75,21. [12] J. Gocerzewicz, H. Mrcikowsk, W. Okrsiński, d K. Tisz, O the percoltio of wter from cylidricl reservoir ito the surroudig soil, Zstosowi Mtemtyki, vol. 16, o. 2, pp , 1978.

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