Oscillatory Quadratures Based on the Tau Method

Transcription

1 Journl of Mthemtics Reserch; Vol. 5, No. 3; 213 ISSN E-ISSN Published by Cndin Center of Science nd Eduction Oscilltory Qudrtures Bsed on the Tu Method Mohmed K. El-Dou 1 1 Applied Sciences Deprtment, College of Technologicl Studies, Kuwit Correspondence: Mohmed K. El-Dou, P.O. Box Shuwikh/B, 7453, Kuwit. E-mil: mk.eldou@pet.edu.kw Received: April 24, 213 Accepted: July 18, 213 Online Published: August 8, 213 doi:1.5539/jmr.v5n3p56 URL: Abstrct In this pper we discuss the integrtion of highly oscilltory univrite nd multivrite functions. Bsed on the recursive formultion of the Tu method we develop numericl qudrtures tht chieve high degree of ccurcy when the frequency in the integrnd tkes moderte s well s very lrge vlues. With our procedures the integrl is obtined in terms of the vlue of the function nd its derivtives t the boundry points. The ccurcy of our results re confirmed through numericl exmples. Keywords: oscilltory integrl, colloction, Tu method 1. Introduction Consider the integrl f (x)e iωg(x) dx, (1) where f (x) nd g(x) re smooth functions in n intervl [, b] nd ω is positive rel number. By tking the rel nd the imginry prts of (1) we obtin integrls with trigonometric kernels: Re(I) = f (x) cos(ωg(x))dx, Im(I) = f (x) sin(ωg(x))dx. The most immedite cndidte for numericlly pproximting integrl (1) might be the stndrd Guss-Christoffel qudrture (Dvis & Rbinowitz, 198), where we interpolte the integrnd t distinct nodes c 1 < c 2 <... < c ν in [, b] by polynomil p(x) of prescribed degree ν 1, nd pproximte I[ω] p(x)dx. (2) When the frequency ω>>1is lrge, the integrnd in (1) oscilltes very rpidly nd I[ω] is clled highly oscilltory. Integrls of this form rise in wide rnge of science nd engineering such s quntum chemistry, imge nlysis, coustics, electrodynmics, computerized tomogrphy nd fluid mechnics etc. The evlution of highly oscilltory integrls ws considered s chllenging problem in the numericl nlysis nd computtionl physics. Unfortuntely, if ω >> 1 the ccurcy of pproximtion (2) obtined by the stndrd qudrture deteriortes rpidly due to the presence of shrp vritions throughout [, b]. The reson behind this deteriortion lies in tht such methods fil to detect the shrp oscilltions exhibited by the integrnd unless the degree of its interpolnt p(x) grows with the frequency. The following numericl experiment shows tht while the exct vlue of the highly oscilltory integrl decys like O(ω 1 ), the Guss-Christoffel qudrture produces pproximte results of O(1). Consider the integrl I [ω] = (3x 2 + 2x + 1)e iω(x3 +x 2 +x) dx. (3) We pproximte I [ω] for frequencies 1 ω 6 using the 2-point Guss-Christoffel qudrture; the results re plotted in Figure 1-right. On the other hnd, using integrtion by prts we find the exct vlue of I [ω], I [ω] = i( 1 + ( ) e3iω ) 1 = O, ω ω 56

2 which is shown in Figure 1-left for the sme frequencies. It is clerly seen from Figure 1 tht, when ω is lrge, the pproximte vlues of I [ω] re of O(1) wheres the exct vlues of I [ω] behves s O(ω 1 ) Figure 1. [left] Plot of (ω, I [ω]) where I [ω] given by (3) is obtined by integrtion by prts. [right] Plot of (ω, Ĩ [ω]) where Ĩ[ω] I [ω] using the 2-point Guss-Christoffel qudrture This test suggests tht ny numericl method of prcticl vlue must be t lest of order O(ω 1 ). The first known numericl qudrture with this order ws developed in Filon (1928). Therein, Filon presented n efficient method for computing integrls of the form (1) with g(x) = x. His pproch consists of dividing the intervl into 2n subintervls of size h, nd then f (x) is interpolted t the endpoints nd midpoint of ech subintervl by qudrtic function. In ech subintervl the integrl becomes polynomil multiplied by the oscilltory kernel sin ωx, which cn be integrted in closed form. This method ws generlized in Luke (1954) by using higher degree polynomils in ech pnel, gin with evenly spced nodes. The computtion of the Filon pproximtion rests on the bility to compute the moments x k e iωx dx. In order to chieve higher ccurcy, Iserles nd Nørsett (25) suggested the pproximtion of f (x) by its Hermite interpolnt by choosing sequence of nodes {x 1, x 2,..., x ν } ssocited with sequence of multiplicities {m k ; k = 1,..., ν}, where x 1 = nd x ν = b. With this choice, nd in the bsence of sttionry points, (ξ is sttionry with order r if g ( j) (ξ) = for j =, 1, 2,..., r but g (r+1) (ξ) ), the error is of O(ω s 1 ) where s: = min{m 1, m ν }. Another efficient method to pproximte (1) tht ws described in Iserles nd Nørsett (25) is the truncted symptotic expnsion of I[ω]: I[ω] j ω j+1, where the coefficients j depend on the function nd the derivtive vlues of f (x) nd g(x) t the points nd b (Stein, 1993). The unknown coefficients { j } depend on the moments xk e iωg(x) dx, nd therefore they cn be obtined explicitly if the moments re known. Other numericl methods for pproximting (1) include qudrtures bsed on the nlytic continution nd numericl steepest descent method (Huybrechs & Vndewlle, 26) nd the exponentilly fitted qudrtures (Ixru & Vnden Berghe, 24; Vn Dele, Vnden Berghe, & Vnden Vyver, 25) where the weights nd the nodes of the qudrtures depend on the frequency of the problem. Mny of the bove mentioned methods, in prticulr the uniform symptotic expnsion nd the Filon pproximtion, requires n exct computtion of the moments. For the prticulr oscilltor g(x) = x, these moments cn be obtined exctly, either through integrtion by prts or by using the incomplete Gmm function Γ (Abrmowitz & Stegun, 1965). However, for irregulr oscilltors g(x) the vlues of the the moments x k e iωg(x) dx re not necessrily computble in closed form nd therefore the Filon-type nd the symptotic methods re not pplicble. Thus it is necessry to find lterntive techniques tht re free of the moments. 57

3 The first numericl method tht pproximtes highly oscilltory integrls without using moments ws developed in Levin (1996). Levin s pproch consists in finding function F(x) such tht d [ ] F(x)e iωg(x) = f (x)e iωg(x). dx Expnding out the derivtives, we find tht F(x) stisfies the non oscilltory differentil eqution So, once F(x) is obtined we immeditely gin (DF)(x) := F (x) + iωg (x)f(x) = f (x). (4) f (t)e iωg(t) dt = F(b)e iωg(b) F()e iωg(). (5) Thus, the problem of evluting the definite integrl I[ω] turns out to be question of pproximting the solution F(x) of the differentil Eqution (4). In the bsence of the sttionry points, F(x) is smooth nd, ccording to Levin (1996), Eqution (4) cn be pproximted efficiently by colloction. The ltter belongs to clss of powerful techniques tht use globl pproch clled spectrl methods. In this pper we will consider solving (4) by nother spectrl technique clled the Tu method (see Ortiz, 1969). Unlike the colloction method, the recursive nture of the Tu method permits to express the pproximte vlue in qudrture form, nd this feture llows us to derive integrtion procedures competitive with the present techniques. This pper is orgnized s follows: In Section 2 we recll the min fetures of the Tu method nd we describe how to construct the Tu pproximnt in terms of specil polynomil bsis clled cnonicl polynomils. Section 3 is devoted to present Tu-bsed qudrtures tht estimte univrite oscilltory integrls. The cse of multivrite integrls is discussed in Section 4. Numericl exmples confirming our results re provided in the lst section. 2. Integrtion with the Tu Method The bsic ide of the Tu method is to perturb the right hnd side of Eqution (4) in wy tht the resulting perturbed eqution cn be solved nlyticlly. More precisely, we introduce in the right hnd side of (4) perturbtion term H N (x) such tht the exct solution F N (x) of the eqution (DF N )(x) := F N (x) + iωg (x)f N (x) = f (x) + H N (x), (6) cn be obtined in closed form. Here H N (x) is polynomil of degree depending on prescribed N 1 nd whose the coefficients re djusted in wy tht F N (x) is found nlyticlly. Usully H N (x) is chosen s liner combintion of the Chebyshev or Legendre polynomils. This is due to the fct tht the equioscilltory behvior of those polynomils leds to uniform distribution of the error throughout the intervl of integrtion (see Ortiz, 1969). But in our context, since the min contribution to the vlue of I comes from the re of the portions neighboring the end points of [, b] (s explined in Huybrechs & Vndewlle, 26), it is convenient to dopt choice for H N (x) tht forces the differentil Eqution (6) nd its first N 1 derivtives to be exct t x = nd x = b. Of the forms tht enjoys this feture is r H N (x) = τ j (x ) j (x )N (x b) N, (7) where {τ j, j =, 1, 2,..., r} re free unknown prmeters tht re computed simultneously with F N (x) nd r is d fixed s explined lter. Clerly (6) nd (7) imply tht k [(DF dx k N )(x) f (x)] x {,b} = for k =, 1, 2,..., N 1s desired. For simplicity, let us tke = nd b = 1. Then (7) becomes r r N H N (x) = τ j x j xn (x 1) N = τ j CN i xn+i+ j ; Ci N = N! i!(n i)!. (8) The Tu method procedure consists of expressing F N (x) in terms of specil polynomils bsis {Q k (x); k } clled cnonicl polynomils bsis. These re defined s follows: For ech k N, let Q k be the exct solution of the differentil eqution (D Q k )(x) = x k. (9) 58 i=

4 From Ortiz (1969), when the coefficients of the differentil opertor D re polynomils, the set of functions { Q k (x); k } re generted by self strting recursive formul. So let us consider first the cse of f (x) nd g (x) being polynomils of degree μ nd ν respectively, f (x) = μ β j x j nd g (x) = ν α j x j. Then, for ll k N, ν Dx k = kx k 1 + iω α j x j+k = kd Q k 1 + iω α j D Q j+k + iωα ν x ν+k, nd since D is liner, we hve Compring the ltter with (9) we find tht D[x k k Q k 1 iω α j Q j+k ] = iωα ν x ν+k. Q ν+k (x) = 1 iωα ν xk k Q k 1 (x) iω α j Q j+k (x), k N. With this formul we cn generte ll the Q k s except possibly { Q, Q 1,..., Q ν 1 }. For exmple, if k = we get ( ) 1 α j Q ν (x) = Q j (x). iωα ν α ν Therefore, ech Q k (x) cn be represented s Q k (x) = Q k (x) + ρ k, j Q j (x), (1) where {Q k ; k N} re clled cnonicl polynomils nd generted by the recursion Q k (x) =, k =, 1, 2,..., ν 1, Q ν+k (x) = 1 iωα ν xk kq k 1 (x) iω α j Q j+k (x), k N. (11) nd {ρ k,l ; k N} re sequences of complex numbers defined s ρ k,l = δ l k (Kronecker s symbol) for k =, 1, 2,..., ν 1 nd l =, 1, 2,..., ν 1 ρ ν+k,l = 1 iωα ν kρ k 1,l iω α j ρ j+k,l, k N nd l =, 1, 2,..., ν 1. (12) Now, since D is liner, the exct solution of (6) cn be formlly expressed in terms of { Q k } s follows: μ r N F N (x) = β j Q j (x) + τ j C i Q N N+i+ j (x) i= μ r N = β j [Q j (x) + ρ j,l Q l (x)] + τ j CN i [Q N+i+ j(x) + ρ N+i+ j,l Q l (x)] l= i= l= μ r N = β j Q j (x) + τ j CN i Q N+i+ j(x) (13) l= μ + β j ρ j,l + i= r N τ j CN i ρ N+i+ j,l Q l (x). (14) i= 59

5 The {τ k ; k =, 1,..., r} re found then by equting the coefficients of { Q l (x); l =, 1, 2,..., ν 1} in (14) to, r N μ τ j CN i ρ N+i+ j,l = β j ρ j,l ; l =, 1, 2,..., ν 1. (15) i= Now if we choose r = ν 1, Eqution (15) becomes squre lgebric system consisting of ν equtions with ν unknowns {τ k ; k =, 1,..., ν 1}, nd hence the Tu method pproximtion F N (x) will be given by (13) F N (x) = μ N β j Q j (x) + τ j CN i Q N+i+ j(x), (16) which is polynomil of degree 2N Construction of Algorithms Once the pproximte solution F N (x) tof(x) is obtined s in (16), we go bck to the integrl (5), nd write i= f (t)e iωg(t) dt F N (1)e iωg(1) F N ()e iωg() I N [ω]. (17) We cn summrize now ll the necessry steps required to compute I N [ω]. 3.1 Algorithm TQ(N) 1) Store {α i } ν i= nd {β i} μ i=. 2) Form {Q k (x)} N k= using (11). 3) Form {ρ k,l ; k =, 1, 2,..., N; l =, 1, 2,..., ν 1} using (12). 4) Form A l, j = N i= Ci N ρ N+i+ j,l, for l, j =, 1, 2,..., ν 1. 5) Form B l = μ β jρ j,l, for l =, 1, 2,..., ν 1. 6) Compute {τ j } ν 1 through solving system (15): ν 1 A l, jτ j = B l ; l =, 1, 2,..., ν 1. 7) Compute F N (1) nd F N () using (16). 8) Compute I N [ω] = F N (1)e iωg(1) F N ()e iωg(). If f (x) nd g (x) re not polynomils, then, due to their smoothness, they cn be pproximted by polynomils with high degrees of ccurcy. In order to be consistent with the form of H N (x) given in (7) we dopt the truncted two-point Tylor series expnsions (see Dvis, 1975): N 1 P 2N 1 (x) = (x ) N k= B k N 1 k! (x b)k + (x b) N where [ ] [ ] A k = dk G(x) dx k (x b) N nd B k = dk G(x) x= dx k (x ) N with G(x) stnding for g (x) or f (x). In terms of {x k ; k N} we cn write g (x) 2N 1 α j x j nd f (x) 2N 1 k= A k k! (x )k, (18), x=b β j x j. (19) Thus we pply Algorithm TQ(N) with μ = ν = 2N 1. For the specil cses where N = 1 nd N = 2, Algorithm TQ(N) tkes the form of qudrtures s explined next: 3.2 Tu Qudrture TQ(1) Let N = 1, = nd b = 1. Then μ = ν = 2N 1 = 1 nd (18) reduces to P 1 (x) = xg(1) (x 1)G() 6

6 which gives, in ccordnce with (19), g (x) (g (1) g ())x + g () α 1 x + α when G = g, f (x) ( f (1) f ())x + f () β 1 x + β when G = f, where α = g (), α 1 = g (1) g (), β = f (), β 1 = f (1) f (). The recursions (11)-(12), in turn, become In prticulr, Q k+1 (x) = ρ k+1, = 1 [ ] x k kq k 1 iωα Q k for k with Q =, iωα 1 1 [ ] kρk 1, iωα ρ k, for k with ρ, = 1. iωα 1 Q 1 (x) = 1, ρ 1, = α, iωα 1 α 1 Q 2 (x) = 1 [ x α ], ρ 2, = 1 [ ] 2 α +. iωα 1 α 1 iωα 1 α 1 In this cse we hve r: = ν 1 = nd therefore the Tu perturbtion must contin one free prmeter τ only, H 1 (x) = τ x(x 1) = τ (x 2 x), nd the liner system (15) reduces to single eqution with one unknown τ, β ρ + β 1 ρ 1 + τ (ρ 2 ρ 1 ) =, which gives τ = β ρ + β 1 ρ 1. ρ 1 ρ 2 Substituting τ in F 1 (x) given by (16) we rrive to [ 1 f (1) f () iω f ()g (1) F 1 (x) = β Q + β 1 Q 1 + τ (Q 2 Q 1 ) = iω g (1) g () iωg ()g (1) Thus the integrl I[ω] cn be pproximted by the qudrture I[ω] := ] [ + f ()g (1) f (1)g () g (1) g () iωg ()g (1) f (x)e iωg(x) dx F 1 (1)e iωg(1) F 1 ()e iωg() = ψ 1 e iωg(1) ψ e iωg() I 1 [ω] (2) where ψ = 1 f (1) f () iω f ()g (1) iω g (1) g () iωg ()g (1) nd ψ 1 = 1 f (1) f () iω f (1)g () iω g (1) g () iωg ()g (1), (21) provided tht the denomintor in (21) is nonzero. It is clerly seen tht qudrture TQ(1) expresses the pproximte integrl I 1 [ω] in terms of the vlues of { f (x), g (x)} evluted t the boundries of the intervl [, 1]. It is lso importnt to point out tht ψ nd ψ 1 s well s I 1 [ω] re of O(ω 1 ) unless g () = org (1) =. 3.3 Tu Qudrture TQ(2) The construction of qudrture for N = 2 proceeds in the sme mnner s bove. Tking N = 2, = nd b = 1, then μ = ν = 3 nd (18)-(19) give f (x) (x 1) 2 (c + c 1 x) + x 2 (d + d 1 (x 1)) g (x) (x 1) 2 ( + 1 x) + x 2 (b + b 1 (x 1)) 61 3 α j x j, 3 β j x j, ] x.

7 where or = g (), b = g (1), 1 = 2g () + g (), b 1 = 2g (1) + g (1), c = f (), d = f (1), c 1 = 2 f () + f (), d 1 = 2 f (1) + f (1), α = g (), α 1 = g (), α 2 = 3g () + 3g (1) 2g () g (1), α 3 = 2g () 2g (1) + g () + g (1), β = f (), β 1 = f (), β 2 = 3 f () + 3 f (1) 2 f () f (1), β 3 = 2 f () 2 f (1) + f () + f (1). Since r: = ν 1 = 3 1 = 2, we need three free prmeters {τ,τ 1,τ 2 }, 2 H 3 (x) = τ j x j x3 (x 1) 3, which re found by solving the 3 3 liner system resulting from (15). Substituting {τ,τ 1,τ 2 } in F 3 (x) given by (16), we compute F 3 (1) nd F 3 () nd rrive to where I[ω] := f (x)e iωg(x) dx 1 D ( ψ1 e iωg(1) ψ e iωg()) I 2 [ω], (22) ψ = 6(c 1 + d 1 ) + (6 c + 2b 1 c 2b c 1 + b 1 c 1 + 6b d + 2b 1 d 2b d 1 )iω +(2b 2 c 2 b c + b 1 c + b 2 c 1)ω 2 b 2 c iω 3, ψ 1 = 6(c 1 + d 1 ) + (6 c 2 1 c + 2 c d + 6b d + 2 d d 1 )iω +(2 b d 2 2 d + 1 b d + 2 d 1)ω 2 2 b d iω 3, D = 6( 1 + b 1 )ω + (2 1 b 6 2 6b2 2 b 1 1 b 1 )ω 2 +(2 b 2 22 b + 1 b b 1)iω b2 ω4, provided tht D. Agin qudrture TQ(2) expresses I 2 [ω] in terms of { f (i) (), g (i) (1); i =, 1, 2}. Here I 2 [ω] = O(ω 1 ) when { f (i) (), g (i) (1); i = 1, 2} re ll nonzero. Finlly we point out tht in order to pply our results to n rbitrry intervl [, b], f (t)e iω g(t) dt, we simply shift [, b]to[, 1] with x = t b nd set f (t)e iω g(t) dt = (b ) f ((b )x + )e iω g((b )x+) dx = (b ) f (x)e iωg(x) dx, where f (x) = f ((b )x + ) nd g(x) = g((b )x + ). 4. Multivrite Oscilltory Functions In this section we consider the integrl f (x)e iωg(x) dx, (23) Ω where f (x) nd g(x) re smooth functions in domin Ω R s = R R... R,(s 1), dx = dx 1 dx 2...dx s nd ω>>1is lrge rel number. The integrtion procedure presented in the previous section cn be redily extended to multivrite highly oscilltory integrls of the form (1) with s 2, see (Olver, 26) nd (Iserles & Nørsett, 26). To this end we need some nottions: For ll k = 1, 2,..., s set x k := (x k, x k+1,...,x s ), 62

8 nd let u k nd v k be functions of x k+1 : u k = u k (x k+1 ) nd v k = v k (x k+1 ), k = 1, 2,..., s 1, with u s nd v s b (constnts), nd Then (23) tkes the explicit form dx k := dx k dx k+1...dx s. vs vs 1 u s u s u s 1... v1 u 1 f (x 1, x 2,...,x s )dx 1 dx 2...dx s = vs vs 1 v1 In order to pproximte I we split it into set of s univrite integrls s follows:... f (x 1 )dx 1. u s 1 u 1 f ( 1) (x 2 ) = f ( 2) (x 3 ) = f ( k) (x k+1 ) = f ( s+1) (x s ) = v1 (x 2 ) u 1 (x 2 ) v2 (x 3 ). u 2 (x 3 ) vk (x k+1 ). u k (x k+1 ) vs 1 (x s ) u s 1 (x s ) vs f (x 1 )dx 1, f ( 1) (x 2 )dx 2, f ( k+1) (x k )dx k, f ( s+2) (x s 1 )dx s 1, u s f ( s+1) (x s )dx s. Now pplying the lgorithm TQ(N), developed in the previous section, to the highly oscilltory integrl f ( 1) (x) we obtin new integrl pproximting f ( 2) (x) with one dimension less. We cn thus iterte the procedure on ech dimension nd obtin { f ( 3) (x), f ( 4) (x),..., f ( s+1) (x)} eventully rriving t the univrite integrl I[ω] tht cn be pproximted using the sme procedure. 5. Numericl Exmples This section is devoted to illustrte the ccurcy of our results through set of exmples. Exmple 1 Let us evlute the definite integrl (x 3 + 4x) e iω(x3 +x 2 +x) dx. (24) The oscilltory behvior of (x 3 + 4x) cos ω(x 3 + x 2 + x), the rel prt of the integrnd, is shown in Figure 2 for ω = 8. We pproximted I[ω] byi N [ω] using Algorithm TQ(N) with N = 1, 2, 3, 4, 5 for different vlues of frequencies ω. The errors committed by TQ(N), E N [ω] := I[ω] I N [ω], re listed in Tble 1 for some selected vlues of the frequency ω. We hve lso plotted in Figure 2 the scled error ω N+1 E N [ω] for N = 1 nd N = 2. One cn esily notice tht Algorithm TQ(N) chieves the symptotic ccurcy s in (Levin, 1996): ω N+1 E N [ω] = O(1). In other words, symptoticlly we hve E N [ω] = O(ω N 1 ). 63

9 Figure 2 (Exmple 1). [left] Plot of (x 3 + 4x) cos 8(x 3 + x 2 + x). [right] Plot of the scled errors ω 2 E 1 [ω] (up) nd ω 3 E 2 [ω] (down) where 1 ω 6 Tble 1 (Exmple 1). Integrl (24) is pproximted by TQ(N), N = 1, 2, 3, 4, 5. List of E N [ω] for 1 ω 6 ω\n E E-3 5.E-4 2.2E E E E-4 6.8E E E E E E E-6 3.8E E E E E E E E E E E E-4 4.9E E E-8 8.5E E E E-8 1.8E E E E E E E E E E E-1 4.6E E E-7 2.3E E E E E E E E E-5 7.5E-8 7.3E E E E-5 5.3E E E E E E E E E E E-8 2.4E E E E-6 2.3E E E E-15 Exmple 2 In our second integrl 1 +x+1) 1/3 dx, (25) x + 1 eiω(x2 f (x) = 1 x + 1 nd g(x) = (x2 + x + 1) 1/3 re not polynomils. So in order to pply Algorithm TQ(N), we replce, ccording to (18)-(19), f (x) nd g (x) by f (x) nd g (x) tht represent their respective Nth truncted two-point Tylor series expnsions t x = nd x = 1. We computed I N [ω] by mens of Algorithm TQ(N) with N = 1, 2, 3, 4, 5 nd different vlues of frequencies ω. The errors E N [ω] re displyed in Tble 2 for some vlues of ω nd Figure 3-right shows the scled error ω 2 E 1 [ω]. Agin, symptoticlly we hve E 1 [ω] = O(ω 2 ). The clcultions were repeted for lrge frequencies ω where 4 ω 16: the bsolute errors re summrized in Tble 3 nd plot of the scled error ω 2 E 1 [ω] for ω = 4, 42, 44,..., 16 is shown in Figure 3-left. The ltter conforms tht our results conserve the symptotic property even when very lrge frequencies re considered. 64

10 Figure 3 (Exmple 2). Plot of the scled error ω 2 E 1 [ω] for 4 ω 8 [left] nd for ω = 4, 42, 44,..., 16 [right] Tble 2 (Exmple 2). Integrl (25) is pproximted by TQ(N), N = 1, 2, 3, 4, 5. List of E N [ω] for 4 ω 1 ω\n E E E E E E E E E E E E E E E E E-5 6.7E E-8 4.4E E E E E E E E-6 1.5E E E E E E E E E E E E E E E E E-1 2.9E E E E E E E E-7 1.2E E-1 9.E E E E E E E E-7 4.9E E E E-5 2.7E-7 4.5E-9 8.8E E E E E E E E E E E E E E E E E-13 Tble 3 (Exmple 2). Integrl (25) is pproximted by TQ(N), N = 1, 2, 3, 4, 5. List of E N [ω] for 4 ω 16 ω\n E E E E E E E E E E E E E-12 4.E E E E E E E E E E E E E E-1 2.5E E E E E E E E E-7 1.1E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-2 65

11 Exmple 3 We consider now the multivrite oscilltory integrl y 2 (e xy + y 2 + 1)e iω(cos( y 2 +x) +1)+x2 dxdy. (26) As explined in section 4, we pply TQ(N) twice: First we integrte with respect to x nd we get new integrnd s function of y. Repeting the sme lgorithm to this new function gives n pproximte vlue for the required integrl I N [ω]. Tble 4 shows the bsolute vlue of the error E N [ω] for some vlues of ω. Figure 4 shows the scled error ω N+2 E N [ω] for N = 1 (down) nd N = 2 (up). Here 5 ω 1. Exmple 4 Our lst exmple is integrted on the unit semi-disk, 1 1 y 2 1 y 2 cos ye iω(x2 +y2 +3x+4y) dxdy. (27) Agin, s in the previous exmple, we pply the lgorithm presented in Section 4 for multivrite integrls. The exct vlues of E N [ω] where 5 ω 2 nd N = 1, 2 re recorded in Tble 4-right. In Figure 4-right we drw the scled error ω N+2 E N [ω] where N = 1 nd N = Figure 4. [left] (Exmple 3). Plot of the scled error ω 3 E 1 [ω] (up) nd ω 4 E 2 [ω] (down) where 5 ω 1. [right] (Exmple 4). Plot of the scled error ω 3 E 1 [ω] (up) nd ω 4 E 2 [ω] (down) where 1 ω 2 66

12 Tble 4 (Exmple 3). [left] Integrl (26) is pproximted by TQ(N), N = 1, 2. List of E N [ω] for 1 ω 1 4. [right] (Exmple 4). Integrl (27) is pproximted by TQ(N), N = 1, 2. List of E N [ω] for 5 ω 2 ω\n E E E E E E E E E E E-9 1.7E E E E E E E E E E-1 1.7E E E E-1 5.2E E E E E E E E E E E E E E E E E-14 ω\n E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Conclusion We hve presented n pproch to evlute highly oscilltory integrls of the form (1) nd (23). The method pplies to integrls involving moderte frequencies of different sizes nd the ccurcy of the method improves when the frequency increses. With our procedures the integrl is obtined in terms of the vlue of the function nd its derivtives t the boundry points. The order of our method is O(ω N 1 ) for univrite nd O(ω N d ) if the domin of the integrl is N dimensionl. Acknowledgments The uthor pprecites finncil support from the Public Authority for Applied Eduction nd Trining in Kuwit. References Abrmowitz, M., & Stegun, I. A. (1965). Hndbook of mthemticl functions with formuls, grphs, nd mthemticl tbles. New York, NY: Dover Publictions. Dvis, P. J. (1975). Interpoltion nd Approximtion. New York, NY: Dover Publictions. Dvis, P. J., & Rbinowitz, P. (198). Methods of numericl integrtion (2nd ed.). Orlndo, FL: Acdemic Press. Filon, L. N. G. (1928). On the qudrture formul for trigonometric integrls. Proc. Roy. Soc. Edinburgh, 49, Huybrechs, D., & Vndewlle, S. (26). On the evlution of highly oscilltory integrls by nlytic continution. SIAM J. Numer. Anl., 44, Iserles, A. (24). On the method of Neumnn series for highly oscilltory equtions. BIT, 44, Iserles, A., & Nørsett, S. P. (25). Efficient qudrture of highly oscilltory integrls using derivtives. Proc. R. Soc. A, 461, Iserles, A., & Nørsett, S. P. (26). Qudrture methods for multivrite highly oscilltory integrls using derivtives. Mth. Comp., 75, Ixru, L. Gr., & Vnden Berghe, G. (24). Exponentil Fitting. Dordrecht: Kluwer Acdemic Publishers. 67

13 Levin, D. (1996). Fst integrtion of rpidly oscilltory functions. J. Comput. App. Mth., 67, Luke, Y. L. (1954). On the computtion of oscilltory integrls. Proc. Cmbridge Phil. Soc., 5, Olver, S. (26). On the qudrture of multivrite highly oscilltory integrls over non-polytope domins. Numer. Mth., 4(13), Ortiz, E. L. (1969). The Tu Method. SIAM J. Numer. Anl., 6, Stein, S. (1993). Hrmonic Anlysis: Rel-Vlued Methods, Orthogonlity nd Oscilltory Integrls. Princeton, NJ: Princeton University Press. Vn Dele, M., Vnden Berghe, G., & Vnde Vyver, H. (25). Exponentilly fitted qudrture rules of Guss type for oscilltory integrnds. Applied Numericl Mthemtics, 53(2-4), Copyrights Copyright for this rticle is retined by the uthor(s), with first publiction rights grnted to the journl. This is n open-ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution license ( 68