Applied Mthemticl Sciences, Vol. 8, 214, no. 31, 1535-1542 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/1.12988/ms.214.43166 Efficient Computtion of Clss of Singulr Oscilltory Integrls by Steepest Descent Method A. Ihsn Hsçelik Gzintep University, Deprtment of Mthemtics 273 Gzintep, Turkey Diln Kılıç Gzintep University, Deprtment of Mthemtics Gzintep, Turkey Copyright c 214 A. Ihsn Hsçelik nd Diln Kılıç. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct As qudrture rule the numericl steepest descent method is very efficient for computing highly oscilltory integrls with nlytic integrnds. In this pper we show tht the method cn lso be pplied to compute oscilltory singulr integrls of the form I[f] = c f(x)(x ) α x b β (c x) δ e iωx dx, (α, β, δ > 1, <b<c) efficiently, where ω 1 nd f is nlytic in sufficiently lrge region contining the intervl of integrtion. We lso present computer code in Mthemtic to compute this type of integrls efficiently. Mthemtics Subject Clssifiction: 65D3,65D32 Keywords: singulr oscilltory integrls, steepest descent method
1536 A. Ihsn Hsçelik nd Diln Kılıç 1 Introduction Consider singulr oscilltory integrl of the form I[f] = c f(x)(x ) α x b β (c x) δ e iωx dx, α, β, δ > 1, ω 1 (1) where <<b<c< nd f(z) is of exponentil order s I(z), non-oscilltory, nd nlytic function on the strip B = {z C : R(z) c, I(z) }. Here f(z) is of exponentil order s I(z) mens tht there exist positive constnts m nd M such tht f(x+iy) Me m y holds for ll y. This type of integrls rises in mny pplictions in engineering, nd pplied sciences. For α = β = δ = the integrl (1) becomes n ordinry Fourier integrl nd therefore cn be efficiently pproximted by one of the methods for highly oscilltory integrls such s Levin-type method, Filon-type method, the symptotic method, or the numericl steepest descent method. For these nd other methods for oscilltory integrls, see, e.g., [1, 2, 4, 5, 6, 7, 8, 9] nd the references therein. If t lest one of the prmeters {α, β, δ} is negtive, neither Levin-type method nor Filon-type method is suitble for the evlution of this integrl. Let ψ(x) =(x ) α x b β (c x) δ. (2) If x [, c] nd the prmeters α, β, δ re negtive, ψ is rel vlued function defined on [, c]\{, b, c}. A modifiction of the steepest descent method [7] proposed by R. Chen in [1] cn not be pplied to the integrl (1) since ψ(z) is nowhere nlytic in the complex plne. In the next section choosing n pproprite pth in the complex plne we give modifiction of the steepest descent method for fst computtion of (1). We lso give Mthemtic [11] code for utomtic implementtion of the method. Note tht if the frequency ω is smll, sy ω < 2, then Guss-Jcobi qudrture rule [3], which cn be pplied over the subintervls [, b] nd [b, c], my be more efficient thn the method presented in this pper. In this cse nd for moderte vlues of ω the double exponentil method [6] my lso be pplied to the integrl on ech of the subintervls [, b] nd [b, c]. 2 An pproprite numericl method for (1) The integrl (1) cn be written s I[f] = b f(x)ψ 1 (x)e iωx dx + c b f(x)ψ 2 (x)e iωx dx = I 1 [f]+i 2 [f],
Efficient computtion of clss of singulr oscilltory integrls 1537 Figure 1: An pproprite integrtion pth for the integrl (1). R Im(z) r +r b b+r Re(z) c r c where ψ 1 (x) =(x ) α (b x) β (c x) δ,ψ 2 (x) =(x ) α (x b) β (c x) δ. Now consider Figure 1 nd closed contour CL = 6 k=1 C k, where C 1 : γ 1 (t) = + re it, t π/2, C 2 : γ 2 (t) =t, + r t b r C 3 : γ 3 (t) =b + re it,π/2 t π, C 4 : γ 4 (t) = + it, t R C 5 : γ 5 (t) =t + ir, t b, C 6 : γ 6 (t) = + it, t R. Lemm 2.1 If f(z) is of exponentil order s I(z) with constnts {m, M} nd nlytic in the hlf-strip B, then we hve nd CL f(z)ψ 1 (z)e iωz dz = (3) I 1 [f] = ( i) β+1 e ibω (b + it) α (c b it) δ f(b + it)t β e ωt dt + i β+1 e iω (b it) β (c it) δ f( + it)t α e ωt dt (4) for ω>m, where the orienttion of CL is tken s shown in Figure 1.
1538 A. Ihsn Hsçelik nd Diln Kılıç Proof. Since the integrnd is nlytic on CL nd inside the region bounded by CL, the first result is direct consequence of Cuchy-Gourst Theorem. For the second result we hve to show tht the integrl over ech qurter circle centered t nd b respectively tends to zero s r, nd tht the integrl over the top side of the left rectngle in Figure 1 tends to zero s R. On the qurter circle centered t we hve f(z)ψ 1 (z)e iωz dz π/2 = C 1 f( + re it )ψ 1 ( + re it )e iω(+reit) ire it dt π/2 r 1+α F (r, t)dt where F is continuous function of r nd t. Since α> 1 nd the integrl of F is continuous with respect to r, the limit of the right hnd side (nd hence the left hnd side) of the bove inequlity for r is zero. The sme conclusion is obtined for the integrl long the qurter circle C 3. On C 5, tking z = t + ir in the corresponding integrl we hve f(z)ψ 1 (z)e iωz dz = C 5 b f(t + ir)ψ 1 (t + ir)e iω(t+ir) dt b e ωr f(t + ir) ψ 1 (t + ir) dt b Me ω R e ωr ψ 1 (t + ir) dt where 2 ω =(ω m) >. Since ψ 1 (t + ir) e ω R for ll t [, b] nd sufficiently lrge R, tking the limit s R we obtin the desired result. Now using the bove results nd the prmetriztion for ech of C 2,C 4 nd C 6 we obtin the second result of Lemm. Q.E.D. A similr result is obtined for the integrl I 2 [f]. In this cse we use the right pnel of Figure 1 s the contour for the integrl. Combining these two results we obtin the result in the following theorem. Theorem 2.2 The integrl given by (1) cn be written in the form I[f] = iα+1 e iω ω α+1 (b it ω )β (c it ω )δ f( + it ω )tα e t dt + [iβ+1 +( i) β+1 ]e iωβ + ( i)δ+1 e iωc ω δ+1 ω β+1 (b + it ω )α (c b it ω )δ f(b + it ω )tβ e t dt (c + it ω )α (c b + it ω )β f(c + it ω )tδ e t dt.
Efficient computtion of clss of singulr oscilltory integrls 1539 Proof. Using Lemm 2.1 nd replcing t by t/ω in the results obtined for I 1 [f] nd I 2 [f] we obtin the result s the sum of these two integrls. Q.E.D. The integrls in Theorem 2.2 cn be efficiently clculted by the generlized Guss-Lguerre rule [3]. If we use the n-point Guss-Lguerre rule for ech integrl, we need 3n integrnd evlutions to get n pproximtion to I[f]. For utomtic clcultion of these integrls by the Guss-Lguerre rule nd to obtin n pproximtion to the originl integrl I[f] we give Mthemtic progrm in the next section. Note tht ech integrl in the theorem cn lso be clculted by modified version of the double exponentil method [1], which is denoted by DoubleExponentilOscilltory in Mthemtic. 3 A Mthemtic progrm for the method Subroutine singnsdm for the method: (***singnsdm:singulr Numericl Steepest Descent Method***) singnsdm[_,b_,c_,lph_,bet_,delt_,prec_,dig_,f_,omeg_,inc_]:= Module[{n=n,Qf=1,err=1}, While[Abs[err]>1^(-prec)\[And]n<2, xn=arry[n[root[lguerrel[n,lph,s],#],dig]&,n]; x[k_]:=xn[[k]]; wex=gmm[n+1+lph]*xn/(gmm[n+1]((n+1)lguerrel[n+1,lph,xn])^2); yn=arry[n[root[lguerrel[n,bet,s],#],dig]&,n]; y[k_]:=yn[[k]]; wey=gmm[n+1+bet]*yn/(gmm[n+1]*((n+1)*lguerrel[n+1,bet,yn])^2); zn=arry[n[root[lguerrel[n,delt,s],#],dig]&,n]; z[k_]:=zn[[k]]; wez=gmm[n+1+delt]*zn/(gmm[n+1]((n+1)lguerrel[n+1,delt,zn])^2); wx[k_]:=wex[[k]]; wy[k_]:=wey[[k]]; wz[k_]:=wez[[k]]; Qf=I^(lph+1)*Exp[I*omeg*]*omeg^(-lph-1)* Sum[wx[k]*(b--I*x[k]/omeg)^(bet)*(c--I*x[k]/omeg)^(delt)* f[+i*x[k]/omeg], {k,1,n}]+(-i)^(delt+1)*exp[i*omeg*c]* omeg^(-delt-1)*sum[wz[k]*(c-+i*z[k]/omeg)^(lph)* (c -b+i*z[k]/omeg)^(bet)*f[c+i*z[k]/omeg],{k,1,n}]+ (I^(bet+1)+(-I)^(bet+1))*Exp[I*omeg*b]*omeg^(-bet-1)* Sum[wy[k](b-+I*y[k]/omeg)^(lph)*(c-b-I*y[k]/omeg)^(delt) *f[b+i*y[k]/omeg],{k,1,n}]; err=abs[qf-qf]; Print["n=",n,", Qf=",Qf]; If[n>n,Print[" AbsError=",err]]; Qf=Qf; n+=inc;]]; (*** INPUT VALUES ***) =1;b=4;c=1;lph=-1/3;bet=-9/1;delt=-1/4;omeg=1; prec=16; (*required precision*)
154 A. Ihsn Hsçelik nd Diln Kılıç dig=32; (*working precision for internl clcultions*) f[t_]:=1*log[t+1]/(t+1); (*given function*) n=floor[3 prec/(2(mx[abs[log1[omeg+1]],1]+2))]; inc=3; (*n nd (n+inc) point Guss-Lguerre rules re used*) (*Qf=result of pproximtion for ech n *) Timing[singNSDM[,b,c,lph,bet,delt,prec,dig,f,omeg,inc]] If we execute the progrm using the bove input vlues, the progrm displys the following output: n=3 Qf=3.441954325663232983485399+1.965789215925227729737 I n=6 Qf=3.441954325663232983485399+1.965789215925227729737 I AbsError=3.*1^-27 {.16, Null} Here AbsError is the mgnitude of difference of two results obtined by the n-point nd (n + inc)-point Guss-Lguerre rules. The bove results show tht the 3-point Guss rule for ech integrl is sufficient to obtin 16-digit ccurte result for the given integrl. Actully, the progrm with the use of 3 3 = 9 integrnd evlution yields n pproximtion to the integrl with error 3. 1 27. The lst row in the output given bove shows the execution time in seconds, for this exmple it is equl to.16. 4 Numericl results nd comprison Consider the integrl 1 1 log (x +1) I 1 (α, β, δ, ω) = (x 1) α x 4 β (1 x) δ e iωx dx (5) 1 x +1 We computed this integrl using singnsdm tking {prec=16, dig=32, inc=1} nd NIntegrte s DoubleExponentil method tking {AccurcyGol=16, WorkingPrecision=32}. Tble 1 shows the required number of integrnd evlutions, N f, nd the execution time in seconds for ω =1, 1 2, 1 3, 1 4. Integrnd evlutions for the error estimtion is lso included in N f. Observe tht for low frequency, ω = 1, the execution time for both methods is nerly equl but for lrger frequencies the double exponentil method (DoubleExponentil) requires much more time nd function evlutions thn our method. This is n expected behvior of the steepest descent method since the symptotic order of the method is equl to 2n, i.e., I[f] Q n [f] =O(ω 2n 1 ). In other words, s frequency increses the number of function evlutions required for the steepest descent method decreses. Note tht singnsdm with {prec=14, dig=16} lso yields correct results, correct to (t lest) 14-digit ccurcy. As the prmeters pproch to 1
Efficient computtion of clss of singulr oscilltory integrls 1541 it gives ccurte results too. We hve observed tht DoubleExponentil in Mthemtic (v.8.) usully fils when the prmeters re in ( 1, 9/1]. We lso computed ech integrl in Theorem 2.2 using NIntegrte s Double- ExponentilOscilltory method [11], which is modified version of the double exponentil method for infinite rnge oscilltory integrls. The results re shown in the lst two rows of Tble 1. Tble 1: Number of integrnd evlutions N f nd execution time required to obtin 16-digit ccurte pproximtion to the integrl I 1 ( 1/2, 2/5, 1/4) by singnsdm nd NIntegrte s DoubleExponetil methods. ω execution time N f Present method 1.312 75 1.31 33 1.31 27 1.16 21 DoubleExponentil 1.344 92 1 1.391 3678 1 1.78 29428 1 85.64 235426 DoubleExponentilOscilltory 1.75 2283 1.75 222 Our method is lso suitble to obtin very high precision pproximtions. For exmple, with {prec=6, dig=11} singnsdm produced 1-digit ccurte pproximtion to I 1 ( 1/2, 9/1, 1/4, 1 6 ), in 14 seconds on our PC. For this computtion singnsdm used 112 3 + 115 3 = 681 integrtion points. 5 Conclusion Using well-known contour integrtion technique we hve developed method for the evlution of certin clss of highly oscilltory singulr integrls. The exmples given in this pper nd our other numericl experiments hve shown tht the proposed method requires less integrnd evlutions nd execution time thn the double exponentil method nd its modifiction. Moreover it is stble, esy to implement on computer, nd does not require ny derivtive computtion. It is obvious tht the proposed method cn be generlized to compute integrls of the form c m I[f] = x b k β k f(x)(x ) α (c x) δ e iωx dx, b k (, c). k=1
1542 A. Ihsn Hsçelik nd Diln Kılıç Finlly, the Mthemtic progrm given in this pper is very useful for the implementtion of the method. Similr progrms cn esily be developed in FORTRAN, C, Mtlb, or Mple. References [1] R. Chen, Fst computtion of clss of highly oscilltory integrls, Appl. Mth. Comput., 227 (214), 494-51. [2] L.N.G. Filon, On qudrture formul for trigonometric integrls. Proc. R. Soc. Edinb., 49 (1928), 38-47. [3] W. Gutschi, Orthogonl Polynomils:Computtion nd Approximtion, Oxford University Press, New York, 24. [4] A.I. Hscelik, Suitble Guss nd Filon-type methods for oscilltory integrls with n lgebric singulrity, Appl. Numer. Mth., 59(29),11-118. [5] A.I. Hscelik, An symptotic Filon-type method for infinite rnge highly oscilltory integrls with exponentil kernel, Appl. Numer. Mth., 63 (213), 1-13. [6] A.I. Hscelik nd L. Ky, Some bounds for certin highly oscilltory integrls, Appl. Mth. Sci., 2/4 (28), 187-193. [7] D. Huybrechs nd S. Vndewlle, On the evlution of highly oscilltory integrls by nlytic continution., SIAM J.Numer Anl., 44 (26), 126-148. [8] D. Levin, Fst integrtion of rpidly oscilltory functions, J. Comput. Appl. Mth., 67 (1996), 95-11. [9] J. Li., X. Wng, T. Wng, nd S. Xio, An improved Levin qudrture method for highly oscilltory integrls, Appl. Numer. Mth., 6 (21), 833-842. [1] T. Oour, M. Mori, The double exponentil formul for oscilltory functions over the hlf-infinite intervl, J. Comput. Appl. Mth., 38 (1991), 353-36. [11] S. Wolfrm, The Mthemtic Book (4th edition), Cmbridge Univ. Pr., Cmbridge, 1999. Also see, http://www.wolfrm.com Received: Jnury 1, 214