NUMERICAL EVALUATION OF HIGHLY OSCILLATORY INTERGRALS WITH WEAK SINGULARITIES

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Ἰάκωβος Κολιάτσος
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1 27 Â 9 ÞÒÙ ËÙ ÒÂÆ 38 3 Æ Sep., 27 Journl On Numericl Methods nd Computer Applictions Vol.38, No.3 Â ÀÃ Ä ÁÅ * 2. ²É± Ñ Ê É Ï, ²É± Ý Ê¹ Ï, 4543» Ê ÔÁ ¾Æ µôá b lnx lnb x x α x ηb x βfxex dx, È < α <, < β <,η,b, fx Ü [,b], ÙÝÑÔÁ. Ä, Æ µôá ²Û Á ¹ÆÁ, ÆÁ Å ², ÆÁÒÅ n Guss-Lguerre ÔÁØÐ., ¾Ê ¾ ÅÌ Ý ², Ä ¾ Ý. ÝÑÐ ¼Ä ¾È ¾. : µ Ý; ÝÑÔÁ; ; Guss-Lguerre ÔÁ; Ê ¾ MR 2 ² : 65D32 NUMERICAL EVALUATION OF HIGHLY OSCILLATORY INTERGRALS WITH WEAK SINGULARITIES Wng Xinhui Qio Hui 2. School of Mechnicl nd Power Engineering, Henn Polytechnic University, Jiozuo 4543, Chin 2. School of Mthemtics nd Informtion Science, Henn Polytechnic University, Jiozuo 4543, Chin Abstrct In this pper, we explore qudrture methods for highly oscilltory integrl b lnx lnb x x α x ηb x βfxex dx, where < α <, < β <,η,b. fx is n nlytic function in [,b]. In the presented method, the highly oscilltory integrls cn be seprted to two prts. One prt is evluted by the symptotic method, nd nother prt is evluted by the n points Guss- Lguerre qudrture. A new constructed function is used to tret the wek singulrity in ech expnsion of symptotic method. The symptotic error is given in inverse powers of the * 26 Â 6 4 Ùº. Ð ² : Æ Ð Þ²Ð ÐÜ NSFRF422 ²É± Ð B24-38.

2 26 ÝÑØÐÊØÐÑÁÅ 27 Â frequency. The vlidity of the presented method hs been demonstrted by the numericl exmples. Keywords: oscilltory function; numericl qudrture; symptotic method; Guss- Lguerre qudrture; wek singulrity 2 Mthemtics Subject Clssifiction: 65D32. ½ º Å ÓÀÍ Ö È Ö Í¹½ Ç. ÉÅ ÓÀÍ, Õ Á ¼ [ 4]. Ö, Å ÓÀÍ Å Đ³½ º, «Ì Ç Æ Đ Ü Å½½Õ. Ú º½Å ÓÀ b lnx lnb x x α x ηb x βfxex dx,. ÇÖ < α <, < β <, η,b, fx Ö Û [,b] Ö½ Ü. ÇÓÀ½ ±Ú ½ Ç, ÇÖÓÀ½Å ½, ¾«ÇÖÓÀ Ç É ½. Ç É ½½Å ÓÀ ½. Erdélyi [5] ØÑÄ Neutrlizer Ü À ÓÀ½ß«. Lyness [6,7] À Ü± ß«. [8]» ØÅ ÓÀ. É Ç É Cuchy É ½½Å Ü, Ø ßÓÀ [9]. Clenshw-Curtis-Filon Å [] Ä Í Û Ç²Ü É ½½Å Ü., Ç Ç ½ Ä ± Ç É ½½Å ÓÀ Ü [ 3]., Â ½ ÐÚÈ ÖÛ Ö ± ÑÄ, È ÖÛ ÎÅË ØÝ Çµ½ÜÐ. É»ÓÀ ß«, ØÇµ½ÜÐ. Ã Ö, Å ÓÀÚ Äß«n Guss-Lguerre ÓÀ ±. ß«Ö, ½ É ½ ÄË Ü ±. ÜÐ ³»Ã ½Çµ½. 2. Ú, Ú ÓÀ»Ó½ß«, ÓÀ»ÓÆ ÇÖ fx, gx ÖÏ Ü,. ¼ 2. [3]., É, Ç I[f] I[f] b σ [f]x fx σ n+ [f]x d dx m fxe gx dx. 2. σ n [f]x, n,,2... g x 2.2 m+ { } e gb g b σ m[f]b eg g σ m[f]. 2.3

3 3 Æ : Ê ÔÁ ¾Æ µôáýñîø 27 ± 2., gx x, Ó 2.3 ß Æ I[f] m ± 2. ¼ß«½ s ± Q A s [f] s m m+{ e gb σ m [f]b e g σ m [f]}. 2.4 m+ { } e gb g b σ m[f]b eg g σ m[f] ¼ 2.2 []. ÉÏ Ü f g, s ±½ Æ 2.5 Q A s [f] I[f] O s,. 2.6 ÔÁÔ 3. ¾ÔÁ 3. ¾ Ú ØÅ ÓÀ½ÜÐÍ. Ú, ¼ ½ ±. ¼ 3.. ÉÓÀ I η,b.fx Ö Û [,b] ½ Î Â Ì ½ Ü. b fx x η ex dx, 3. Fx fx x η, 3.2 ÉÓÀ 3., ¼¹ ½ I m+ d m dx mfbeb dm dx mfe +iπe η fη, 3.3 m. Ú, É Õ½»ÓÀ, Ç Γ +Γ 2+Γ 3+Γ 4+Γ 5+Γ 6 fz z η ez dz. 3.4

4 28 ÝÑØÐÊØÐÑÁÅ 27 Â, Γ 2 «¼ Ü, [2] Õ, Ç ½ Γ 4+Γ 5+Γ 6 µ. Γ 2 fz z η ez dz fz z η ez dz iπe η fη. 3.5 m. ± 3., ÓÀ lim ǫ η+ǫ η ǫ Ö½ Cuchy É ½. m+ d m dz mfbeb dm dz mfe. 3.6 e x x η dx ½Ð iπeη fη, ÓÀ 3. 2 ÔÁÔ 3.3 ¾ÔÁ Å 3.. ÉÓÀ I[f] b lnx lnb x x α x ηb x βfxex dx, 3.7 ÇÖ, < α <, < β <,η,b, fx Ö Û [,b] Ö½ Ü. lnb x Fx x ηb x βfx, lnx Gx x α x η fx, Hx lnx lnb x x α b x β fx, ρ [F]x Fx, ρ k+ [F]x x α d lnx dx ϕ [G]x Gx, ϕ k+ [G]x b x β d lnb x dx ρ k [F]x ρ k [F] x α,k,,2... lnx ϕ k [G]x ϕ k [G]b b x β,k,,2... lnb x

5 3 Æ : Ê ÔÁ ¾Æ µôáýñîø 29, I[f] + + k+ lnc i α e α kρ k[f]+ i β e b β c α [ρ k[f]c ρ k [F]]e c ln it t α e t dt ie c ln it t β e t dt+ie c d k lnc+ip k+ dp k c+ip α p k+ lnb c b c β [ϕ k[g]c ϕ k [G]b]e c k+ d k lnb c ip dp k b c ip β kϕ k[g]b+iπe η Hη. 3.. Ú ÀÄ ± 3., 3.7 ß Æ ½»Ó I[f] b Ý, ÊÓ 3.2 Ö, ÓÀ½ Cuchy É ½., 3.2 Ö½ÓÀ ß Æ ÇÓÀ ÇÖ, c,b,c η. p Hx x η ex dx+iπe η Hη, 3.2 I [f] c lnx lnb x x α x ηb x fxe x dx, β I 2 [f] b lnx lnb x c fxe x dx, x α x ηb x β ÐÚ Å ÓÀ I [f], ÀÄ ± 2. Ç I [f] c c { lnx lnb x x α x ηb x βfxex dx lnx x α [F[x] F[]]ex dx+f[] c } c lnx x α ex dx lnx x α [ρ [F]x ρ [F]]e x c { } lnx e x d x α [ρ [F]x ρ [F]] +ρ [F] { } c lnx x α [ρ [F]x ρ [F]]e x c c lnx x α ρ [F]xe x dx+ lnx x α ρ [F]e x dx ρ [F] c lnx c lnx x α ex dx+ρ [F] x α ex dx k+ c lnx x α [ρ k[f]x ρ k [F]]e x + k c ρ k [F] c 3.3 lnx x α ex dx lnx x α ex dx. 3.4

6 22 ÝÑØÐÊØÐÑÁÅ 27 Â ÀÄ Hópitl, ρ k [F]x ρ k [F] x α e x lnx x d dx ρ k[f]x αx α lnx x α e x lnx 2 x d dx ρ k[f]x e x αα x α α x α 2lnx x d dx ρ k[f]x αα x α α 2 x α ex x x α d dx ρ k[f]x αα α 2 e x. 3.5 x, ÓÀ c lnx x e x dx»óà ±, 2 Õ. α c lnx lnz x α ex dx Γ 2+Γ 3+Γ 4+Γ 5 z α ez dz, 3.6 z +ip, lnz Γ 4 z α ez dz lim r r i α e i α e α ln+ip +ip α e+ip idp lnip p α e p dp ln it t α e t dt 3.7, z +r e iθ, ³ r Ç lnz Γ 5 z α ez dz π i 2 ln+r e iθ iθ +r e iθ α e+re r e iθ dθ π 2 lnr e iθ ie r e iθ α r ereiθ e iθ dθ π e r α 2 lnr e iθ e rsinθ dθ π e r α 2 lnr dθ e r α lnr π, lnz Γ 3 z α ez dz c lnx+ir x+ir α ex+ir dx, R. 3.9

7 3 Æ : Ê ÔÁ ¾Æ µôáýñîø 22 ¾«, z c+ip, Ç Γ 2 lnz z α ez dz ½ Ç, I [f] + i α e lnc+ip c+ip α ec+ip idp ie c lnc+ip c+ip α e p dp ie c ie c k+ lnc α k+ k+ p dk lnc+ip e dp k c+ip α p dk lnc+ip e dp k c+ip α c α [ρ k[f]c ρ k [F]]e c ln it t α e t dt ie c. 3.2 p d k lnc+ip k+ dp k c+ip α p kρ k[f]. 3.2 ±, I 2 [f] + k+ lnb c i β e b β b c β [ϕ k[g]c ϕ k [G]b]e c ln it t β e t dt+ie c d k lnb c ip k+ dp k b c ip β p kϕ k[g]b 3.22 Ý, ÓÀ ¼. ln it e t dt ln it t β t e t dt n Guss-Lguerre ÓÀÍ [4]. ± α. ± 3. Ö, Ü fx Å Ï Û [,b]. ¼ 3.2. ÉÓÀ 3.7, Ç ±½ß«Æ s Ö ±Ü.. ÓÀ error O s, lnxln x x.5 x.5 x /3 sinxex dx Ô c.8, 4 Guss-Lguerre ÓÀÄ Í ÀÓÀ. 3 4 ÀÅ Õ» s s Þ, ½Ø ½. 2 Ö Þ ½ Í., ÖÛÐ c ½ Ç ½Ã ³., 2 Æ Đ ½ Å. Ã ³» ½ Æ Çµ½.

8 222 ÝÑØÐÊØÐÑÁÅ 27 Â 3 s 4 s «³ c, s c c.5 c.5 c.25 c.35 c e e e e e-5 c c.55 c.65 c.75 c.85 c e-5.285e e e e-5 c c.5-e-4 c.5-e-8 c.5 c.5+e-4 c.5+e e e-5.373e e-5

9 3 Æ : Ê ÔÁ ¾Æ µôáýñîø Đµ± Æ s 5 Ñ i i ØÐÑ i i.22e e-5.76e e-5 2 Ñ i i ØÐÑ i i.2862e e-6.222e e-6 4. É»ÓÀ ±,» Ø ± Ç É ½ Cuchy É ½½Å ÓÀ. Ã Ö, Ë ÜÄ ± ß«½ É ½. Ü fx ½ ½ ÏÅ É Û [,b], ß«Ó. Å ÓÀ ±Ú ÀÆ À, À Ä ß«±, ¾ ÀÑÄ n Guss-Lguerre ÓÀ. ß«ÖÃ ½ Ü. ÜÐ ³»Ã ½Çµ½. ¹ [] Iserles A nd Nørsett S. On qudrture methods for highly oscilltory integrls nd their implementtion[j]. BIT, 24, 44: [2] Filon L N G. On qudrture formul for trigonometric integrls[j]. Proc. Roy. Soc. Edinburgh, 928, 49: [3] Levin D. Procedures for Computmg One nd Two Dimensionl Integrls of Functions With Rpid Irregulr Oscilltions[J]. Mth. comp., 982, 38: [4] Huybrechs D, Vndewlle S. On the evlution of highly oscilltory integrls by nlytic continution[j]. SIAM J. Numer. Anl., 26, 44: [5] Erdélyi A. Asymptotic representtions of Fourier integrls nd the method of sttionry phse[j]. J. Soc. Ind. Appl. Mth., 955, 3: [6] Lyness J N. Numericl evlution of fixed-mplitude vrible-phse integrl[j]. Numer. Algorithms. 28, 49: [7] Lyness J N, Lottes J W. Asymptotic expnsions for oscilltory integrls using inverse functions[j]. BIT Numer. Mth., 29, 49: [8] Kng H C, Xing S H. Efficient integrtion for clss of highly oscilltory integrls[j]. Appl. Mth. Comp., 2, 28:

10 224 ÝÑØÐÊØÐÑÁÅ 27 Â [9] Fng C H. Efficient methods for highly oscilltory integrls with wek nd Cuchy singulrities[j]. Int. J. Computer Mth., 25, [] Kng H C, Ling C. Computtion of integrls with oscilltory singulr fctors of lgebric nd logrithmic type[j]. J. Comput. Appl. Mth., 25, 285: [] Wng H Y, Xing S H. On the evlutions of cuchy principl vlue integrls of oscilltory functions[j]. J. Comput. Appl. Mth., 2, 234: 95-. [2] MilovnoviøG V. Numericl clcultion of integrls involving oscilltory nd singulr kernels nd some pplictions of qudrtures[j]. Comput. Mth. Appl., 998, 36: [3] Iserles A. nd Nørsett S. Efficient qudrture of highly oscilltory integrls using derivtives[j]. Proc. Royl Soc. Lond. Ser. A Mth. Phys. Eng. Sci., 25, 46: [4] Abrmowitz M, Stegun I A. Hndbook of Mthemticl Functions, Ntionl Bureu of Stndrds, Wshington, DC, 964.

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