Numerical Green s Function Method Based on the DE Transformation

Transcription

1 Jpn J. Indust. Appl. Mth., 23 (2006, Are 2 Numericl Green s Function Method Bsed on the DE Trnsformtion Mstke Mori nd Toshihiko Echigo Deprtment of Mthemticl Sciences, Tokyo Denki University, Htoym, Hiki-gun, Sitm , Jpn Received September 2, 2005 Revised Jnury 16, 2006 A method for numericl solution of boundry vlue problems with ordinry differentil eqution bsed on the method of Green s function incorported with the double exponentil trnsformtion is presented. The method proposed does not require solving system of liner equtions nd gives n pproximte solution of very high ccurcy with smll number of function evlutions. The error of the method is O (exp ( C 1 N/ log(c 2 N where N is prmeter representing the number of function evlutions nd C 1 nd C 2 re some positive constnts. Numericl exmples lso prove the high efficiency of the method. An lterntive method vi n integrl eqution is presented which cn be used when the Green s function corresponding to the given eqution is not vilble. Key words: double exponentil trnsformtion, DE trnsformtion, Green s function, boundry vlue problem, integrl eqution 1. Introduction There hve been severl methods for numericl solution of boundry vlue problems with ordinry differentil eqution. The shooting method is trditionl one nd the implicit finite difference method is lso one of the useful tools. In 2002 Sugihr [8] nd in 2005 Nurmuhmmd et l. [5] proposed sinc colloction methods bsed on the double exponentil trnsformtion. In the present pper, method for numericl solution of boundry vlue problems bsed on the clssicl method of Green s function incorported with the double exponentil trnsformtion is presented. Although the method of Green s function is clssicl one for nlyticl mnipultion of solution of boundry vlue problems with differentil eqution, the method presented here gives n pproximte solution of very high ccurcy with smll number of function evlutions. The differentil eqution we consider here is L[u]+f(x =0, < x < b, (1.1 where u(x is the function to be determined nd L[u] is self-djoint opertor ssocited with the Sturm-Liouville eigen-vlue problem defined s L[u] d { p(x du } + q(xu. (1.2 dx dx We ssume here tht p(x, q(x ndf(x re known nd nlytic on (, b, nd f(x my be singulr t x =, b provided tht the eqution is defined properly. We lso

2 194 M. Mori nd T. Echigo ssume tht p(x > 0on(, b. For the moment we ssume tht the boundry condition is homogeneous, i.e. u( =0, u(b =0. (1.3 However, if p(x vnishestx = or x = b we cn impose there n inhomogeneous boundry condition s will be shown lter in Exmple 2. u( = finite 0, or u(b = finite 0 ( The Method of Green s Function We strt with short revisit to the method of Green s function [1]. We consider here the following boundry vlue problem with homogeneous differentil eqution with the sme L[u] s given in (1.2: { L[y] =0 y( =0, y(b =0. (2.1 It is well known tht the Green s function K(x, ξ corresponding to the opertor L[y] in (2.1 with the homogeneous boundry condition is defined s follows: 1. For fixed ξ, K(x, ξ is continuous function of x nd stisfies the homogeneous boundry condition ( The first nd the second derivtives of K(x, ξ with respect to x re continuous on (, b except x = ξ nd dk/dx stisfies ( dk dx x=ξ+0 ( dk = 1 dx x=ξ 0 p(ξ. ( K(x, ξ stisfies the differentil eqution (2.1 on (, b except x = ξ s function of x. Let y 1 nd y 2 be fundmentl solutions of L(y = 0 which stisfy y 1 ( =0nd y 2 (b = 0, respectively. Then the Green s function cn be written { y1 (ξy 2 (x, ξ x K(x, ξ = y 1 (xy 2 (ξ, x ξ. (2.3 Under these preliminries the solution of the boundry vlue problem of inhomogeneous eqution (1.1 is expressed s follows [1]: u(x = b K(x, ξf(ξdξ. (2.4

3 Green s Function Method Bsed on the DE Trnsformtion The DE Formul for Indefinite Integrls The derivtive of the integrnd in (2.4 hs discontinuity t ξ = x nd we cn not pply the double exponentil formul for definite integrls over (, b to (2.4 [10]. However, we cn write (2.4 in terms of sum of two indefinite integrls with n nlytic integrnd over ech intervl of integrtion: u(x = x K(x, ξf(ξdξ + b x K(x, ξf(ξdξ. (3.1 And hence it is quite nturl tht we employ numericl integrtor tht gives good result for ech of the indefinite integrl whose integrnd is regulr on ech intervl. In 2003 Muhmmd nd Mori [3] nd in 2004 Tnk, Sugihr nd Murot [11] proposed double exponentil formul for indefinite integrl which works well for such kind of integrls. We first consider here n indefinite integrl of function g(ξ over(, x corresponding to the first term in the right-hnd side of (3.1: I 1 (x = x g(ξdξ, < x < b. (3.2 Here we ssume tht g(ξ is regulr over (, b except ξ = nd b. If we pply the vrible trnsformtion to (3.2 we hve ξ = ψ(t = b 2 I 1 (x = τ We ssume here tht g(ψ(tψ (t stisfies ( π tnh 2 sinh t + b + 2 (3.3 g(ψ(tψ (tdt, τ = ψ 1 (x. (3.4 g(ψ(tψ (t is regulr in the strip Im t <dfor some d>0 (3.5 nd g(x =O (( (x (b x 1+α s x, b (3.6 with some positive constnt α. Trnsformtion (3.3 is first proposed by Tkhsi nd Mori in 1974 [10] for numericl definite integrtion nd is clled the double exponentil trnsformtion, bbrevited s the DE trnsformtion, becuse the integrnd g(ψ(tψ (t fter the trnsformtion decys double exponentilly, i.e., ( ( g(ψ(tψ π(α ε (t =O exp exp t, s t ± (3.7 2 for n rbitrrily smll positive number ε.

4 196 M. Mori nd T. Echigo Under the ssumptions given bove with some dditionl mild ones g(ψ(tψ (t cnbeexpndedintermsofthesincfunction[9,6,3]withmeshsizeh: g(ψ(tψ (t = j= e n = sin(πt/h 2πi ( t g(ψ(jhψ (jhsinc C n where the sinc function sinc(t is defined h j g(ψ(τψ (τ (τ tsin(πτ/h dτ, + lim n e n, (3.8 sin πt sinc(t = (3.9 πt nd lim n e n is the error term in which the pth C n surrounds ll the simple poles τ = jh, j =0, ±1,...,±n nd τ = t in the positive direction but not the singulrities of g(ψ(τψ (τ. We substitute (3.8 into (3.4 nd truncte the infinite summtion t ±N in such wy tht N = 1 2d log h α h, α = α ε (3.10 holds. Reltion (3.10 comes from the requirement tht the discretiztion error lim n e n due to the sinc expnsion nd the error due to the trunction of the infinite summtion (3.8 be the sme order of mgnitude. Then we hve the double exponentil formul, bbrevited s the DE formul, for indefinite integrl [3] I 1 (x =h g(ψ(jhψ (jh ( π Si π ψ 1 (x πj + E N1, (3.11 h where Si(t is the sine integrl defined by Si(t = t 0 sin τ τ dτ (3.12 nd E N1 is the error term given by ( ( πdn E N1 = O exp log(2dn/α (3.13 which shows nerly exponentil decy of the errror of the DE formul in terms of N. Herewefirstgveh nd then determined N by (3.10. However, if we wnt to give first N then to determine h, it should be determined h = 1 N log 2dN α, α = α ε. (3.14 Although in (3.10 nd (3.14 ε need to be n rbitrrily smll positive number from the theoreticl view point, in ctul computtion we cn tke ε =0.

5 Green s Function Method Bsed on the DE Trnsformtion 197 Next we consider the indefinite integrl of g(ξ over(x, b corresponding to the second term of the right-hnd side of (3.1: I 2 (x = b x g(ξdξ, < x < b. (3.15 Completely in the sme wy s in I 1 (x we obtin the DE formul for indefinite integrl (3.15: I 2 (x =h g(ψ(jhψ (jh 2 1 ( π Si π ψ 1 (x πj + E N2, (3.16 h where E N2 isthesmese N1 in (3.13. Note tht the only difference between (3.11 nd (3.16 is the sign of the term Si /π. 4. The DE Formul for the Method of Green s Function Now we return to the method of Green s function. The integrl we wnt to evlute is (3.1, i.e. where u(x = x K(x, ξf(ξdξ + b x K(x, ξf(ξdξ = J 1 (x+j 2 (x (4.1 J 1 (x =y 2 (x J 2 (x =y 1 (x x b x y 1 (ξf(ξdξ, (4.2 y 2 (ξf(ξdξ (4.3 from (2.3. If we pproximte J 1 (x ndj 2 (x by (3.11 nd (3.16, respectively, we immeditely hve u(x =y 2 (x h + y 1 (x h + E N, y 1 (ψ(jhf(ψ(jhψ (jh ( π Si π ψ 1 (x πj h y 2 (ψ(jhf(ψ(jhψ (jh 2 1 ( π Si π ψ 1 (x πj h (4.4 where E N is the sme s E N1 in (3.13. This is the DE formul for numericl solution of the boundry vlue problem (1.1 with homogeneous boundry condition (1.3. Since we cn compute y 1 (ψ(jhf(ψ(jhψ (jhndy 2 (ψ(jhf(ψ(jhψ (jh

6 198 M. Mori nd T. Echigo beforehnd for ech j, wht we hve only to compute for given x is y 1 (x, y 2 (x nd 1/2 ± Si ( πψ 1 (x/h πj /π nd their product sum. Thus, this formul for x consists of evlutions of simple functions nd their sum, so tht the present method is suitble for prllel computtion. Also note tht if x is equl to one of the sinc points, i.e., x = ψ(kh for some integer k, Si ( πψ 1 (x/h πj =Si ( π(k j (4.5 holds nd computtion of Si becomes very simple. 5. Numericl Exmples In this section we give some numericl exmples. The intervl of definition of the problems re (0, 1 in ll the exmples. For ech exmple we first chose N =4, 8, 16, 32,... nd computed h by (3.14. For ech N we evluted u(x by mens of the formul (4.4 for x =0.01, 0.02, 0.03,..., 0.97, 0.98, 0.99 (5.1 nd picked up the mximum bsolute vlue of the error. Since in the integrnd of (4.4 the singulr points which lie nerest to the rel xis re the poles of ψ (t = π 4 cosh t/ cosh2 (π/2sinht with the distnce π/2 [10], we set d = π/2 inllthe exmples. In order to show the high efficiency of the present formul we crried out numericl computtion with qudruple precision rithmetic. Exmple 1. d 2 u dx 2 3 ( x 1 2 +(1 x 1 2 =0 4 u(0 = 0, u(1 = 0. (5.2 Here we tke L[u] = d2 u dx 2, (5.3 nd the Green s function corresponding to (5.3 is i.e., y 1 (x =x, y 2 (x =1 x, (5.4 { ξ(1 x, ξ x K(x, ξ = x(1 ξ, x ξ. (5.5 The exct solution of this problem is u(x =x 3 2 +(1 x (5.6

7 Green s Function Method Bsed on the DE Trnsformtion 199 In this exmple, s ξ tends to 0 f(ξ = 3/4 ( ξ 1/2 +(1 ξ 1/2 = O(ξ 1/2 from (5.2 nd y 1 (ξ =O(ξ from (5.4, so tht g(ξ =y 2 (xf(ξy 1 (ξ =O(ξ 1/2 = O(ξ 1+α, α =3/2 holds. In similr wy we see tht s ξ tends to 1 g(ξ = O((1 ξ 1/2. In this wy we chose α =3/2 for (3.6 in this exmple. In other exmples we determined α in the sme wy. We chose N =4, 8, 16, 32, 64, 76, computed numericl solution for x s given in (5.1 nd plotted the mximum bsolute error of the numericl solution s function of N in Fig. 1. Although f(x of this problem is singulr t x =0nd1, the result is very good. Actully, the error decys lmost exponentilly s given in (3.13 nd ttins bout with N = 76. Fig. 1. The mximum error of Exmple 1. Exmple 2. x d2 u dx 2 + du dx + x =0 (5.7 u(0 = finite, u(1 = 0. Inthisexmplewetke L[u] =x d2 u dx 2 + du dx = d ( x du. (5.8 dx dx The Green s function corresponding to (5.8 is y 1 (x =1, y 2 (x = log x, (5.9 i.e., { log x, ξ x K(x, ξ = log ξ, x ξ. (5.10

8 200 M. Mori nd T. Echigo The exct solution of this problem is u(x = 1 4 (1 x2. (5.11 In this exmple p(x =x in (1.2 nd it vnishes t x = 0, so tht we cn impose boundry condition u(0 = finite 0. Actully we see from (5.11 tht u(0 = 1/4. We tke α = 2 in this exmple. We chose N =4, 8, 16, 32, 64, 76, computed numericl solution for x s given in (5.1 nd plotted the mximum bsolute error s function of N in Fig. 2. In this exmple lso the error behvior given by (3.13 is observed. Fig. 2. The mximum error of Exmple 2. Exmple 3. Next exmple is from [7, p. 550]: d 2 u dx 2 ν2 u ν 2 cos 2 πx 2π 2 cos 2πx =0, u(0 = 0, u(1 = 0. (5.12 Although the eqution with ν = 20 is considered in [7] we compute here numericl solutions with ν = 1, ν = 10ndν = 20, nd compre the ccurcy. In this exmple we tke L[u] = d2 u dx 2 ν2 u. (5.13 The Green s function corresponding to (5.13 is y 1 (x = sinh νx ν sinh ν, y 2(x =sinhν(1 x, (5.14

9 Green s Function Method Bsed on the DE Trnsformtion 201 i.e., K(x, ξ = The exct solution of this problem is sinh νξ sinh ν(1 x, ξ x ν sinh ν sinh ν(1 ξsinhνx, x ξ. ν sinh ν (5.15 u(x = exp( ν 1+exp( ν exp(νx+ 1 1+exp( ν exp( νx cos2 πx. (5.16 We tke α = 2 in this exmple. We chose N =4, 8, 16, 32, 64, 128, computed numericl solution for x s given in (5.1 nd plotted the mximum bsolute error s function of N in Fig. 3. Fig. 3. The mximum error of Exmple 3. If we try to solve this eqution with ν = 20 by mens of one of the conventionl difference methods we usully find it difficult to obtin good solution becuse of shrp growth nd shrp decy of the solution due to the ltent fctors exp(νx nd exp( νx in (5.16. In the present method, on the other hnd, while with smll N we fil to obtin good result, we observe in Fig. 3 n exponentil rte of convergence of the error expressed s (3.13 s N becomes lrge. For smller ν the sitution is much better. See the error curves corresponding to ν = 10ndν = 1 in Fig. 3. The reson why the present method fils to obtin good result when ν is lrge nd N is smll cn be explined s follows. First note tht the support of the function 1/2+Si(πψ 1 (x/h πj/π in (3.11 is the entire intervl (, b, lthough the mplitude of 1/2+Si(πψ 1 (x/h πj/π is smll if x<ψ(jh [3]. And hence

10 202 M. Mori nd T. Echigo the formul for indefinite integrtion (3.11 smples vlues not only from inside the intervl (, x but lso from outside the intervl (, x. This is one of the significnt chrcteristics of the formul (3.11. As for (4.4, the first term in the right-hnd side smples the vlues of y 1 (ψ(jh even when ψ 1 (x/h < j. If ν is lrge, while the shpe of the Green s function K(x, ξ isveryshrp,y 1 (ψ(jh becomes quite lrge for lrge j, lthough ψ (jh nd other fctors lower the effect of y 1 (ψ(jh to some extent. Thus, when ν is lrge nd N is smll, the mjor prt of the smple points re locted outside the effective support of the Green s function nd the distribution of the smple points does not follow properly the shrp pek of the Green s function, which deteriortes the ccurcy of the formul. Sitution is the sme for the second term in the right-hnd side of (4.4. As N becomes lrge the distribution of the smple points comes to follow properly the shpe of the Green s function nd the ccurcy chnges for the better quickly. 6. Integrl Eqution It is not lwys possible to find relevnt Green s function corresponding to the given differentil opertor. In such cse we seprte the given opertor into sum of L[u] whose Green s function is known nd other terms including u. Inthis section we consider boundry vlue problem with liner differentil eqution { L[u]+ρ(xu + f(x =0, u( =0, u(b =0 (6.1 where the Green s function corresponding to L[u] is known but not known corresponding to L[u] + ρ(xu. If we regrd ρ(xu + f(x s n inhomogeneous term we see from (2.4 tht the eqution (6.1 cn be trnsformed into the following equivlent integrl eqution: b ( u(x = K(x, ξ ρ(ξu(ξ+f(ξ dξ. (6.2 This eqution seems to be formlly Fredholm integrl eqution. However, s seen from (3.1 we should regrd (6.2 s sum of two Volterr integrl equtions, i.e. we should divide the integrl over (, b into the integrl over (, x nd the other one over (x, b: u(x = x ( K(x, ξ ρ(ξu(ξ+f(ξ dξ + b x ( K(x, ξ ρ(ξu(ξ+f(ξ dξ. (6.3 In order to solve this integrl eqution numericlly we employ the method proposed by Muhmmd et l. [4]. Incidentlly we should note here tht generl theory of the method of Green s function bsed on qudrture is developed s integrl equtions methods in the book by H. B. Keller [2] 1. Now, first we pply the DE 1 The uthors re indebted to the referee for bringing this book to their notice.

11 Green s Function Method Bsed on the DE Trnsformtion 203 formul (4.4 to the right-hnd side nd hve u(x =y 2 (x h + y 1 (x h ( y 1 (ψ(jh ρ(ψ(jhu(ψ(jh + f(ψ(jh ψ (jh ( π ψ 1 (x πj h π Si ( y 2 (ψ(jh ρ(ψ(jhu(ψ(jh + f(ψ(jh ψ (jh 2 1 ( π Si π ψ 1 (x πj h + E N. (6.4 Then, in order to derive n eqution whose solution gives n pproximtion to u(x j, j =0, ±1,...,±N we pply the colloction method bsed on the sinc points x = ψ(kh, k =0, ±1, ±2,...,±N nd obtin [4] u k = y 2 (ψ(kh h + y 1 (ψ(kh h ( y 1 (ψ(jh ρ(ψ(jhu j + f(ψ(jh ψ (jh π Si( π(k j ( y 2 (ψ(jh ρ(ψ(jhu j + f(ψ(jh ψ (jh 2 1 π Si( π(k j, k =0, ±1, ±2,...,±N (6.5 where u i is n pproximte vlue to the solution u(x tthesincpointx = ψ(ih. Here we used the reltion (4.5. This eqution cn be written in mtrix form: (I hmu = hb, (6.6 where I is the identity mtrix nd M kj = y 2 (ψ(khy 1 (ψ(jhρ(ψ(jhψ (jh π Si( π(k j + y 1 (ψ(khy 2 (ψ(jhρ(ψ(jhψ (jh 2 1 π Si( π(k j, (6.7 u =(u N,u N+1,...,u N 1,u N T, (6.8 b =(b N,b N+1,...,b N 1,b N T, (6.9

12 204 M. Mori nd T. Echigo b k = y 2 (ψ(kh + y 1 (ψ(kh y 1 (ψ(jhf(ψ(jhψ (jh π Si( π(k j y 2 (ψ(jhf(ψ(jhψ (jh 2 1 π Si( π(k j. (6.10 This is liner system of lgebric equtions with respect to u j s nd we cn usully solve it. If we wnt to get n pproximte vlue of the solution t n rbitrry x we cn compute it by (6.4 without E n in which u(ψ(jh is replced by u j. Exmple 4. We gin consider the problem (5.12, i.e., d 2 u dx 2 ν2 u ν 2 cos 2 πx 2π 2 cos 2πx =0, u(0 = 0, u(1 = 0, (6.11 in which we tke here L[u] = d2 u dx 2, (6.12 nd hence the Green s function is given by (5.5. We choose ν =1ndν =20 nd ssign the sme vlues to α, N nd h s those given in Exmple 3. We solved lgebric eqution (6.6 using the LU decomposition. We evluted pproximte solution for the sme x s in (5.1 nd plotted the mximum bsolute error s Fig. 4. The mximum error of Exmple 4.

13 Green s Function Method Bsed on the DE Trnsformtion 205 function of N in Fig. 4 for ν =1ndν = 20. We gin see high efficiency of this method vi the integrl eqution. If we compre the error grph for ν = 20 in Fig. 4 with tht in Fig. 3 we see tht the ccurcy of the result presented in Fig. 4 is much better thn tht given in Fig. 3 with the method of Green s function. On the other hnd, for ν = 1 both grphs lmost overlp with ech other. While the method of Green s function presented in Section 4 does not require solving liner system of lgebric equtions, the method vi the integrl eqution requires solving one. In this respect, if the Green s function corresponding to the entire opertor L[u] of (1.1 is vilble the method of Green s function incorported with the DE trnsformtion is recommended, in prticulr for mild problems. References [ 1 ] R. Cournt nd D. Hilbert, Methods of Mthemticl Physics (Vol.1. Interscience Publishers, INC., New York, 1937/1953, p.351. [ 2 ] H.B. Keller, Numericl Methods for Two-Point Boundry-Vlue Problem. Blisdel, 1968, Chpter 4. [ 3 ] M. Muhmmd nd M. Mori, Double exponentil formuls for numericl indefinite integrtion. J. Comput. Appl. Mth., 161 (2003, [ 4 ] M. Muhmmd, A. Nurmuhmmd, M. Mori nd M. Sugihr, Numericl solution of integrl equtions by mens of the Sinc colloction method bsed on the double exponentil trnsformtion. J. Comput. Appl. Mth., 177 (2005, [ 5 ] A. Nurmuhmmd, M. Muhmmd, M. Mori nd M. Sugihr, Double exponentil trnsformtion in the Sinc-colloction method for the boundry vlue problem of fourth-order ordinry differentil eqution. J. Comput. Appl. Mth., 182 (2005, [ 6 ] F. Stenger, Numericl Methods Bsed on Sinc nd Anlytic Functions. Springer, Berlin, New York, [ 7 ] J. Stoer nd R. Bulirsch, Introduction to Numericl Anlysis. Trnslted by R. Brtels, et l., Springer-Verlg, New York, Tokyo, [ 8 ] M. Sugihr, Double exponentil trnsformtion in the Sinc-colloction method for twopoint boundry vlue problems. J. Comput. Appl. Mth., 149 (2002, [ 9 ] H. Tkhsi, Complex function theory nd numericl nlysis. Publ. RIMS Kyoto Univ., 41 (2005, [10] H. Tkhsi nd M. Mori, Double exponentil formul for numericl integrtion. Publ. RIMS Kyoto Univ., 9 (1974, [11] K. Tnk, M. Sugihr nd K. Murot, Numericl indefinite integrtion by double exponentil sinc method. Mth. Comp., 74 (2004,