A Representation of the Exact Solution of Generalized Lane-Emden Equations Using a New Analytical...

7/28/2019 A Representation of the Exact Solution of Generalized Lane-Emden Equations Using a New Analytical Method

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Hndw Pshn CopotonAstct nd Appd AnyssVom 2013, Atc ID 378593, 10 pshttp://dx.do.o/10.1155/2013/378593

Research ArticleA Representation of the Exact Solution of GeneralizedLane-Emden Equations Using a New Analytical Method

Omar Abu Arqub,1Ahmad El-Ajou,1A. Sami Bataineh,1 and I. Hashim2

1 Department o Mathematics, Faculty o Science, Al Balqa Applied University, Salt 19117, Jordan2 School o Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Cospondnc shod ddssd to I. Hshm; shk [email protected]

Rcvd 10 Ap 2013; Accptd 31 My 2013

Acdmc Edto: Mssmo F

Copyht 2013 Om A Aq t . Ts s n opn ccss tc dsttd nd th Ctv Commons AtttonLcns, whch pmts nstctd s, dstton, nd podcton n ny mdm, povdd th on wok s popyctd.

A nw nytc mthod s ppd to sn nt-v Ln-Emdn-typ poms, nd th fctvnss nd pomnc oth mthod s stdd. T poposd mthod otns yo xpnson o th soton, nd whn th soton s poynom,o mthod podcs th xct soton. It s osvd tht th mthod s sy to mpmnt, v o hndn snphnomn, yds xcnt sts t mnmm comptton cost, nd qs ss tm. Comptton sts o sv tstpoms psntd to dmonstt th vty nd pctc snss o th mthod. T sts v tht th mthod svy fctv, sthtowd, nd smp.

1. Introduction

Snc th nnn o st stophyscs, th nvsttono st stcts hs n cnt pom. T hvn contnos fots to ddc th d pos o thpss, dnsty, nd mss o st, nd on o th ky ststht cm ot o ths fots s th Ln-Emdn qton,whch dscs th dnsty po o sos st. Mth-mtcy, th Ln-Emdn qton s scond-od sn-

odny dfnt qton. In stophyscs, th Ln-Emdn qton s ssnty Posson qton o thvtton potnt o s-vttn, sphcy sym-mtc poytopc d.

T Ln-Emdn qton hs n sd to mod sv- phnomn n mthmtc physcs, thmodynmcs,d mchncs, ndstophyscs,sch s th thoy o ststct, th thm hvo o sphc cod o s,sothms sphs, ndth thoy o thmonc cnts[913]. Ln-Emdn-typ qtons w st pshd yLn [14]; thy w xpod n mo dt y Emdn n1870 [15], who consdd th thm hvo o sphccod o s tht cts nd th mt ttcton o ts

mocs nd s sjct to th cssc ws o thmo-dynmcs. T d s kndy qstd to ps [917]to know mo dts ot Ln-Emdn-typ qtons,ncdn th hstoy, vtons, nd ppctons.

In th psnt pp, w ntodc smp nw nytcmthod w c th sd-pow-ss (RPS) mthod [18]to dscov ss sotons to n nd nonn Ln-Emdn qtons. T RPSmthods fctv ndsy to sto sov Ln-Emdn qtons wthot nzton, pt-

ton, o dsctzton. Ts mthod constcts n ppox-mt nytc soton n th om o poynom. Bysn th concpt o sd o, w otn sssoton,whch n pctc tnds to tnctd ss soton.

T RPS mthod hs th oown chctstcs [18]:st, t otns yo xpnson o th soton, nd s st, th xct soton s otnd whnv t s poy-nom. Moov, th sotons nd o ts dvtvs ppc o ch ty pont n vn ntv. Scond,th RPS mthod hs sm comptton qmnts ndhh pcson, nd thmo t qs ss tm.

In th psnt pp, th RPS mthod s sd to otn symoc ppoxmt soton o nzd Ln-Emdn


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2 Astct nd Appd Anyss

qtons n th oown om tht ssmd to hv nq soton n th ntv o ntton:

+ 1 1 2 ( ) + 2 2

+ (, ,

) = 0, 0, 0 + ,

(1)

whch s sjct to oth th nt condtons,

0 = 0, 0 = 1, (2)nd on o th oown constnt-condtons css:

cs I: 10 = 0, 10 = 0,cs II: 10 = 0 , 20 = 0,

wh, , nonn nytc nctons, , n-ytc nctons on 0, 0 + , s n nknown nctono n ndpndnt v tht s to dtmnd, nd0, ,

R wth

> 0. Tohot ths pp, w ssm

tht s n nytc ncton on th vn ntv.As spc css, whn = , ,, = , 1 R, 1 = , 2 = 1,0 = 0, nd o spc oms o , w otnsv w-known oms o th Ln-Emdn qtons. Foxmp, whn = , N, ,, =0, 0 = 1, nd 1 = 0, w otn th om o (1) nd (2)tht s th stndd Ln-Emdn qton; ths qton wsony sd to mod th thm hvo o sphccod o s tht cts nd th mt ttcton o tsmocs nd s sjct to th cssc ws o thmody-

nmcs [10, 16]. Howv, whn = (), ,,

= 0,

0

= 0, nd

1

= 0, th otnd mod cn

sd to vw sothm s sphs, wh th tmptmns constnt [10, 17]. Fo thooh dscsson o thomton o th Ln-Emdn qtons nd th co-spondn physc hvo o th modd systms, thd s d to [917].

In most css, th Ln-Emdn qton dos not wyshv sotons tht cn otnd sn nytc mthods.In ct, mny o physc nd nnn phnomntht ncontd most mposs to sov y thstchnq; hnc, ths poms mst ttckd y v-os ppoxmt nd nmc mthods. To, somthos hv poposd nmc mthods to ppoxmtth sotons o spc cs o (1) nd (2). Fo xmp, th

Adomn dcomposton mthod hs n ppd to sovth Ln-Emdn qton + / + + = 0 s dscd n [3]. In [1], th thos dvopd thoptm homotopy symptotc mthod to sov th snqton + / + + = 0. Add-tony, n [2], th thos povdd th Hmt nctonscoocton mthod to th nvstt th Ln-Emdnqton + / + + = 0.Fthmo, th homotopy ptton mthod s cdotn[4] to sov th qton+/+ =0. Rcnty, th Bss coocton mthod ws poposd tosov th n Ln-Emdn qton + / +

+ = 0n [19].

Howv, non o th pvos stds popos mthod-c wy to sov (1) nd (2). Moov, th pvos stdsq mo fot to chv th sts, nd sy thy ony std o spc om o (1) nd (2). Howv, thppctons o oth vsons o ss sotons to n ndnonn poms cn ond n [2025], nd, to dscn

th nmc sovty o dfnt ctos o sndfnt qtons, on cn const [26].T otn o th pp s s oows: n th nxt scton,

w psnt th omton o th RPS mthod. Scton 3covs th convnc thom. In Scton 4, nmcxmps vn to stt th cpty o th poposdmthod. Ts pp nds n Scton 5 wth som concdnmks.

2. The Formulation of the RPS Method

In ths scton, w mpoy th RPS mthod to nd sssoton to th nzd Ln-Emdn qton (1) tht

s sjct to vn nt condtons qton (2). Fst, womt nd nyz th RPS mthod to sov sch po-ms.

T RPS mthod conssts o xpssn th soton o (1)nd (2) s pow-ss xpnson ot th nt pont = 0. o chv o o, w sppos tht ths sotonstk th om = =0 wh th tms oppoxmtons = 0, = 0, 1, 2, . . ..

Ovosy, whn = 0, 1 cs 0, 1 stsy thnt condtons (2) s 0 = 0 = 00 nd 0 =1 = 10, w hv n nt ss o th ppoxmton o, nmy,nt = 0+00. Incontst, w choos nt s th nt ss o n ppoxmtono, thn w cn cct o = 2, 3, 4, . . . ndppoxmt th soton o (1) nd (2) y th oownth-tnctd ss:

= =0

0. (3)Po to ppyn th RPS mthod, w wt th sn

qtons (1) nd (2) n th oown om:

+ ( ) +

+ (, ,

) = 0,(4)

wh = 12, = 1, = 21nd = 12. Sstttn th th tnctd ss nto (4) ds to th oown dnton o th thsd ncton:

Rs = =2

1 02

+ =1

01


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Astct nd Appd Anyss 3

+ =0

0 +

, =0

0, =1

01 ,(5)

nd, thmo, w otn th oown th sdncton Rs = mRs. It sy to s thtRs = 0 o ch 0, 0 + . Ts, Rss nnty dfnt ncton t = 0. Fthmo,1/1Rs0 = 1/1Rs0 = 0. In ct,ths ton s ndmnt n th RPS mthod nd tsppctons.

Now, n od to otn th scond ppoxmt soton,

w st = 2 nd 2 = 2=0 0. Tn wdfntt oth sds o (5) wth spct to nd ssttt = 0 to otn th oown:

Rs2 0= 22 0 + 22 0 1 1+0 0 0, 0, 1 + 1

0 0, 0, 1+ 22 1 0, 0, 1

+

0

0

, 0

, 1

+

0

1

+ 00 + 1 0 00 .

(6)

Usn th cts tht /Rs0 = /Rs20 =0 nd tht 0 = 0 = 0, w know tht (6) vs thoown v o 2:2 = 12 0 + 0/11

[00, 0, 1 + 01 + 00].(7)

Ts, sn th scond tnctd ss, th scondppoxmt soton o (1) nd (2) cn wttn s

2 = 0 + 1 0 12! 00, 0, 1 + 01 + 00 0 + 0/11 02.

(8)

Smy, to nd th thd ppoxmt soton, w st = 3 nd 3 = 3=0 0. Tn w dfnttoth sds o (5) wth spct to nd ssttt = 0 tootn th oown v o3:

3

= 1

62

0 + 0/11 22 0 + 2 0 0 0, 0, 1 + 1

0 0, 0, 1+ 21 1 0, 0, 1

+ 00, 0, 1 + 21 0 0 0

+ 422 0

2

21 1 + 42

0 1 1+ 1 0 .

(9)

T ov st s vd d to th ct tht/Rs30 = 0. Hnc, sn th thd tnctd ss,th thd ppoxmt soton o (1) nd (2) cn wttns

3 = 0 + 1 0 1

2 00, 0, 1 + 01 + 00 0 + 0/11 02 13! 22 0 + + 4220

2

21 1+ 42 0 1 1 + 1

0

2

0

+ 0

1

1

1

03.(10)

Ts pocd cn ptd t th ty odcocnts o RPS sotons o (1) nd (2) otnd.Moov, hh cccy cn chvd y vtnmo componnts o th soton.

3. Convergence Theorem and Error Analysis

In ths scton, w stdy th convnc o th psntmthod to cpt th hvo o th soton. Awds,


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o nctons ntodcd to stdy th cccy nd -cncy o th mthod. Acty, contnos ppoxmtonsto th soton w otnd.

yos thom ows s to psnt y nnctonsxcty n tms o poynoms wth known, spc-d, nd ondd o. T nxt thom w nt

convnc to th xct nytc soton o (1) nd (2).

Teorem 1. Suppose that is the exact solution or(1) and(2). Ten theapproximate solution obtained by theRPS methodis in act the aylor expansion o.Proo. Assm tht th ppoxmt soton o (1) nd (2)s s oows:

= 0 + 1 0 + 2 02 + 3 03 + .(11)

In od to pov th thom, t s noh to show tht th

cocnts n (11) tk th om = 1!() 0 , = 0,1 , 2, . . . , (12)

wh s th xct soton o (1) nd (2). Cy, o = 0 nd = 1 th nt condtons (2) v 0 = 0 nd1 = 0, spctvy. Moov, o = 2, dfnttoth sds o (4) wth spct to nd ssttt = 0 tootn

0

V

+ 0

0

V

V

= 0,V

=

0

.(13)Indd, om (11), on cn wt

= 0 + 0 0 + 2 02+ 3 03 + .

(14)

By sstttn (14) nto (4), dfnttn oth sds o thstn qton wth spct to , nd thn sttn = 0,w otn

01 + 220 1 1 = 0, 1 = 0 .

(15)

By compn (13) nd (15), t sy to s tht 2 =1/2!0. Hnc, ccodn to (14) th ppoxmton o(1) nd (2) s

= 0 + 0 0 + 12 0 02+ 3 03 + .

(16)

Fthmo, o = 3, dfnttn oth sds o (4)twcwth spct to nd thn sstttn = 0 yds thoown st:

0 ( 0)2 2

V2 V

+ [2 0 0 + 2 0 0] V V+ 0 V = 0, V= 0 .

(17)

By sstttn (16) nto (4), dfnttn oth sds o thstn qton twc wth spct to , nd thn sttn = 0, w otn

0222 221 1

+ [

0

42

+ 0

63

]1

1

+ 01 = 0, V = 0 .(18)

By compn (17) nd (18), w cn concd tht 3 =1/3!0. Ts, ccodn to (16), w cn wt thppoxmton o (1) nd (2) s

= 0 + 0 0 + 12 0 02+ 13! 0 03 + .

(19)

By contnn th ov pocd, w cn sy pov ( 13)

o = 4, 5, 6, . . .. Ts, th poo o th thom s compt.Corollary 2. I is a polynomial, then theRPS method willobtain the exact solution.

It w convnnt to hv notton o th on th ppoxmton . Accodny, tRm dnot th dfnc twn nd ts thyo poynom, whch s otnd om th RPS mthod;tht s, t

Rm

=

=

=+1() 0

! 0

.(20)T nctons Rm cd th th mnd o

th RPS ppoxmton o. In ct, t on hppns thtth mnds Rm com sm nd ppoch zos ppochs nnty. T concpt o cccy s tohow cosy comptd o msd v s wth tht v. o show th cccy o th psnt mthod, wpot th typs o o nctons. T st on s th xcto, Ext, whch s dnd s oows:

Ext :=

= Rm

. (21)


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able 1: A nmc compson o th xct o ncton Ext to Exmp 1 o dfnt vs o n 0,10. Exct soton Rnc [1] Rnc [2] Psnt mthod0 0 0 0 0

0.01 0.00000099 6.295572 1019 5.790000 108 00.1 0.0009 5.83469 1013 8.409000 107 00.5 0.0625 4.937685 10

9

2.195800 106

01 0 1.079378 108 8.284000 107 02 8 1.614569 104 1.732000 107 05 500 1.80785 10+2 1.909000 107 010 9000 1.894851 10+6 3.391999 104 0

Smy, th consctv o, whch s dnotd y Con, ndth sd o, whch s dnotd y Rs, dnd y

Con := +1 = Rm+1 Rm ,Rs

:=

22

+

+ ( ) + , ,

,(22)

spctvy, wh , nd th th-odppoxmton o tht s otnd y th RPS mthod.An xcnt ccont o th stdy o o nyss, whchncds ts dntons, vts, ppctons, nd mthod odvtons, cn ond n [27].

4. Numerical Results and Discussion

T poposd mthod povds n nytc ppoxmtsoton n tms o n nnt pow ss. Howv, ths pctc nd to vt ths soton nd to otnnmc vs om th nnt pow ss. T con-sqnt ss tncton nd th cospondn pctcpocd zd to ccompsh ths tsk. T tnctontnsoms th othws nytc sts nto n xctsoton, whch s vtd to nt d o cccy.

In ths scton, w consd sx xmps to dmonsttth pomnc nd cncy o th psnt tchnq.Tohot ths pp, o th symoc nd nmccompttons pomd sn th Mp 13 sowpck.4.1. Example 1. Consd th oown n nonhomo-nos Ln-Emdn qton:

+ 8 + = 5 4 + 442 30,0 < < ,

(23)

whch s sjct to th nt condtons

0 = 0, 0 = 0. (24)

As w mntond , w sct th st two tmso th ppoxmtons s 0 = 0 nd 1 = 0, thn thth-tnctd ss hs th om

= =2

0 = 22 + 33 + 44

+ + .(25)

o nd th vs o th cocnts , = 2, 3, 4, . . .,w mpoy o RPS othm. To, w constct thsd ncton s oows:

Rs =

=2

1 02

+ 8 =2

01 + 2 =2

0

(5

4

+ 442

30) . (26)Consqnty, th 4th-od ppoxmton o th RPS

soton o (23)nd (6) ccodn to ths sd ncton ss oows:

3 = 3 + 4, (27)whch s wth Cooy 2. It sy to dmonstt thtch o th cocnts o 5 n xpnson (25) vn-shs. In oth wods, =0 = 4=0 . Ts, thnytc ppoxmt soton to (23) nd (6) s dntc to

th xct soton = 43. 1 shows compsontwn th sotos o o mthod tht w otnd

om 4th-od ppoxmton, th optm homotopysymptotc mthod [1], nd th Hmt nctons cooc-ton mthod [2]. Fom th t, t cn sn tht th RPSmthod povds s wth n cct ppoxmt sotonto (23) nd (6). In ct, th sts potd n ths tconm th fctvnss nd cccy o o mthod.

4.2. Example 2. Consd th oown nonn homo-nos Ln-Emdn qton:

+ 8 + 9 + 2 n = 0,0 < < ,

(28)


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1.41.210.80.60.40.20

2

Exact

1.5

1

0.5

0

0.5

1

1.5

2

y

(x)

y5 (x)

y10 (x)

y15 (x)

y20 (x)

x

Figure 1: Pots o th RPS soton to ( 28) nd (29) whn =5,10,15,20 nd th xct soton s on 0,/2.


0 = 1, 0 = 0. (29)Assm tht th nt ss ppoxmton (whch s th

1st ppoxmton) hs th om 1 = 1. Tn, th 10thtnctd ss o th RPS soton o o (28) nd (29)s s oows:

10 = 1 2 2 + 2

8 4 3

486 + 4

3848 5

384010

=5

=02 2 .

(30)

Ts, th xct soton o (28) nd (29) hs nom tht concds wth th xct soton

= =0

2 2 = (/2)

2 . (31)Lt s cy ot n o nyss o th RPS mthod o

ths xmp. F 1 shows th xct soton nd tho ttd ppoxmtons

o

= 5,10, 15, 20. Ts

ph xhts th convnc o th ppoxmt sotonsto th xct soton wth spct to th od o th sotons.

In F 2, w pot th xct o nctons Ext whn = 5, 10, 15, 20, 30, whch ppoch th xs = 0 s thnm o ttons ncss. Ts ph shows tht thxct os com sm s th od o th sotonsncss, tht s, s w poss thoh mo ttons.Ts o ndctos conm th convnc o th RPSmthod wth spct to th od o th sotons. FomF 2, t s sy o th d to comp th nw st oth RPS mthod wth th xct soton. Indd, ths phshows tht th cnt mthod hs n ppopt conv-nc t.

4E 04

3E 04

2E 04

1E 04

0E + 00

Exacterror

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x

y5 (x)

y10 (x)

y15 (x)

y20 (x)

y30 (x)

Figure 2: Pots o th xct o nctons o (28) nd (29) whn

= 5, 10,15, 20 on 0,/2.


+ 2 + 4(2() + (1/2)()) = 0, 0 < < ,(32)


0 = 0,

0 = 0. (33)A w ppy th RPS mthod to sov (32) nd (33), w

constct th sd ncton s oows:

Rs = =2

1 02

+ 2 =1

01

+ 4 2=0 (0) + (1/2)=0 (0) ,(34)

wh =2 0 s th th-tnctd ss thtppoxmts th soton . As w mntond , w sct th st two tms o th ppoxmtons s 0 =0 nd 1 = 0 (whch wod mpy tht 1 = 2 = 0), thnth st w tms o th ppoxmtons o th RPS sotono (32) nd (33)

2 = 2, 3 = 0, 4 = 4,5 = 0, 6 = 2

36, . . . . (35)


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able 2: T vs o th sd o ncton Rs to Exmp 4 whn = 10, 15,20, 25 o dfnt vs o n 0,2. Rs10 Rs15 Rs20 Rs25

0 0 0 0 0

0.1 4.528461 1013 4.528461 1013 4.528461 1013 4.528461 10130.2 1.250167 1012 3.996803 1015 3.996803 1015 3.996803 10150.5 1.785672 10

9

6.611378 1014

1.045275 1013

1.045275 1013

1 3.539988 106 8.587641 109 4.061307 1012 1.395550 10131.5 2.884109 104 3.208217 106 1.902825 108 1.854792 10102 6.215665 103 1.865257 104 7.253518 106 1.909733 107

able 3: A nmc compson o th ppoxmt soton to Exmp 4 o dfnt vs o n 0,2. Rnc [3] Rnc [2] Rnc [4] Psnt mthod0 1 1 1 1

0.1 0.9985979358 0.9986051425 0.9985979358 0.9985979274

0.2 0.9943962733 0.9944062706 0.9943962733 0.9943962649

0.5 0.9651777886 0.9651881683 0.9651777886 0.9651777802

1 0.8636811027 0.8636881301 0.8636811027 0.8636811256

1.5 0.7050419247 0.7050524103 0.7050419247 0.7050452522

2 0.5063720330 0.5064687568 0.5063720330 0.5064651631

Fthmo, w coctth ov sts, thn th 10th-tnctd ss o th RPS soton o s vn s10 = 22 + 4 236 + 128 2510

= 2(2)1 12(2)2 + 13(2)

3 14(2)4

+15(2

)5

.

(36)

Ts, th xct soton o (32) nd (33) hs th nom tht concds wth th xct soton

= 2=1

1+12 = 2 n (1 + 2) . (37)In most - sttons, th Ln-Emdn qton s

too compctd to sov xcty, nd, s st, th s pctc nd to ppoxmt th soton. In th nxt twoxmps, th xct soton cnnot ond nytcy.


+ 2 sn = 0, 0 < < , (38)whch s sjct to th nt condtons

0 = 1, 0 = 0. (39)As w mntond , w sct th nt ss

ppoxmton s 1 = 1, thn th yo ss xpnsono th soton o (38) nd (39) s s oows:

= 1 + 22 + 33 + 44 + . (40)

Consqnty, th 10th-od ppoxmton o th RPSsoton o (38) nd (39) s

10 = 1 sn 16 2 + sn 2240 4 sn 37560 sn 15040 6 61 sn 413063680 13 sn 21632960 8

+ 629sn

53592512000 1319sn

33592512000 + 41sn

116329600010

.(41)O nxt o s to show how th th v n th th-

tnctd ss (3) fcts th ppoxmt sotons. In 2, th sd o hs n cctd o vos

vs o n 0,2 to ms th xtnt o mnttwn th th-od ppoxmt RPS sotons whn =10,15,20,25. As st, 2 stts th pd conv-nc o th RPS mthod y ncsn th ods o ppox-mton. o show th cncy o th RPS mthod, nmccompsons so stdd. 3 shows compsono tht s otnd y th 10th-od ppoxmtono th RPS mthod wth thos sts tht w otnd

y th Adomn dcomposton mthod [3], th Hmtnctons coocton mthod [2], nd th homotopy pt-ton mthod [4]. An, w nd tht o mthod hs sm d o cccy to ths oth mthods.

4.5. Example 5. Consd th oown homonos non-n Ln-Emdn qton:

+ 2 () = 0, 0 < < , (42)whch s sjct to th nt condtons

0 = 0, 0 = 0. (43)


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able 4: A nmc compson o th ppoxmt soton to Exmp 5 whn = 10 o dfnt vs o n 0,1. Rnc [5] Rnc [6] Rnc [7] Rnc [8] Psnt mthod0 0 0 0 0 0

0.1 0.0016 0.0166 0.0016 0.0027 0.0017

0.2 0.0065 0.0333 0.0066 0.0038 0.0067

0.3 0.0145 0.0500 0.0149 0.0152 0.01490.4 0.0253 0.0666 0.0266 0.0341 0.0265

0.5 0.0385 0.0833 0.0416 0.0456 0.0412

0.6 0.0536 0.1000 0.0598 0.0601 0.0589

0.7 0.0700 0.1166 0.0813 0.0935 0.0797

0.8 0.0870 0.1333 0.1060 0.1399 0.1034

0.9 0.1038 0.1500 0.1338 0.1786 0.1298

1 0.1199 0.1666 0.1646 0.2005 0.1588

Hstocy, ths typ o Ln-Emdn qton wsdvd y Bonno [28] n 1956 to dsc wht nowcommony known s Bonno-Et [28, 29] s sphs.Ts s sphs sothm s sphs tht hv nmddd n psszd mdm t th mxmm possmss tht ows hydosttc qm. T dvton ssd on wok y Et [29], nd hnc, th qtons on d to s th Ln-Emdn qton o th scondknd (whch dpnds on n xponnt nonnty). Fo dvton o th Ln-Emdn qton o th scond knd,th d s kndy qstd to ps [3033].

In ct, ths mod pps n Rchdsons thoy othmonc cnts whn th dnsty nd ctc oc on cton s n th nhohood o hot ody n thm qm[10] mst dtmnd. Fo thooh

dscsson o th omton o (42) nd (43) nd th phys-c hvo o th msson o ctcty om hot ods,s [10, 11]. It shod osvd tht ths qton s non-n nd hs no nytc soton.

As w mntond , w sct th nt ss

ppoxmton s 2 = 0, thn th yo ss xpnsono sotons to (42) nd (43) s s oows:

= 22 + 33 + 44 + . (44)Consqnty, th 10th-od ppoxmton o th RPS

soton o (42) nd (43) ccodn to ths nt ss s

10

= 162

11204

118906

6116329608

+ 62922453200010.(45)

As n th pvos xmp, xct sotons do not xst oth Ln-Emdn qtons (42) nd (43). Ts, n 4 wcomp o sts to th sts om th tt. Somo ths sts w otnd n [5] y constctn nytcppoxmtons sd on E tnsom ss; oths wotnd n [6] y sn two ccton mthods to mpovth convnc ov th stndd yo ss sts. Inddton, th tho n [7] ppd th cton ppoxm-ton tchnq nd th thos n [8] ppd th Bok

poynoms xpnson schm. In th ov t, t cn sn tht o sts om th RPS mthod pncpywth th mthods o [5, 7]. In ddton, w nd tht osts w wth mthod [6]. Howv, o sm,wndthtth sts o [8] w wth th mthod o [7], nd s, w nd tht ths sts wth mthod [6]. Tsconcson s son, s th cton-ppoxmton-tchnq soton n [7] s o ow od, nd hnc, t s

vd o cos to = 0. In contst, th soton n [6]nvovs Pd ppoxmton, whch cn mpov th ono convnc.

In th nxt xmp, w show tht th RPS mthod scp o podcn th xct soton to nw vsono th Ln-Emdn qton. Fthmo, w show thtth consctv o s s ndcto n th tton

posss, nd moov, ths o cn sd to stdy thstct nyss o th RPS mthod.

4.6. Example 6. Consd th oown nonn nonho-monos sn nt-v pom:

= sn 1 cos ( ) cos

sn 1 sn1 + 2

+ ()+() + ,0 < <

(46)


1 = 1, 1 = 1, (47)wh s chosn so tht th xct soton s =sn

1 +cos

1.


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able 5: T vs o th consctv o ncton Con to Exmp 6 whn = 10, 15,20, 25 o dfnt vs o n 1,3. Con10 Con15 Con20 Con25

1 0 0 0 0

1.2 5.130672 1016 3.132278 1025 4.104743 1035 1.664029 10451.4 1.050762 1012 2.052770 1020 8.608271 1029 1.116711 10371.6 9.088831 10

11

1.348343 1017

4.293706 1025

4.229738 1033

1.8 2.151960 109 1.345303 1015 1.805285 1022 7.494120 10302 2.505211 108 4.779477 1014 1.957294 1020 2.479596 10272.2 1.861393 107 8.836501 1013 9.004555 1019 2.838529 10252.4 1.014501 106 1.040948 1011 2.292687 1017 1.562104 10232.6 4.407214 106 8.816580 1011 3.785957 1016 5.029219 10222.8 1.610059 105 5.804175 1010 4.491369 1015 1.075143 10203 5.130672 105 3.132278 109 4.104743 1014 1.664029 1019

I w sct th nt ss ppoxmton s 2 = 1/2 12, thn th 10th-tnctd ss o th RPSsoton o (46) nd (47) w

10 = 1 + 1 122 13

6 + 14

24+ 15120 + 1

6

720 + 17

5040+ 1840320 + 1

9

362880 + 110

3628800 .(48)

Fthmo, w spt th ov ppoxmtonsodd nd vn tms, thn t s sy to dscov tht th xctsoton o (46) nd (47) hs th n om tht concdswth th xct soton

10 = 1 136 + 15

120+ 175040 + 1

9

362880 + + 1 122 + 1

4

24 + 16

720+ 18

40320+ 110

3628800+ .

(49)

Ts, th ppoxmt soton o (46) nd (47) hs thn om tht concds wth th xct soton

= sn 1 + cos 1 . (50)Remark. Wh on cnnot know th xct o wth-ot known th soton, n most css th consctvo cn sd s ndcto n th ttonposss. In 5, th vs o th consctv o

nctons Con , = 1,2, 3 o two consctv ppox-mt sotons hv n cctd o vos vs o

n

0,1wth stp sz o

0.1; th o ws to ms

th dfnc twn th consctv sotons tht wotnd om th 10th-od RPS sotons o (46) nd(47). Howv, th comptton sts povd nmc

stmt o th convnc o th RPS mthod. Indd, t sc tht th cccy tht s otnd sn th psntmthod s dvncd y sn n ppoxmton wth ony w ddton tms.In ddton,w cnconcdtht hhcccy cn chvd y vtn mo componnts oth soton. Ts, w tmnt th tton n o mthod.

5. Conclusion

T o o th psnt wok ws to dvop n cnt ndcct mthod to sov th Ln-Emdn-typ qtons osn nt-v poms.W cn concd thtth RPSmthod s pow nd cnt tchnq tht nds n

ppoxmt soton to n nd nonn Ln-Emdnqtons. T poposd othm podcd pdy con-

vnt ss wth sy compt componnts snsymoc comptton sow. T sts otnd y thRPS mthod vy fctv nd convnnt n n ndnonn css cs thy q ss comptton woknd tm. Ts convnnt t conms o thtth cncy o o tchnq w v t mch tppcty n th t o n csss o n ndnonn sn poms.

Acknowledgments

Ts wok ws comptd dn th vst o th tho A.Sm Btnh (ASB) to th Unvst Knsn Mys(UKM), n JnAst 2013, s vstn sch omthmtcs. T thos I. Hshm ndA. S. Btnh t-y cknowd th Gnt povdd y UKM ot o thUnvsty Rsch Fnd DIP-2012-12.

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