Numerical Inversion of the Abel Integral Equation using Homotopy Perturbation Method

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1 Numeical Invesion of the Abel Integal Equation using Homotopy Petubation Method Sunil Kuma and Om P Singh Depatment of Applied Mathematics Institute of Technology Banaas Hindu Univesity Vaanasi -15 India Repint equests to O P S; singhom@gmailcom Z Natufosch 65a (1); eceived Novembe 11 9 / evised Decembe 1 9 Many poblems in physics like econstuction of the adially distibuted emissivity fom the line-ofsight pojected intensity the 3-D image econstuction fom cone-beam pojections in computeized tomogaphy etc lead natually in the case of adial symmety to the study of Abel s type integal equation The aim of this communication is to popose a new algoithm fo the Abel invesion by using the homotopy petubation method Key wods: Abel s Invesion; Homotopy Petubation Method 1 Intoduction Abel s integal equation [1] occus in many banches of science The ealiest application due to Mach [] aose in the study of compessible flows aound axially symmetic bodies Usually physical quantities accessible to measuement ae quite often elated to physically impotant but expeimentally inaccessible ones by Abel s integal equation In flame and plasma diagnostics the Abel integal equation 1 ε() I(y)= d y 1 (1) y y elates the emission coefficient distibution function of optically thin cylindically symmetic extended adiation souce to the line-of-sight adiance measued in the laboatoy Obtaining the physically elevant quantity fom the measued one equies theefoe the invesion of the Abel integal and in case the object does not have adial symmety it equies in pincipal the invesion of Radon tansfom Equation (1) is inveted analytically by [3] ε()= 1 π 1 1 di(y) dy 1 () y d(y) As the pocess of estimating the solution function (emissivity) ε() if the data function (pojected intensity) I(y) is given appoximately and only at a discete set of data points is ill-posed since even vey small high fequency eos in the measued I(y) suchas will aise fom expeimental eos photon counting noise and noise in the electonics might cause lage eos in the econstucted solution ε() This is due to the fact that the fomula () equies diffeentiating the measued data In fact two explicit analytic invesion fomulae wee given by Abel [1] but thei diect application amplifies the expeimental noise inheent in the adiance data significantly [] In 198 a thid analytic but deivative fee invesion fomula was obtained by Deutsch and Beniaminy [5] to avoid this poblem In addition many numeical invesion methods [5 1] have been developed with vaying degee of success with the inheent limitations of all measued data Consequently the diect use of () is esticted and numeical methods become impotant The aim of the pesent pape is to pesent and analyse a new algoithm fo the numeical solution of Abel s integal equation by using the homotopy petubation method (HPM) Basic Idea of Homotopy Petubation Method In this method using the homotopy technique of topology a homotopy is constucted with an embedding paamete p [1] which is consideed as a small paamete This method became vey popula amongst the scientists and enginees even though it involves continuous defomation of a simple poblem into the moe difficult poblem unde consideation Most of the petubation methods depend on the existence of a small petubation paamete but many non / 1 / $ 6 c 1 Velag de Zeitschift fü Natufoschung Tübingen

2 678 S Kuma and O P Singh Numeical Invesion of Abel Integal Equation linea poblems have no small petubation paamete at all Many new methods have been poposed in the late nineties to solve such nonlinea equations devoid of such small paametes [15 18] Late 199s saw a suge in applications of homotopy theoy in the scientific and engineeing computations [19 ] When the homotopy theoy is coupled with the petubation theoy it povides a poweful mathematical tool [3 5] A eview of ecently developed methods of nonlinea analysis can be found in [6] To illustate the basic concept of the homotopy petubation method (HPM) conside the following nonlinea functional equation: A(u)= f () conditions B Ω with the bounday ( u u ) = Ω n (3) whee A is a geneal functional opeato B is a bounday opeato f () is a known analytic function and Ω is the bounday of the domain Ω The opeato A is decomposed as A = L + N wheel is the linea and N is the nonlinea opeato Hence (3) can be witten as L(u)+N(u) f ()= Ω We constuct a homotopy v( p) : Ω [1] R satisfying H(v p)=(1 p)[l(v) L(u )] + p[a(v) f ()] = p [1] Ω () Hence H(v p)=l(v) L(u )+pl(u )+p[n(v) f ()] (5) = whee u is an initial appoximation fo the solution of (3) As H(v)=L(v) L(u ) and H(v1)=A(v) f () (6) it shows that H(v p) continuously taces an implicitly defined cuve fom a stating point H(u ) to a solution H(v1) The embedding paamete p inceases monotonously fom zeo to one as the tivial linea pat L(u)= defoms continuously to the oiginal poblem A(u)= f () The embedding paamete p [1] can be consideed as an expanding paamete [19] to obtain v = v + pv 1 + p v + (7) The solution is obtained by taking the limit as p tends to 1 in (7) Hence u = lim v = v + v 1 + v + (8) p 1 The seies (8) conveges fo most cases and the ate of convegence depends on A(u) f () [19] 3 Method of Solution By change of vaiables Abel s integal equation (1) educes to I ( 1 ε( ) y)= d y 1 y y which may be witten as 1 η() I 1 (y)= d whee I 1 (y)=i ( ) y y y and η()=ε ( ) (9) To solve (9) by HPM we conside the following convex homotopy: [ 1 ] L(y) (1 p)l()+p dy I 1 () = (1) y We seek the solution of (1) in the following fom: L()= i= p i L i () (11) whee L i () i = 13 ae the functions to be detemined We use the following iteative scheme to evaluate L i () Substituting (11) in (1) and equating the coefficients of p with the same powe one gets p : p 1 : p : p 3 : L ()= L 1 ()=I 1 () L ()=L 1 () 1 L 1 (y) y dy 1 L (y) L 3 ()=L () dy y p n+1 : L n+1 ()=L n () 1 Hence the solution of (9) is given by η()=lim L()= p 1 i= L n (y) y dy (1) L i () (13)

3 S Kuma and O P Singh Numeical Invesion of Abel Integal Equation 679 Numeical Examples The simplicity and accuacy of the poposed method is illustated by the following numeical examples by computing the absolute eo E() = η() η() whee η() is the exact solution and η() is an appoximate solution of the poblem obtained by tuncating the seies (13) Example 1: Conside the Abel integal equation of the fist kind 1 η() d = 16 y y 15 (1 y)5/ (19 + 7y) (1) fo y 1 with the exact solution η()=(1 ) (1 + 1) Using (1) we compute the vaious iteates L i () and they ae given as follows: p : L ()= p 1 : L 1 ()= (1 )5/ (19 + 7) p : L ()= π 3 ( 1)3 ( + 9) (1 )5/ (19 + 7) Figues 1 and show the compaison between the exact solution η() (solid line) and the appoximate solution η() (dotted line) obtained by tuncating (13) at the level n = 3 and the absolute eo E() espectively The othe fou examples ae diffeent in the sense that thei invesions ae evaluated using the following fom of Abel s integal equation: y η() I 1 (y)= d y Example : We now take the following Abel integal equation: y η() d = πy y (15) Fig 1 (colou online) Exact (solid line) and the appoximate solution (dotted line) of the Abel integal equation (1) in Example 1 Fig 3 (colou online) Exact (solid line) and appoximate solutions (dotted line) of the Abel integal equation of fist kind (15) in Example Fig Absolute eo E() fo Example 1 Fig Absolute eo E() fo Example

4 68 S Kuma and O P Singh Numeical Invesion of Abel Integal Equation with the exact solution η() = The vaious iteates L i () ae given as follows: p : L ()= p 1 : L 1 ()= π p : L ()= π 3 ( 1)3 ( + 9) p 3 : p : L 3 ()= π π3/ + π 3 L ()= π π3/ + 3π π 5/ 15 The seies (13) is tuncated at level n = to obtain Figues 3 and conveying the same infomation fo Example as Figues 1 and did fo Example 1 Example 3: In this example we conside the following Abel integal equation: y η() y d = 16y5/ 15 (16) with the exact solution η()= Using (1) we compute the vaious iteates L i () and they ae given as follows: p : L ()= p 1 : L 1 ()= / p : L ()= / π3 3 Fig 5 (colou online) Exact (solid line) and appoximate solution (dotted line) of the Abel integal equation of fist kind (16) in Example 3 Fig 6 Absolute eo E() fo Example 3 p 3 : L 3 ()= / π π7/ p : L (x)= x5/ πx πx7/ π x 1 The coesponding gaphs ae shown in Figues 5 and 6 with the level of tuncation being n = 36 fo Example 3 Example : Conside the Abel integal equation y [{ } { η() d = 8y3/ 1 5 y 3 π F 1 7 ] }y (17) with the exact solution η()=efi(x) wheeefi(x) is the imaginay eo function The vaious iteates L i () ae given as follows: p : L ()= [{ } { p 1 : L 1 ()= 83/ π F 1 7 ] } p : L ()= ( π 1 e + ) π efi(x) [{ } { + 83/ π F 1 7 ] } p 3 : L 3 ()= π + e π + π π efi(x) [{ { ] 1 3 π F } } [{ } { πx 5/ 1 7 F / 3 π F [{ } { ] } 5 ] }

5 S Kuma and O P Singh Numeical Invesion of Abel Integal Equation 681 Fig 7 (colou online) Exact (solid line) and appoximate solution (dotted line) of the Abel integal equation of fist kind (17) in Example Fig 9 (colou online) Exact (solid line) and appoximate solution (dotted line) of the Abel integal equation of fist kind (18) in Example 5 Fig 8 Absolute eo E() fo Example We have dawn Figues 7 and 8 by taking n = 8 fo Example Example 5: Conside the Abel integal equation of the fist kind y η() d = π ( yj y ) ( y 3/ J5/ ) (18) y with the exact solution η()=j ()wheej () is the Bessel function of the fist kind The vaious iteates L i () ae given as follows: p : L ()= p 1 : L 1 ()= π ( J ) ( ) 3/ J5/ p : L ()= π ( J ) ( ) 3/ J5/ [{ } { } π F 3 ] Fig 1 Absolute eo E() fo Example 5 p 3 : L 3 ()= π + π ( J ( ) 3/ ) J5/ + π [{ 1F 1 } { 1 3 } ] } { ] 5 } 3 π3/ 1F [{ 1 π3 1 1 F [{ } { } ] Calculating the tems up to L 3 () wehavedawnthe Figues 9 and 1 showing the compaison and the absolute eo between the exact and appoximate solutions espectively fo Example 5 5 Conclusions Fom the given numeical examples and the Figues 1 1 we conclude that the HPM is a vey poweful and simple tool fo solving Abel s integal equations of the fist kind

6 68 S Kuma and O P Singh Numeical Invesion of Abel Integal Equation Acknowledgements The fist autho acknowledges the financial suppot fom Rajiv Gandhi National Fellowship of the Univesity Gant Commission New Delhi India unde the JRF scheme The authos would like to thank the anonymous efeees fo thei helpful comments [1] N H Abel J Reine Angew Math (186) [] L Mach Wien Akad Be Math Phys Klasse (1896) [3] F G Ticomi Integal Equations Intescience New Yok 1975 [] G N Minebo and M E Levy SIAM J Nume Anal (1969) [5] M Deutsch and I Beniaminy Appl Phys Lett 1 7 (198) [6] MDeutschApplPhysLett 37 (1983) [7] S A Yousefi Appl Math Comput (6) [8] I Beniaminy and M Deutsch Comput Phys Commun 7 15 (198) [9] D A Muio D G Hinestoza and C E Mejia Comput Math Appl 3 3 (199) [1] S Bhattachaya and B N Mandal Appl Math Sci 1773 (8) [11] S Ma H Gao L Wu and G Zhang J Quant Spectosc Radiat Tansfe (8) [1] S Ma H Gao G Zhang and L Wu J Quant Spectosc Radiat Tansfe (7) [13] O P Singh V K Singh and R K Pandey Int J Nonlinea Sci Nume Simul (9) [1] O P Singh V K Singh and R K Pandey J Quant Spectosc RadiatTansfe (1) [15] D D Ganji and A Rajabi Int Commun Heat Mass Tanfe (6) [16] J H He Int J Nonlinea Mech (1999) [17] S J Liao Int J Nonlinea Mech (1995) [18] S J Liao Eng Anal Bounday Element 91 (1997) [19] J H He Comput Meth Appl Mech Eng (1999) [] J H He Int J Nonlinea Mech () [1] C Hillemeie Int J Optim Theoy Appl (1) [] J H He Commun Nonlinea Sci Nume Simul 3 9 (1998) [3] J H He Appl Math Comput () [] J H He Appl MathComput () [5] D D Ganji G A Afouzi H Hosseinzadeh and R A Talaposhti Phys Lett A 371 (7) [6] J H He Int J Nonlinea Sci Nume Simul 1 51 ()

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