Jordan Journal of Mathematics and Statistics (JJMS) 11(1), 2018, pp 1-12

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1 Jordn Journl of Mthemtics nd Sttistics (JJMS) 11(1), 218, pp 1-12 HOMOTOPY REGULARIZATION METHOD TO SOLVE THE SINGULAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND MOHAMMAD ALI FARIBORZI ARAGHI (1) AND SAMAD NOEIAGHDAM (2) Abstrct. In this pper, by combining the regulriztion method (RM) nd the homotopy nlysis method (HAM) new pproch is presented to solve the generlized Abel s integrl equtions (AIEs) of the first kind which is clled the HAMregulriztion method (HRM). The RM is pplied to trnsform the first kind integrl equtions (FKIEs) to the second kind (SK) which depends on the regulriztion prmeter nd the HAM is used to find the solution of AIEs of the SK. The vlidtion of results re illustrted by presenting convergence nlysis theorem. Also, the efficiency nd ccurcy of the HRM re shown by solving three exmples. 1. Introduction Mny pplicble problems were presented in severl fields of sciences nd engineering such s mechnics, physics, plsm dignostics, physicl electronics, nucler physics, optics, strophysics nd mthemticl physics 2, 8, 9, 11, 19] which they cn be trnsformed to the liner nd non-liner forms of the AIEs 21, 22]. In the recent yers, severl computtionl nd semi-nlyticl schemes to solve the singulr problems of the Abel s kind were presented 4, 15, 25, 26]. Furthermore, the 2 Mthemtics Subject Clssifiction. 65R2, 45E1. Key words nd phrses. Homotopy nlysis method, Regulriztion method, Abel s singulr integrl eqution of the first kind. Copyright c Denship of Reserch nd Grdute Studies, Yrmouk University, Irbid, Jordn. Received: April 24, 217 Accepted: Nov. 26,

2 2 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM Chebyshev polynomils 16], the spectrl method 18] nd the Legendre wvelets 24] were pplied to estimte the solution of the first nd second kinds AIEs. The HAM is one of the semi-nlyticl nd flexible methods which ws presented by Lio 12, 13] nd it ws pplied to solve different problems 6, 7, 15, 17]. The RM ws constructed by Tikhonov 2] nd it ws extended by others 5, 14, 23]. In this pper, in order to solve the first kind generlized AIEs of the liner or nonliner form, the HAM nd the RM re combined. The RM is used to convert the first kind AIEs to the SK. When we pply the HAM to solve the AIE, the pproximte solution will be obtined. But, by using the HRM, the exct solution cn be produced. According to 23], this method is depend on the prmeter τ nd the exct solution of IE cn be obtined when τ. The orgniztion of this pper is in the following form: the HRM is introduced to solve the AIE of the first kind in section 2. In this section, simple technique is pplied to trnsform the non-liner IEs to the liner form. The convergence theorem is presented in section 3. In section 4, some pplicble exmples of singulr IEs re solved bsed on the proposed lgorithm. Finlly, section 5 is conclusion. 2. The HAM-Regulriztion method The non-liner form of generlized AIE of the first kind is introduced s 21, 22]: (2.1) g(z) = P (ψ(s)) ds, < β < 1, (p(z) p(s)) β where is given rel vlue, p nd g re known, P (ψ(s)) is nonliner function of ψ(s) tht ψ is n unknown function. Furthermore p is differentible, monotone nd strict incresing function in, b], nd p for every s in the intervl. By using the trnsformtion (2.2) ω(z) = P (ψ(z))

3 HOMOTOPY REGULARIZATION METHOD TO SOLVE... 3 the non-liner IE (2.1) cn be chnged to the liner form s follows: (2.3) g(z) = ω(s) ds, < β < 1. (p(z) p(s)) β Now, the RM is pplied to convert Eq. (2.3) to the SK (2.4) τω τ (z) = g(z) ω κ (s) (p(z) p(s)) β ds, which is depend on the regulriztion prmeter τ. Now, we cn write (2.5) ω τ (z) = 1 τ g(z) 1 τ nd the HAM is employed to solve Eq. (2.5). Let (2.6) N Wτ (z; l) ] = W τ (z; l) 1 τ g(z) + 1 τ ω τ (s) (p(z) p(s)) β ds, W τ (s; l) (p(z) p(s)) β ds, is n opertor of non-liner form where the homotopy prmeter is shown by l, 1] nd W τ (z; l) is n unknown function. The following zero order deformtion eqution cn be produced by the prevlent homotopy method 12, 13] s (2.7) (1 l) L W τ (z; l) ω τ, (z)] = l Ĥ(z) N W τ (z; l)], where the convergence-control prmeter is demonstrted by, the uxiliry function is displyed by Ĥ(z), the uxiliry opertor of liner form is shown by L, the primry conjecture of ω τ (z) is shown by ω τ, (z) nd W τ (z; l) is n unknown function. By putting l =, 1 into Eq. (2.7), since nd Ĥ(z) we hve W τ (z; ) = ω τ, (z), nd W τ (z; 1) = ω τ (z). Now, by using Tylor s theorem, we get (2.8) W τ (z; l) = ω τ, (z) + where (2.9) ω τ,d (z) = 1 d! ω τ,d (z)l d, d W τ (z; l) r d. l=

4 4 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM In order to show the convergence of series (2.8) t l = 1, we ssume tht, Ĥ(z), ω τ,(z) nd L re so correctly selected. Therefore, the following series solution cn be obtined: (2.1) ω τ,d (z) = W τ (z; 1) = ω τ, (z) + ω τ,d (z). According to 5, 12, 13, 15], the d-th order deformtion eqution is defined s follows: (2.11) L ωτ,d (z) υ d ω τ,d 1 (z) ] = Ĥ(z) R d (ω τ,d 1 ) ], where (2.12) R d (ω τ,d 1 ) = ω τ,d 1 (z) + 1 τ ω τ,d 1 (s) (p(z) p(s)) β ds (1 υ d) 1 τ g(z), nd (2.13) υ d =, d 1, 1, d > 1. By choosing Lω τ = ω τ, Ĥ(z) = 1 nd = 1, the following recursive reltion is produced (2.14) ω τ, (z) = 1 τ g(z), ω τ,d (z) = 1 τ ω τ,d 1 (s) ds, d 1. (p(z) p(s)) β The j-th order of regulrized pproximte solution cn be estimted by (2.15) ω τ,j (z) = j ω τ,d (z), d= nd the solution of IE (2.3) cn be obtined by τ. Finlly, the trnsformtion (2.16) ψ = P 1 (ω),

5 HOMOTOPY REGULARIZATION METHOD TO SOLVE... 5 is pplied to find the exct solution of Eq. (2.1). 3. Theoreticl discussion nd lgorithm By presenting the following theorem, the convergence of the HRM to solve the AIE of the first kind (2.3) is discussed. The existence nd uniqueness theorems were demonstrted in 1, 3, 1]. Theorem 1. Let the series solution (3.1) V τ (z) = ω τ, (z) + ω τ,d (z), is convergent where ω τ,d (z) is produced by the d-th order deformtion eqution (2.11) then the regulrized SK IE (2.5) hs n exct solution which is the solution of the series (3.1). Proof Since the series (3.1) is convergent, then (3.2) lim d ω τ,d (z) =. The liner opertor L is pplied to left hnd side of d-th order deformtion eqution (2.11) s follows (3.3) where we hve ] L ω τ,d (z) υ d ω τ,d 1 (z) = L ( ωτ,d (z) υ d ω τ,d 1 (z) )], (3.4) q ] ω τ,d (z) υ d ω τ,d 1 (z) = ω τ,1 (z)+(ω τ,2 (z) ω τ,1 (z))+...+(ω τ,q (z) ω τ,q 1 (z)) = ω τ,q (z), nd (3.5) ω τ,d (z) υ d ω τ,d 1 (z) ] = lim q ω τ,q (z) =.

6 6 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM Therefore Eq. (3.3) is equl with zero nd by using Eq. (2.11) the following reltion cn be obtined ] (3.6) L ω τ,d (z) υ d ω τ,d 1 (z) = Ĥ(z) R d 1 (ωτ,d 1 (z)) =. Since, Ĥ(z) re not equl with zero thus R d 1(ωτ,d 1 (z)) = nd we get R d 1 (ω τ,d 1 (z)) (3.7) = ω τ,d 1 (z) + 1 τ = 1 τ g(z) + ω τ,d (z) + 1 τ d= ω τ,d 1 (s) (p(z) p(s)) ds (1 υ d) 1 ] β τ g(z) d= ω τ,d(s) (p(z) p(s)) β ds = 1 τ g(z) + V τ(z) + 1 τ V τ (s) (p(z) p(s)) β ds. Now, by using Eqs. (3.6) nd (3.7) we cn write (3.8) V τ (z) = 1 τ g(z) 1 τ V τ (s) (p(z) p(s)) β ds, therefore V τ (z) is the exct solution of Eq. (2.5). According to 2] when τ the solution of regulrized AIE of the SK (2.5) is convergent to the solution of the first kind singulr IE (2.3). In order to implement the exmples, the following lgorithm is presented bsed on the proposed pproch. Algorithm: Step 1: Trnsform the AIE (2.1) to the liner form by using Eq. (2.2). Step 2: Apply the RM to construct the regulrized AIE of the SK (2.5). Step 3: Use Eqs. (2.14) to clculte the regulrized series solution (2.15). Step 4: Find the exct solution of liner IE (2.3) by τ.

7 HOMOTOPY REGULARIZATION METHOD TO SOLVE... 7 Step 5: Apply the trnsformtion (2.16) to find the exct solution of non-liner IE (2.1). 4. Smple exmples In this section, some exmples of the first kind generlized AIEs re solved. By using the proposed technique, t first, the non-liner given AIE is converted to the liner form. Then, the HRM is pplied to clculte the exct solution bsed on the proposed lgorithm. The progrms hve been provided by Mthemtic 1. Exmple 1: Let the following non-liner AIE of the first kind 21] (4.1) π + z = The trnsformtion sin 1 (ψ(s)) ds, < z < 2. (z 2 s 2 ) 1 2 (4.2) ω(z) = sin 1 (ψ(s)), ψ(z) = sin(ω(z)), nd the RM, converts Eq. (4.1) to the liner SK IE (4.3) ω τ (z) = 1 τ (π + z) 1 τ ω τ(s) ds. (z 2 s 2 ) 1 2 By using recursive formul (2.14) which is obtined from HRM, we get ω τ, (z) = 1 τ g(z) = π + z, τ (4.4) ω τ,1 (z) = 1 τ ω τ,2 (z) = 1 τ. ω τ,m (z) = 1 τ ω τ, (s) (z 2 s 2 ) 1 2 ω τ,1 (s) (z 2 s 2 ) 1 2 ω τ,m 1 (s) ds. (z 2 s 2 ) 1 2 ds = π2 + 2z 2τ 2, ds = π3 + 4z 4τ 3,

8 8 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM Therefore, the solution of liner IE (4.3) is obtined s follows: (4.5) ω τ (z) = z τ + π τ z τ 2 π2 2τ 2 + z τ 3 + π3 4τ 3 + = z τ 1 1 τ + 1 τ 1 ] 2 τ + + π 3 τ 1 π ( π ) 2 ( π ) ] 3 2τ + + 2τ 2τ = z ( ) ( ) π. τ + 1 2τ + π We know the obtined solution of the HRM is convergent to the exct solution when τ. So, we obtin (4.6) ω(z) = lim τ ω τ (z) = z + 2, nd by substituting ω(z) in (4.2) the exct solution of nonliner IE (4.1) cn be obtined s (4.7) ψ(z) = sin(z + 2). Exmple 2: Consider the non-liner AIE 21] (4.8) 4 z 3 z 3 ln(ψ(s)) 2 = ds. z s By trnsformtion ω(z) = ln(ψ(z)), ψ(z) = e ω(z), nd pplying the RM, the FKIE (4.8) cn be converted to the liner SK AIE s (4.9) ω τ (z) = 4 3τ z 3 1 z ω τ (s) 2 ds. τ z s

9 HOMOTOPY REGULARIZATION METHOD TO SOLVE... 9 By pplying the HAM nd the following recursive formul ω τ, (z) = 1 τ g(z) = 4 3τ z 3 2, (4.1) ω τ,1 (z) = 1 τ ω τ,2 (z) = 1 τ. ω τ,m (z) = 1 τ ω τ, (s) z s ds = πz2 2τ 2, ω τ,1 (s) z 3 2 ds = 8πz τ 3, ω τ,m 1 (s) ds, the regulrized series solution of IE (4.9) cn be produced s (4.11) ω τ (z) = ω τ,m (z) = 4 3τ z 3 πz 2 2 2τ + 8πz 5 ( τ + = 1.33z τ z 1 2 m= Since ω(z) = lim τ ω τ (z) = z therefore, the solution of Eq. (4.8) is obtined in the following form (4.12) ψ(z) = e ω(z) = e z. Exmple 3: In this exmple, the following generlized AIE 21] z 3 2 ). (4.13) 3 1 z 1 3 = ψ 3 (s) ds, (z 4 s 4 ) 1 6 is considered. The trnsformtion ω(z) = ψ 3 (z) nd ψ(z) = ω 1 3 (z) re chnged the non-liner IE (4.13) to the liner form s (4.14) ω τ (z) = 3 1τ z τ ω τ (s) ds. (z 4 s 4 ) 1 6 By pplying the formul (2.14), the series solution of HRM cn be constructed s follows: (4.15) ω τ (z) = ω τ,m (z) = m= 1 3z 3 1τ 3z 11 3 Γ 5 6 4τ 2 Γ ] Γ 13 ] ] 12 + πz4 Γ ] 5 6 Γ 13 ] 12 16τ 3 Γ ] + = z τ + 3 z

10 1 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM Now, in order to clculte the exct solution of SK IE (4.14) we find 3 (4.16) ω(z) = lim ω τ (z) = lim z τ τ τ + 3 z = z 3. Thus ψ(z) = ω(z) 1 3 = z is the solution of Eq. (4.13). 5. Conclusions Generlized AIEs re specil form of Volterr IEs which is modelled in sciences nd engineering problems. In this work, n pplicble nd efficient pproch bsed on the HAM nd the RM ws pplied to solve the first kind AIEs. In this pproch, the RM is combined s n uxiliry scheme with the HAM to find the exct solution. Also, the convergence theorem ws proved to illustrte the solution obtined from the proposed lgorithm is stisfied to the exct solution of the AIE. Finlly, some exmples of the non-liner generlized AIEs were solved bsed on the proposed lgorithm. As furture works, solving the Volterr integro-differentil equtions, system of IEs nd high dimensionl IEs with singulr kernel by using the HRM re suggested. Acknowledgement The uthors would like to thnk the nonymous reviewers for their crful reding nd lso the Islmic Azd University, Centrl Tehrn Brnch for the support of this project. References 1] K.E. Atkinson, An exictnce theorem for Abel integrl equtions, SIAM J. MATH. ANAL. 5 (1974). 2] CT. H. Bker, The numericl tretment of integrl equtions. Oxford: Clrendon Press; ] E. Buckwr, Existence nd uniqueness of solutions of Abel integrl equtions with power-lw non-linerities, Nonliner Anl. 63 (25)

11 HOMOTOPY REGULARIZATION METHOD TO SOLVE ] M. A. Friborzi Arghi nd S. Yzdni, A Method to Approximte Solution of the First kind Abel Integrl Eqution using Nvot s Qudrture nd Simpson s rule, Int. J. Industril Mthemtics, 1 (29) ] M.A. Friborzi Arghi nd S. Noeighdm, Fiboncci-regulriztion method for solving Cuchy integrl equtions of the first kind, Ain Shms Eng J,8(217) ] M.A. Friborzi Arghi nd S. Noeighdm, A novel technique bsed on the homotopy nlysis method to solve the first kind Cuchy integrl equtions rising in the theory of irfoils, Journl of Interpoltion nd Approximtion in Scientific Computing 216 No.1 (216) ] M.A. Friborzi Arghi nd S. Noeighdm, Homotopy nlysis trnsform method for solving generlized Abel s fuzzy integrl equtions of the first kind, IEEE (216). DOI: 1.119/CFIS ] MA. Golberg, Numericl solution of integrl equtions. New York: Plenum Press; ] R. Gorenflo nd S. Vessell, Abel Integrl Equtions, Springer, Berlin, ] H. Hochstt, Integrl equtions, A Wiley- Interscience Publiction, New York, ] E.L. Kosrev, Applictions of integrl equtions of the first kind in experiment physics. Comput. Phys. Commun. 2 (198) ] S.J. Lio, The proposed homotopy nlysis techniques for the solution of nonliner problems, Ph.D. Thesis, Shnghi Jio Tong University, Shnghi, 1992 (in English). 13] S.J. Lio, Beyond Perturbtion: Introduction to Homotopy Anlysis Method, Chpmn & Hll/CRC Press, Boc Rton, ] A. Molbhrmi, An lgorithm bsed on the regulriztion nd integrl men vlue methods for the Fredholm integrl equtions of the first kind, Appl. Mth. Modell. 37 (213) ] S. Noeighdm, E. Zrei nd H. Brzegr Kelishmi, Homotopy nlysis trnsform method for solving Abel s integrl equtions of the first kind, Ain Shms Eng J 7 (216) ] P. Piessens, Computing integrl trnsforms nd solving integrl equtions using Chebyshev polynomil pproximtions, J. Comput. Appl. Mth. 121 (2) ] E. A. Rwshdeh, H. M. Jrdt nd F. Awwdeh, Homotopy nlysis method for delyintegro-differentil equtions, Jordn Journl of Mthemtics nd Sttistics 2(1) ] A. Shoj, A.R. Vhidi nd E. Bbolin, A spectrl itertive method for solving nonliner singulr Volterr integrl equtions of Abel type, Applied Numericl Mthemtics, 112 (217) 79 9.

12 12 MOHAMMAD ALI FARIBORZI ARAGHI AND SAMAD NOEIAGHDAM 19] V.Sizikov nd D. Sidorov, Generlized qudrture for solving singulr integrl equtions of Abel type in ppliction to infrred tomogrphy, Applied Numericl Mthemtics, 16 (216) ] A.N. Tikhonov, On the solution of incorrectly posed problem nd the method of regulriztion, Soviet Mth. 4 (1963) ] A.M. Wzwz, Liner nd Nonliner Integrl Equtions, Methods nd Applictions, Springer, Heidelberg, Dordrecht, London, New York, ] A.M. Wzwz, A First Course in Integrl Equtions, World Scientific, New Jersey, ] A.M. Wzwz, The regulriztion-homotopy method for the liner nd non-liner Fredholm integrl equtions of the first kind, Commun. Numer. Anl. 211 (211) ] S.A. Yousefi, Numericl solution of Abel s integrl eqution by using Legendre wvelets, Appl. Mth. Comput. 175 (26) ] Ş. Yüzbşi nd M. Krcyir, A Glerkin-like pproch to solve high-order integro-differentil equtions with wekly singulr kernel, Kuwit Journl of Science, 43 (216) ] Ş. Yüzbşi nd M. Sezer, A numericl method to solve clss of liner integro-differentil equtions with wekly singulr kernel, Mth. Meth. Appl. Sci. 35 (212) (1) Deprtment of Mthemtics, Centrl Tehrn Brnch, Islmic Azd University, Tehrn, Irn. E-mil ddress: friborzi.rghi@gmil.com; m friborzi@iuctb.c.ir. (2) Deprtment of Mthemtics, Ardbil Brnch, Islmic Azd University, Ardbil, Irn. E-mil ddress: smdnoeighdm@gmil.com, s.noeighdm.sci@iuctb.c.ir.