Modified Iterative Method For the Solution of Fredholm Integral Equations of the Second Kind via Matrices

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1 Modified Ieraive Mehod For he Soluion of Fredholm Inegral Equaions of he Second Kind via Marices Shoukralla, E. S 1, Saber. Nermein. A 2 and EL-Serafi, S. A. 3 1s Auhor, Prof. Dr, faculy of engineering and echnology, Fuure Universiy, Egyp shoukrala@homail.com 2s Auhor, Pos Graduae suden, Faculy of Elecronic Universiy, Egyp engnermeinn#homail.com 3s Auhor, Prof. Dr, Faculy of Elecronic Universiy, Egyp said-elserafi@yahoo.com Absrac The Tradiional Ieraive echnique is modified by expanding boh he kernel and he known funcions ino Maclaurin polynomials of he same degree. The given modified mehod gives a very simple form for he ieraed kernels via he well - known Hilber marix. Thus, he ieraive soluions of an inegral equaion of he second kind can be reduced o he soluion of a marix equaion, whereas only one coefficien marix is required o be compued. Therefore, compuaional complexiy can be considerably reduced and much compuaional ime can be saved. The convergence of he given ieraed soluion is sudied and he hree condiions are given concluded resuls, figures, and Tables of calculaions are observed during he soluion of some numerical examples using MaLab. Keywords: Inegral Equaions, Ieraive Mehods, Approximae Soluions. Marix Treamen 1. Inroducion Inegral equaion is encounered in a variey of applicaions in poenial heory, geophysics, elecriciy and magneism, radiaion, and conrol sysems [1]. Many mehods of solving Fredholm inegral equaion of he second kind have been developed in recen years [1,3,6,7,13], such as quadraure mehod, collocaion mehod and Galerkin mehod, expansion mehod, produc-inegraion mehod, deferred correcion mehod, graded mesh mehod, and Perov Galerkin mehod. In addiion, he ieraed kernel mehod is a radiional mehod Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 95

2 for solving he second kind. However, his mehod also requires a huge size of calculaions. The objecive of his paper is o esablish a promising ieraive algorihm ha can be easily programmed on Malab, and give numerical examples wih ables and figures of he obained resuls. Compuaional modificaions are performed on he ieraive algorihm presened in [9], where he procedure beginning by replacing he kernel of an inegral equaion approximaely by a degenerae kernel [4,5,8] in a marix form using Maclaurin polynomial of degree n, whereas he daa funcion is approximaed by Maclaurin polynomial of he same degree n [2]. Owing o he simpliciy of some operaional marix of inegraion, he ieraed kernels is represened in a very simple form via Hilber marix. This simplifies he presen ieraive algorihm and reduces he problem of compuing ieraive soluions o he compuaion of only one marix. Despie of he advanages of mehods [4,5] here was apparen higher cos comparing wih he presen mehod, ha minimizes he compuaional effor and smooh he round-off errors ou. The convergence of he given modified ieraive soluions is sudied and hree condiions for he exisence of he soluions are given. Due o he simple form of he obained ieraed kernels, which is sraighforward and convenien for compuaion, The presen mehod may be generalized o solve boh second kind and well-posed singular inegral equaions of he firs kind [10,11]. 2. Modified Ieraive Mehod Consider he Fredholm inegral equaion of he second kind x K x, y ydy f x q 1 (1) where he funcion f x and he kernel K x, y are given. The kernel, in he square x, y in he xy plane. The funcion x unknown required soluion. soluion The given ieraed Algorihm sars wih an iniial approximaion K x y is defined is he 0 x o he x of inegral equaion (1) and hen generaes a sequence of soluions q x q 0 ha converges o x q q-1 such ha f x -f x q f x < δ; δ > 0 Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 96

3 0 Afer he iniial soluion f x is chosen, he sequences of approximae soluions can be generaed by compuing β q q-1 f x = f x + K x, y f ydy α q 1 (2) 0 If we begin by x 0, hen we ge where he ieraed kernels where q q s x f x K x, y f ydy q 1 (3) s1 K s x, y can be found by he recurrence form s1 s K x, y K x, z K z, y dz ; s 2 Approximae he kernel x y 1,, K x y K x y q 1 (4) K, using Maclaurin polynomial for funcions of wo variables of degree n, hen we can pu i in he marix form K x, y P y L n KL n P x where K k ij 1s an n +1 n +1 coefficiens marix whose enries are defined by (5) and he marix The marix i j K x, y ; i j n k i j ij x y i, j o, n xy, 0,0 0 ; i+ j n n L 1s an n 1 n 1 x marix defined by Ln = diag... 0! 1! n! P of order n 1 n 1 is defined o be (6) (7) Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 97

4 P x p o x p... p 1 x n x (8) where i pi x = x for i = 0;n Now, we define he Maclaurin operaional inegraion marix,h, o be H P yp ydy Where = β BβH Bβ - α BαH Bα 2 n 2 n Bα = diag1 α α... α ;Bβ = diag 1 β β... β Here H is he well - known Hilber marix of order (n +1) n +1 wih elemens H ij = i + j Subsiuing (5) ino (4) and by virue of (9) he ieraed kernels become (s-1) s K x, y = P x L n K L n H L n K L n P y Also, approximaing he daa funcion where f (9) (10) (11) x in Maclourin polynomial of degree n yields f x = P x F ; F = f o f 1... fn (12) i 1 d f x f i = ; i = 0,n i! i dx x=0 Now, Subsiuing (11) and (12) ino (3) we find ha q f x = P x F q β s -1 + P x K H K Py P yfdy s=1α Again using he operaional marix given by (9), we ge (13) Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 98

5 where Le q q s f x = P x I n+1 + K H F 1 s=1 K = Ln K Ln q Q is he n +1 1 ieraive coefficiens vecor defined by q q (14) q s Q a ij = I n+1 + K H (15) s=1 Then we have If C = K H,hen he approximae soluion of he following hree condiions is saisfied Where ρcc is he specral radius of Tha is, o find he ieraive soluions compue he marix K given by (6). q q f x = P x Q F q f x converges o he exac x n C = max C ij < 1; 1 i n j=1 n C = max C ij < 1; 1 j n i=1 2 C = ρ CC < 1 2 CC. if one q x of inegral equaion (1) i is required only o 3. Compuaional Resuls In his par we ry o apply he old ieraed kernel echnique and he modified echnique via marices o approximae he soluion of Fredholm inegral equaion of he second kind. We will use suiable algorihms and MATLAB sofware, hen we will compare he Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 99

6 exac soluion wih he approximae one using suiable number of n poins and finally graph he resuls. The compuaion is made on a personal compuer using MATLAB Example(1) ( x ) 1 x ( ) d Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page Whose exac soluion is given by ( x) 1 2x Table (1) he approximae soluion using he old ieraed echnique N ( x ) 1 x x x x x x x x x x x x x x x x x x x x x x x x x x

7 x x x x x x x x x x x x x x x x x x The CPU ime is equal o Seconds. As he above able show, we reached he exac soluion afer 44 ieraions. Table (2) The exac and approximae soluion of he old ieraed echnique N x The exac sol The approximae sol The error n Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 101

8 Figure(1) Table (3) he approximae soluion using he modified echnique via marices N ( x ) N ( x ) 1 1.0*x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x The CPU ime is equal o Seconds As he above able show, we reached he exac soluion afer 18 ieraions. Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 102

9 Table (4) The exac and approximae soluion of he modified echnique via marices N x n The exac sol The approximae sol The error e e e e e e e e e e-06 Figure(2) Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 103

10 Example(2) Whose exac soluion is given by ( x) e x ( x ) e 1 ( ) d Table (5) he approximae soluion using he old ieraed echnique N ( x ) 1 exp(x) exp(x) xp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) x Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page

11 29 exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) exp(x) The CPU ime is equal o Seconds. As he above able show, we reached he exac soluion afer 41ieraions. Table (6) The exac and approximae soluion of he old ieraed echnique N x n The exac sol The approximae sol The error e e e e e e e e e e e-13 Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 105

12 Figure(3) Table (7) he approximae soluion using he modified echnique via marices N ( x ) *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 106

13 *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x *x^ *x^ *x^ *x The CPU ime is equal o Seconds As he above able show, we reached he exac soluion afer 20 ieraions. Table (8) The exac and approximae soluion of he modified echnique via marices N x n The exac sol The approximae sol The error Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 107

14 Figure(4) 4. Conclusion A simple Ieraive Algorihm for he soluion of Fredholm Inegral Equaions of he second kind has been presened. The given mehod gives a very simple form for he ieraed kernels via he well - known Hilber marix. Thus, he ieraive soluions of an inegral equaion of he second kind can be reduced o he soluion of a marix equaion, whereas only one coefficien marix is required o be compued. Therefore, compuaional complexiy can be considerably reduced and much compuaional ime can be saved. The new proposed approach needs a small number of ieraions o provide an exac resul, ha proofs he power of he presened Algorihm, and simulaes o find ou he relaion beween he inegral equaions and Hilber Marix. Briefly words he Advanages of he given modified mehod Maye be concluded as Decrease he CPU ime In a specacular way. less number of ieraions o reach he ieraive soluion Easy o compue because only he kernel marix is required o find he ieraive soluion I is more powerful if he kernel funcion is complicaed as in our new mehod we differeniae he kernel o ge he soluion bu in he old one we inegrae Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 108

15 References [1] Andrei D. Polyanin & Alexander V. Manzhirov, "Handbook of Inegral Equaions ",CRC Press [2] Burden Richard L. & Faires J. Douglas "Numerical Analysis", PWS Publishing Company, Boson,2010. [3] C.T.H. Baker, The Numerical Treamen of Inegral Equaions, Clarendom Press, Oxford, 4 h Ediion [4] Guangqing L., Gnaneshwar N., Ieraion mehods for Fredholm inegral equaions of he second kind,compuers & Mahemaics wih Applicaions, Volume 53, Issue 6, March 2007, Pages [5] Graham I.G., S. Joe, I.H. Sloan, Ieraed Galerkin versus ieraed collocaion for inegral equaions of second kind, IMA J. Numer. Anal., 5 (1985), pp [6] k.e. Akinson, The numerical Soluion of Inegral Equaions of he second kind, Cambridge Universiy pree, Cambridge, [7] M.C.De Bonis,C.Lauria,"Numerical Treamen of Second Kind Fredholm Inegral Equaions Sysems on Bounded Inervals, Journal of Compuaional and Applied Mahemaics, 2008, Pages [8] H. Kaneko, Y. Xu, Super convergence of he ieraed Galerkin mehods for Hammersein equaions, SIAM J. Numer. Anal., 33 (1996), pp [9] Shoukralla, E. S., EL-Serafi, S. A., Saber, Nermein A,"A marix Ieraive echnique for he soluion of Fredholm Inegral Equaions of he second kind" "Elecronic Journal of Mahemaical Analysis and Applicaions" Vol. 4(1) Jan. 2016, pp ISSN: [10] Shoukralla, E. S., "A Algorihm For The Soluion Of a Cerain Singular Inegral Equaion Of The Firs Kind ", Inern. J. Compuer Mah., Vol. 69, pp , Gordon and Breach Science, England, June [11] Shoukralla, E. S., "Approximae Soluion o weakly singular inegral equaions", J. Appl. Mah. Modeling, Elsevier, England, Vol. 20, pp , November [12] S.S. Bellale and S.B. Birajdar, On Exisence Theorem for Nonlinear Funcional Two Poin Boundary Value Problems in Banach Algebras, In. Jr. Universal Mahemaics and Mahemaical Sciences 1(1) (2015), [13] S.S. Bellale, S.B. Birajdar and D.S. Palimkar, Exisence Theorem for Absrac Measure Delay Inegro-Differenial Equaions, Applied and Compuaional Mahemaics 4(4) (2015), , doi: /j.acm [14] Verlane A. F., Cazakov V. C.,"Inegral Equaions", Nauka Domka, Kiev, USSR, [15] Z. Chen, C.A. Micchelli, Y. Xu, Fas collocaion mehods for second kind inegral equaions, SIAM J. Numer. Anal., 40 (2002), pp [16] Sajane B.A., New Exensions of Finehankel Type Transformaion on Generalized Funcion, Inernaional Journal of Mahemaics and Mahemaical Sciences, 1(1) (2015) Shoukralla E.S 1, Saber Nermein A. 2 and ElSerai S.A. 3 Page 109