Numerical Solution of Second Kind Linear Fredholm Integral Equations Using QSGS Iterative Method with High-Order Newton-Cotes Quadrature Schemes

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1 Malaysia Joural of Mathematical Scieces 5(1): (2011) Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes 1 Mohaa Sudaram Muthuvalu ad 2 Jumat Sulaima 1,2 School of Sciece ad Techology, Uiversiti Malaysia Sabah, Locked Bag 2073, Kota Kiabalu, Sabah, Malaysia sudaram_at2@yahoo.com ad jumat@ums.edu.my ABSTRACT The mai purpose of this paper is to examie the effectiveess of the Quarter-Sweep Gauss-Seidel (QSGS) method i solvig the dese liear systems geerated from the discretizatio of the liear Fredholm itegral equatios of the secod kid. I additio, the applicatios of the various orders of closed Newto-Cotes quadrature discretizatio schemes will be ivestigated i order to form liear systems. Furthermore, the basic formulatio ad implemetatio of the proposed method are also preseted. The umerical results of test examples are also icluded i order to verify the performace of the proposed method. Keywords: Liear Fredholm equatios, Newto-Cotes quadrature, Gauss-Seidel, Quartersweep iteratio INTRODUCTION Itegral equatios have bee oe of the pricipal tools i various areas of sciece such as applied mathematics, physics, biology ad egieerig. O the other had, itegral equatios are ecoutered i umerous applicatios i may fields icludig cotiuum mechaics, potetial theory, geophysics, electricity ad magetism, kietic theory of gases, hereditary pheomea i physics ad biology, reewal theory, quatum mechaics, radiatio, optimizatio, optimal cotrol systems, commuicatio theory, mathematical ecoomics, populatio geetics, queuig theory, medicie, mathematical problems of radiative equilibrium, particle trasport problems of astrophysics ad reactor theory, acoustics, fluid mechaics, steady state heat coductio, fracture mechaics, ad radiative heat trasfer problems (Wag (2006)). From few types of the itegral equatios, the most frequetly ivestigated itegral equatios are Fredholm liear equatios ad its oliear couterpart. However, i this paper, liear Fredholm itegral equatios of the secod kid are cosidered.

2 Mohaa Sudaram Muthuvalu & Jumat Sulaima Geerally, secod kid liear itegral equatios of Fredholm type i the geeric form ca be defied as follows ( ) (, ) ( ) = ( ), [ a, b] λ y x K x t y t dt f x Γ Γ = λ 0 (1) where the parameter λ, kerel K ad free term f are give, ad y is the ukow fuctio to be determied. Kerel K is called Fredholm kerel if the kerel i Equatio (1) is cotiuous o the square S = { a x b, a t b} or at least square itegrable o this square ad it is also assumed to be absolutely itegrable ad satisfy other properties that are sufficiet to imply the Fredholm alterative theorem. Meawhile, Equatio (1) also ca be rewritte i the equivalet operator form ( λ κ ) y f where the itegral operator defie as follows κ y t = (2) = K x t y t dt. (3) ( ) (, ) ( ) Γ Theorem (Fredholm Alterative) (Atkiso (1997)) Let χ be a Baach space ad let κ : χ χ be compact. The the equatio ( λ κ ) y f =, λ 0 has a uique solutio x χ if ad oly if the homogeeous equatio ( λ κ ) z 0 such a case, the operator λ κ : χ χ = has oly the trivial solutio z = 0. I 1 1 has a bouded iverse ( λ κ ) 1 oto. Defiitio (Compact operators) (Atkiso (1997)) Let χ ad Υ be ormed vector space ad let κ : χ Υ be liear. The κ is compact if the set { κ x x x 1} has compact closure i Υ. This is equivalet to sayig that for every bouded sequece { x } χ, the sequeces { κ x } has a subsequece that is coverget to some poits i Υ. Compact operators are also called completely cotiuous operators. 86 Malaysia Joural of Mathematical Scieces

3 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes I may applicatio areas, umerical approaches were used widely to solve Fredholm itegral equatios of the secod kid tha the aalytical method. To solve Equatio (2) umerically, we either seek to determie a approximate solutio by usig the quadrature method (Laurie (2001); Li (2003); Muthuvalu ad Sulaima, (2008b; 2009)) ( λ κ ) I y = f (4) where I is the idetity matrix ad κ is the approximatio of the κ which is obtaied by discretizatio of κ by a poit quadrature method, or use the projectio method ( λ κ ) P y = P f (5) where y V ad P : C V is a projectio operator i a chose fiite dimesioal space V ; see Kaeko (1989), Che et al. (2002), Malekejad ad Kajai (2003), Asady et al. (2005), Kajai ad Vecheh (2005), Xiao et al. (2006), Che et al. (2007), Log ad Nelakati (2007), ad Oladejo et al. (2008). Such discretizatios of itegral equatios lead to dese liear systems ad ca be prohibitively expesive to solve as, the order of the liear system icreases. Thus, iterative methods are the atural optios for efficiet solutios. Cosequetly, the cocept of the half-sweep iterative method has bee proposed by Abdullah (1991) via the Explicit Decoupled Group (EDG) method to solve two-dimesioal Poisso equatios. Half-sweep iteratio is also kow as the complexity reductio approach (Hasa et al. (2007)). Sice the implemetatio of half-sweep iteratios will oly cosider half of all iterior ode poits i a solutio domai. Followig to that, further studies o the applicatios of the half-sweep iterative methods have bee reviewed by Yousif ad Evas (1995), Abdullah ad Ali (1996), Othma et al. (2000), Muthuvalu ad Sulaima (2008a; 2008b; 2009; 2011), Sulaima et al. (2004a; 2007; 2008a) ad Abdullah et al. (2006). I 2000, Othma ad Abdullah exteded the cocept of half-sweep iteratio by itroducig quarter-sweep iterative method via the Modified Explicit Group (MEG) iterative method to solve two-dimesioal Poisso equatios. Further studies to verify the effectiveess of the quarter-sweep iterative methods have bee carried out by Othma ad Abdullah (2001), Malaysia Joural of Mathematical Scieces 87

4 Mohaa Sudaram Muthuvalu & Jumat Sulaima Hasa et al. (2005), Sulaima et al. (2004b), Hasa et al. (2008), Sulaima et al. (2008b), ad Sulaima et al. (2010). However, i this paper, we examied the applicatios of the half- ad quarter-sweep iteratio cocepts with Gauss-Seidel (GS) iterative method by usig approximatio equatio based o Newto-Cotes quadrature schemes for solvig problem (1). The stadard GS iterative method is also called as the Full-Sweep Gauss-Seidel (FSGS) method. Meawhile, combiatios of the GS method with half- ad quarter-sweep iteratios are called as Half-Sweep Gauss-Seidel (HSGS) ad Quarter-Sweep Gauss-Seidel (QSGS) methods respectively. The remaider of this paper is orgaized i followig way. I ext sectio, the formulatio of the full-, half- ad quarter-sweep quadrature approximatio equatios based o repeated Newto-Cotes schemes will be elaborated. The latter sectio of this paper will discuss the formulatios of the FSGS, HSGS ad QSGS iterative methods ad some umerical results will be show to assert the effectiveess of the proposed method. Besides that, aalysis o computatioal complexity is also give ad the cocludig remarks are give i fial sectio. QUARTER-SWEEP QUADRATURE APPROXIMATION EQUATION As explaied i previous sectio, discretizatio method based o quadrature schemes was used to costruct approximatio equatios for problem (1) by replacig the itegral to fiite sums. Geerally, quadrature method ca be defied as follows b a ( ) = j ( j ) + ε ( ) y t dt A y t y (6) j= 0 where t j ( j 0,1,2,, ) itegratio iterval [ a, b ], A j ( j 0,1,2,, ) do ot deped o the fuctio y ( t ) ad ( y) = is the abscissas of the partitio poits of the = is umerical coefficiets that ε is the trucatio error of Equatio (6). Meawhile, Figure 1 shows the fiite grid etworks i order to form the full-, half- ad quarter-sweep quadrature approximatio equatios. 88 Malaysia Joural of Mathematical Scieces

5 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes Based o Figure 1, the full-, half- ad quarter-sweep iterative methods will compute approximate values oto ode poits of type oly util the covergece criterio is reached. Accordig to Abdullah (1991) ad, Othma ad Abdullah (2000), the approximatio solutios for the remaiig poits are calculated by usig direct methods. h (a) 2h (b) 4h (c) Figure 1: (a), (b) ad (c) show distributio of uiform ode poits for the full-, half- ad quarter-sweep cases respectively However, i 2009, Muthuvalu ad Sulaima carried out a study to ivestigate the applicatios of the half-sweep iteratio i solvig dese liear system geerated from the discretizatio of the secod kid Fredholm itegral equatios usig high-order Newto-Cotes schemes. From the results obtaied, it has show that applicatios of the half-sweep iteratio with high-order Newto-Cotes discretizatio schemes reduce the accuracy of the umerical solutios ad it is due to the computatioal techique for calculatig the remaiig poits by usig direct method. Malaysia Joural of Mathematical Scieces 89

6 Mohaa Sudaram Muthuvalu & Jumat Sulaima Thus, i this paper, we will use secod-order Lagrage iterpolatio method to compute the remaiig poits for both half- ad quarter-sweep iteratios i order to overcome the problem metioed i Muthuvalu ad Sulaima (2009). Formulatios to compute the remaiig poits usig secod order Lagrage iterpolatio for half- ad quarter-sweep iteratios are defied i Equatios (7) ad (8) respectively as follows yi 1 + yi+ 1 yi+ 3, i = 1,3,5,, yi = yi 1 + yi+ 1 yi 3, i = (7) ε ( y) yi 2 + yi+ 2 yi+ 6, i = 2,6,10,, yi 2 + yi+ 2 yi 6, i = yi =. (8) yi 1 + yi+ 1 yi+ 3, i = 1,3,5,, yi 1 + yi+ 1 yi 3, i = By applyig Equatio (6) ito Equatio (1) ad eglectig the error,, a system of liear equatios ca be formed for approximatio values of y ( t ). The followig liear system geerated usig quadrature method ca be easily show i matrix form as follows where M y = f (9) ~ ~ λ A K A K A K A K A K λ A K A K A K M = A K A K λ A K A K A K A K A K A K 0 0,0 p 0, p 2p 0,2p 0, 0 p,0 p p, p 2 p p,2 p p, 0 2 p,0 p 2 p, p 2p 2 p,2 p 2 p, 0,0 p, p 2 p,2 p λ, + 1 x + 1 p p, 90 Malaysia Joural of Mathematical Scieces

7 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes T y = y0 y p y2 p y 2 p y p y ~, T f = f0 f p f2 p f 2 p f p f ~. I order to facilitate the formulatio of the full-, half- ad quartersweep quadrature approximatio equatios for problem (1), further discussio will be restricted oto Newto-Cotes quadrature method, which is based o iterpolatio formulas with equally spaced data. I this paper, three differet schemes i Newto-Cotes quadrature method such as repeated trapezoidal (RT), repeated Simpso s 1 3 (RS1) ad repeated Simpso s 3 8 (RS2) schemes will be applied to discretize the problem (1). RT, RS1 ad RS2 are first, secod ad third order schemes respectively. Further discussios o Newto-Cotes quadrature method to solve Fredholm itegral equatios ca be foud i Atkiso (1997), ad Muthuvalu ad Sulaima (2009). Based o RT, RS1 ad RS2 schemes, umerical coefficiets satisfy followig relatios respectively. A j will 1 ph, j = 0, Aj = 2 ph, otherwise (10) 1 ph, j = 0, 3 4 Aj = ph, j = p,3 p,5 p,, p 3 2 ph, otherwise 3 (11) Malaysia Joural of Mathematical Scieces 91

8 Mohaa Sudaram Muthuvalu & Jumat Sulaima 3 ph, j = 0, 8 3 Aj = ph, j = 3 p,6 p,9 p,, 3 p (12) 4 9 ph, otherwise 8 where the costat step-size, h is defied as follows b a h = (13) ad is the umber of subitervals i the iterval [ a, b ]. Meawhile, the value of p, which correspods to 1, 2 ad 4, represets the full-, half- ad quarter-sweep cases respectively. FORMULATION OF THE ITERATIVE METHODS As afore-metioed, FSGS, HSGS ad QSGS iterative methods will be applied to solve liear system geerated from the discretizatio of the problem (1), as show i Equatio (9). Let matrix M be decomposed ito M = D L U (14) where D, L ad U are diagoal, strictly lower triagular ad strictly upper triagular matrices respectively. Thus, the geeral scheme for FSGS, HSGS ad QSGS iterative methods ca be writte as ( ) ( k + 1) 1 ( k) y = D L U y + f ~ ~ ~. (15) Actually, the iterative methods attempt to fid a solutio by repeatedly solvig the liear system usig approximatios to the vector y ad cotiue util the solutio is withi a predetermied acceptable boud o the error. Based o Abdullah (1991) ad, Othma ad Abdullah (2000), the geeral algorithm for FSGS, HSGS ad QSGS iterative methods to solve problem (1) would be geerally described i Algorithm Malaysia Joural of Mathematical Scieces ~

9 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes Algorithm 1: FSGS, HSGS ad QSGS methods For i = 0, p, 2 p,, 2 p, p, ad j = 0, p, 2 p,, 2 p, p, calculate ( k) fi + AjKi, j y j λ Ai Ki, i, i = 0 j= p p ( k+ 1) ( k+ 1) yi fi + AjKi, j y j λ Ai Ki, i, i = j= 0 i p ( k + 1) ( k) fi + AjKi, j yj + AjKi, j y j λ Ai Ki, i, i = otherwise j= 0 j= i+ p NUMERICAL SIMULATIONS I order to compare the performaces of the iterative methods, several experimets were carried out o the followig Fredholm itegral equatios problems. Example 1 (Wag (2006)) Cosider the itegral equatio 1 2 ( ) ( ) ( ) 4 (16) 0 y x xt x y t dt = x 2 ad the exact solutio of problem (16) is give by y( x) 24x 9x =. Example 2 (Polyai ad Mazhirov (1998)) Cosider the itegral equatio ( ) ( + ) ( ) = y x x t y t dt x x x. (17) Malaysia Joural of Mathematical Scieces 93

10 Exact solutio of problem (17) is Mohaa Sudaram Muthuvalu & Jumat Sulaima = y( x) x 5x x x There are three parameters cosidered i umerical compariso such as umber of iteratios, executio time ad maximum absolute error. Throughout the experimets, the covergece test cosidered the tolerace 10 error of the ε = 10. The experimets were carried out o several differet mesh sizes such as 240, 480, 960, 1920, 3840 ad Results of umerical simulatios, which were obtaied from implemetatios of the FSGS, HSGS ad QSGS iterative methods for Examples 1 ad 2, have bee recorded i Tables 1 ad 2 respectively. TABLE 1: Compariso of a umber of iteratios, executio time (secods) ad maximum absolute error for the iterative methods for Example 1 Mesh Size Number of iteratios Executio time (secods) Maximum absolute error Methods FSGS HSGS QSGS FSGS HSGS QSGS FSGS HSGS QSGS RT E E E-2 RS E E E-10 RS E E E-10 RT E E E-3 RS E E E-10 RS E E E-10 RT E E E-3 RS E E E-10 RS E E E-10 RT E E E-4 RS E E E-10 RS E E E-10 RT E E E-4 RS E E E-10 RS E E E-10 RT E E E-5 RS E E E-10 RS E E E Malaysia Joural of Mathematical Scieces

11 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes TABLE 2: Compariso of a umber of iteratios, executio time (secods) ad maximum absolute error for the iterative methods for Example 2 Mesh Size Number of iteratios Executio time (secods) Maximum absolute error Methods FSGS HSGS QSGS FSGS HSGS QSGS FSGS HSGS QSGS RT E E E-2 RS E E E-6 RS E E E-6 RT E E E-3 RS E E E-7 RS E E E-7 RT E E E-3 RS E E E-8 RS E E E-8 RT E E E-4 RS E E E-10 RS E E E-9 RT E E E-4 RS E E E-11 RS E E E-11 RT E E E-5 RS E E E-10 RS E E E-10 COMPUTATIONAL COMPLEXITY ANALYSIS I order to measure the computatioal complexity of the iterative methods, a estimatio of the amout of the computatioal works required for both methods have bee coducted. The computatioal works are estimated by cosiderig the arithmetic operatios performed per iteratio. Based o Algorithm 1, it ca be observed that there are + 1 p additios/subtractios (ADD/SUB) ad multiplicatios/divisios p (MUL/DIV) i computig a value for each ode poit i the solutio domai. From the order of the coefficiet matrix, M i Equatio (9), the Malaysia Joural of Mathematical Scieces 95

12 Mohaa Sudaram Muthuvalu & Jumat Sulaima total umber of arithmetic operatios per iteratio for the FSGS, HSGS ad QSGS iterative methods has bee summarized i Table 3. TABLE 3: Total umber of arithmetic operatios per iteratio for FSGS, HSGS ad QSGS methods Methods Arithmetic Operatio ADD/SUB MUL/DIV FSGS ( + 1) 2 2( + 1) 2 HSGS QSGS CONCLUSIONS I this paper, we preset applicatios of the half- ad quarter-sweep iterative methods for solvig dese liear systems arisig from the discretizatio of the secod kid liear Fredholm itegral equatios by usig three differet orders of Newto-Cotes quadrature discretizatio schemes such as RT, RS1 ad RS2 schemes. It has show that the quadrature approximatio equatios based o Newto-Cotes schemes ca be easily formulated ad rewritte i geeral form as show i Equatio (9). Through umerical results obtaied for both Examples 1 ad 2 (refer Tables 1 ad 2), it shows that umber of iteratios for HSGS ad QSGS methods are early same compared to the FSGS method. Through the observatio i Tables 1 ad 2, HSGS ad QSGS iterative methods reduce the executio time compared to the FSGS method. Computatioal time for FSGS, HSGS ad QSGS iterative methods with RS1 ad RS2 schemes are icreased compared to the iterative methods with RT scheme. It is due to the computatioal complexity of the high-order discretizatio schemes. I terms of accuracy of umerical solutios obtaied, RS1 ad RS2 schemes are more accurate tha the RT scheme. Besides that, applicatios of secod order Lagrage iterpolatio to compute remaiig poits maaged to overcome the problem discussed i Muthuvalu ad Sulaima (2009). Overall, the umerical results show that the QSGS method is a better method compared to the FSGS ad HSGS methods i the sese of umber of iteratios ad executio time. This is maily because of computatioal 96 Malaysia Joural of Mathematical Scieces

13 Numerical Solutio of Secod Kid Liear Fredholm Itegral Equatios Usig QSGS Iterative Method with High-Order Newto-Cotes Quadrature Schemes complexity of the QSGS method which is approximately 50% ad 75% less tha HSGS ad FSGS methods respectively (refer Table 3). ACKNOWLEDGEMENT The authors ackowledge the Postgraduate Research Grat, Uiversiti Malaysia Sabah (GPS0003-SG-1/2009) for the completio of this article. REFERENCES Abdullah, A. R The four poit Explicit Decoupled Group (EDG) method: A fast Poisso solver. It. J. Comput. Math., 38: Abdullah, A. R. ad Ali, N. H. M A comparative study of parallel strategies for the solutio of elliptic pde s. Parallel Algorithms ad Applicatios, 10: Abdullah, M. H., Sulaima, J. ad Othma, A A umerical assessmet o water quality model usig the Half-Sweep Explicit Group methods. Gadig, 10: Asady, B., Kajai, M. T., Vecheh, A. H. ad Heydari, A Solvig secod kid itegral equatios with hybrid Fourier ad block-pulse fuctios. Appl. Math. Comput., 160: Atkiso, K. E The Numerical Solutio of Itegral Equatios of the Secod Kid, Uited Kigdom: Cambridge Uiversity Press. Che, Z., Micchelli, C. A. ad Xu, Y Fast collocatio methods for secod kid itegral equatios. SIAM J. Numer. Aal., 40(1): Che, Z., Wu, B. ad Xu, Y Fast umerical collocatio solutios of itegral equatios. Commu. Pure Appl. Aal., 6(3): Hasa, M.K., Othma, M., Abbas, Z., Sulaima, J. ad Ahmad, F Parallel solutio of high speed low order FDTD o 2D free space wave propagatio. I O. Gervasi ad M. Gavrilova (Eds.), Malaysia Joural of Mathematical Scieces 97

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16 Mohaa Sudaram Muthuvalu & Jumat Sulaima Sulaima, J., Hasa, M. K. ad Othma, M Red-Black Half-Sweep iterative method usig triagle fiite elemet approximatio for 2D Poisso equatios. I Y. Shi et al. (Eds.), Computatioal Sciece, Lecture Notes i Computer Sciece (LNCS 4487): Spriger- Verlag, Berli. Sulaima, J., Hasa, M. K. ad Othma, M MEGSOR iterative method for the triagle elemet solutio of 2D Poisso equatios. Procedia Computer Sciece, 1: Sulaima, J., Othma, M. ad Hasa, M. K. 2004b. Quarter-Sweep Iterative Alteratig Decompositio Explicit algorithm applied to diffusio equatios. It. J. Comput. Math., 81(12): Sulaima, J., Othma, M. ad Hasa, M. K. 2008a. Half-Sweep Algebraic Multigrid (HSAMG) method applied to diffusio equatios. I Modelig, Simulatio ad Optimizatio of Complex Processes: Spriger-Verlag, Berli. Sulaima, J., Saudi, A., Abdullah, M. H., Hasa, M. K. ad Othma, M. 2008b. Quarter-Sweep Arithmetic Mea algorithm for water quality model. I H.B. Zama, T.M.T. Sembok, K.V. Rijsberge, L. Zadeh, P. Bruza, T. Shih ad M. N. Taib, Proceedigs of the Iteratioal Symposium o Iformatio Techology: Wag, W A ew mechaical algorithm for solvig the secod kid of Fredholm itegral equatio. Appl. Math. Comput., 172: Xiao, J. Y., We, L. H. ad Zhag, D Solvig secod kid Fredholm itegral equatios by periodic wavelet Galerki method. Appl. Math. Comput., 175: Yousif, W. S. ad Evas, D. J Explicit De-coupled Group iterative methods ad their implemetatios. Parallel Algorithms ad Applicatios, 7: Malaysia Joural of Mathematical Scieces