Numerical Solution of Fredholm Integral Equations of the Second Kind by using 2-Point Explicit Group Successive Over-Relaxation Iterative Method

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1 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 umercl Soluton of Fredholm Integrl Equtons of the Second Knd by usng -Pont Eplct Group Successve Over-Relton Itertve Method Mohn Sundrm Muthuvlu, Elyr Aruchunn, Jumt Sulmn nd Mohmmd Mehd Rshd Abstrct In ths pper, we ntroduce nd nlyse the performnce of -Pont Eplct Group Successve Over-Relton () tertve method for the soluton of dense lner systems tht rse from second knd Fredholm ntegrl equtons. The dervton nd mplementton of the proposed method re descrbed. We present results of some test emples nd computtonl complety nlyss to llustrte the effcency of the proposed method. Keywords Fredholm ntegrl equtons, Eplct Group method, Successve Over-Relton pproch, Composte closed ewton- Cotes scheme, Dense lner system. I. ITRODUCTIO TEGRAL equtons (IEs) hve been one of the prncpl Imthemtcl models n vrous res of scence nd engneerng. The IEs re encountered n numerous pplctons ncludng contnuum mechncs, potentl theory, geophyscs, electrcty nd mgnetsm, knetc theory of gses, heredtry phenomen n physcs nd bology, renewl theory, quntum mechncs, rdton, optmzton, optml control systems, communcton theory, mthemtcl economcs, populton genetcs, queung theory, medcne, mthemtcl problems of rdtve equlbrum, prtcle trnsport problems of strophyscs nd rector theory, coustcs, flud mechncs, stedy stte het conducton, frcture mechncs nd rdtve het trnsfer problems ([], [], [3], [4]). Consequently, n ths pper, type of IEs.e. lner Fredholm ntegrl equtons of M. S. Muthuvlu s wth the Deprtment of Fundmentl nd Appled Scences, Fculty of Scence nd Informton Technology, Unverst Teknolog PETROAS, 36 Bndr Ser Iskndr, Perk, Mlys (correspondng uthor; phone: ; e-ml: msmuthuvlu@gml.com). E. Aruchunn s wth the Deprtment of Mthemtcs nd Sttstcs, Curtn Unversty, Perth WA6845, Austrl (e-ml: eruchunn@yhoo.com). J. Sulmn s wth the School of Scence nd Technology, Unverst Mlys Sbh, Jln UMS, 884 Kot Knblu, Sbh, Mlys (e-ml: umt@ums.edu.my). M. M. Rshd s wth the Mechncl Engneerng Deprtment, Unversty of Mchgn-Shngh Jo Tong Unversty Jont Insttute, Shngh Jo Tong Unversty, Shngh, Peoples Republc of Chn nd Mechncl Engneerng Deprtment, Engneerng Fculty of Bu-Al Sn Unversty, Hmedn, Irn (e-ml: mm_rshd@stu.edu.cn). the second knd s consdered. The generl form of lner Fredholm ntegrl equtons of the second knd cn be defned s follows ϕ ( ) K(, t) ϕ( t) dt = f, [ α,β ] β α. () The rght-hnd sde functon f nd kernel K re gven. Menwhle, ϕ s the unknown functon to be determned. The kernel functon K s ssumed to be bsolutely ntegrble nd stsfy the Fredholm lterntve theorem [5]. The pplcton of numercl methods for solvng the problem () s the focus of ths pper. There s huge lterture on numercl methods for solvng problem (), for nstnce refer [6]-[]. The mplementtons of numercl methods on problem () mostly led to dense lner systems. Thus, effcent tertve solvers re requred to solve the resultng dense lner systems. Recently, fmly of block tertve methods known s Eplct Group (EG) tertve methods hs been ppled wdely n solvng vrous types of lner systems. Thus, n ths pper, performnce of n tertve method under EG methods.e. - Pont Eplct Group Successve Over-Relton () wll be nvestgted n solvng frst order composte closed ewton-cotes qudrture (-CCC) lgebrc equtons. The performnce of the method on -CCC lgebrc equtons s comprtvely studed by ther pplcton n solvng problem (). The concept of the method s derved by combnng the stndrd -Pont Eplct Group (- EG) method wth Successve Over-Relton (SOR) pproch. umercl performnce of the method wll be compred wth the stndrd -EG method. The stndrd -EG method s lso known s -Pont Eplct Group Guss-Sedel () method. Ths pper s orgnsed n s mn sectons. Secton II eplns the dervton of -CCC lgebrc equtons for problem () followed by the formultons of the nd methods n Secton III. The computtonl complety nlyss nd numercl results from the smultons re gven n Secton IV nd V respectvely. Fnlly, concludng remrks re gven n Secton VI. ISS:

2 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 II. -CCC ALGEBRAIC EQUATIOS In ths secton, dscretston of the problem () by usng - CCC scheme s dscussed. An pplcton of the -CCC scheme for problem () leds to -CCC lgebrc equtons whch wll be solved by usng nd α, β dvded unformly nto methods. ow, let the ntervl [ ] subntervls nd the dscrete set of ponts of nd t gven by + h =,,,,,, nd = α = α h (,,,,,, ) t + = respectvely, where the constnt step sze, h s defned s follows β α h =. () Before further eplntons, the followng nottons.e., K, K(, t ), ϕ ϕ ( ), ϕ ϕ ( t ) nd f f ( ) wll be ppled subsequently for smplcty. An pplcton of the -CCC scheme reduces problem () nto lgebrc equtons s follows ([8], []) ϕ w K, ϕ = f (3) for =,,,,,,. The soluton ppromton of the ect soluton ϕ to () nd ϕ s n w s the weghts of -CCC scheme tht stsfes the followng condton h, =, w =. (4) h, otherwse Followng the conventonl process, the generted -CCC lgebrc equtons (3) cn be wrtten s the followng mtr form where [ ] ( + ) ( + ) A ϕ = f, (5) A =, R s rel nd dense coeffcent mtr wth elements w K,, =, =. (6) w K,, III. AD ITERATIVE METHODS As fore-mentoned, the formulton nd mplementton of the nd methods for solvng the generted -CCC lgebrc equtons wll be dscussed. ow, let consder ny group of two ponts.e., nd tht re used smultneously to clculte the vlues of ϕ bsed on lgebrc equtons (3). Therefore, t pont, the soluton s ppromted by ϕ w K, ϕ = f (.e. equton (3)). (7) Wheres, t pont the soluton s gven by ϕ w K, ϕ = f +. (8) ow, the equtons (7) nd (8) cn be wrtten smultneously n the mtr form s follows, +,,, ϕ ϕ = f f,, ϕ ϕ, ϕ, ϕ (9) where coeffcent mtr wth sze ( ) cn be esly nverted. Thus, the equton (9) cn be wrtten n eplct form s ϕ ϕ = B, +,,, f f +,, ϕ ϕ, ϕ, ϕ () B = det,,,,. Ths smplfes to the formule where B = ( )( ) ( )( ) wth ϕ = B ϕ C,, ( C), ( D) C +, D () = f, ϕ, ϕ () ISS:

3 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 nd D =, ϕ, ϕ. (3) = = f for nd methods respectvely. By consderng formultons of nd methods, lgorthm for both cses.e. complete grouped (Cse ) nd ncomplete grouped (wth one sngle pont ungrouped) (Cse ) re descrbed n Algorthms nd respectvely. Hence, the tertve scheme for method s gven by ( k + ) ϕ = B ϕ, for =,,4,, 3,, where nd D = f C = f =, ( C), ( D) C +, D,, ϕ = (4) ( k + ) ( k ϕ ) (5),, ϕ ( k + ) ( k ϕ ). (6) By ddng n ccelerted prmeter, ω nto formule (), the tertve scheme for method cn be rewrtten s ( k + ) ( k ) ϕ = ( ) ϕ ω, ω + B, ϕ ϕ ( C), ( D) C +, D (7) for =,,4,, 3,, where C nd D re s shown n equtons (5) nd (6) respectvely. When ω =, the - EGSOR method reduced trvlly to the method. For n even subntervls,, the number of dscrete node ponts s odd.e., +, whch results n one ungrouped pont. Therefore, the ungrouped pont.e.,, wll be computed bsed on the followng pont tertons nd = k + k + ϕ f, ϕ (8), = + k + k ω k + ϕ ω ϕ f, ϕ (9), Algorthm : nd methods for Cse. Set ll the prmeters. Iterton cycle for =,,4,, 3, ( k + ) ( k ) ϕ + + C + D ϕ ω,, ω + B, C +, D ϕ ϕ. Convergence test. If the converge crteron.e., the mmum ( k + ) norm ϕ k ϕ ε (where ε s the convergence v. crteron) s stsfed, go to Step v. Otherwse, go to Step. Stop. Algorthm : nd methods for Cse. Set ll the prmeters. Iterton cycle for =,,4,, 3, ( k + ) ( k ) ϕ + + C + D ϕ ω,, ω + B, C +, D ϕ ϕ for = + k + k ω k + ϕ ω ϕ f, ϕ,. Convergence test. If the converge crteron.e., the mmum ( k + ) norm ϕ k ϕ ε (where ε s the convergence v. crteron) s stsfed, go to Step v. Otherwse, go to Step. Stop. IV. COMPUTATIOAL COMPLEXITY AALYSIS An estmton mount of the computtonl work hs been conducted n order to evlute the computtonl complety of method. The computtonl work s estmted by consderng the rthmetc opertons performed per terton. In estmtng the computtonl work, the vlues of, n A nd B re stored beforehnd. Bsed on Algorthm (for Cse ), the totl rthmetc opertons (ecludng the convergence test) nvolved for methods s ( + + ) ADD / SUB + ( ) MUL / DIV ISS:

4 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 per terton. Menwhle, for Cse, the number of rthmetc opertons requred s ( + + ) ADD / SUB + ( ) MUL / DIV per terton. The ADD / SUB nd MUL / DIV represent ddtons/subtrctons nd multplctons/dvsons opertons respectvely. V. SIMULATIO RESULTS The followng two lner Fredholm ntegrl equtons of the second knd re used s the test problems n order to compre the performnce of the methods. Test Problem [] Consder the Fredholm ntegrl equton of the second knd ϕ ( ) ( 4t ) ϕ( t) dt =, [,] nd the ect soluton s gven by ϕ = 4 9., () Test Problem [8] Consder the Fredholm ntegrl equton of the second knd ϕ ( + t ) ϕ( t) dt = , [,], () wth the ect soluton ϕ = For the numercl smultons, three crter re used for comprtve nlyss.e. k - umber of tertons, CPU - CPU tme (n seconds) when the converged soluton s obtned, RMSE - Root men squre error [3]. The vlue of ntl dtum, ϕ, s set to zero for both the test problems. The computtons re performed on personl computer wth Intel(R) Core(TM) 3- CPU nd 4.GB RAM, nd the progrmmng codes re compled by usng C lnguge. Throughout the smultons, the convergence test consdered s ε = nd tested on eght dfferent vlues of.e. 6,, 4, 48, 96, 9, 384 nd 768. Menwhle, the epermentl vlues of ω were obtned wthn ±. by runnng the progrms for dfferent vlues of ω nd choosng the one tht gve the mnmum number of tertons. For the cse of more thn one ω (bsed on mnmum number of tertons), the optmum vlue of ω s chosen by consderng the mnmum RMSE. The numercl results of the tested methods for test problems nd re presented n Tbles I nd II respectvely. TABLE I. UMERICAL RESULTS OF TEST PROBLEM Methods k CPU RMSE ( ω =.53) ( ω =.54) ( ω =.54) ( ω =.54) ( ω =.54) ( ω =.54) ( ω =.55) ( ω =.55) TABLE II. UMERICAL RESULTS OF TEST PROBLEM Methods k CPU RMSE ( ω =.7) ( ω =.8) 55 3 ( ω =.8) 3 ( ω =.8) 3 ( ω =.9) 3 ( ω =.9) 3 ( ω =.9) 3 ( ω =.9) ISS:

5 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 The followng tbles show the ppromton solutons of ϕ t some dscrete ponts for test problems nd. TABLE III. UMERICAL SOLUTIOS FOR CASE = 6 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (5.864E-3) (5.864E-3) (.37E-).37E (.44837E-) (.44837E-) (.8596E-) (.8596E-) (.343E-) (.343E-) (.53588E-) (.53588E-) (.88E-) (.88E-) (3.634E-) (3.634E-) (3.647E-) (3.647E-) (3.4459E-) (3.4459E-) TABLE IV. UMERICAL SOLUTIOS FOR CASE = OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (.349E-3) (.349E-3) (.5977E-3) (.5977E-3) (3.6457E-3) (3.6457E-3) (4.693E-3) (4.693E-3) (5.54E-3) (5.54E-3) (6.386E-3) (6.386E-3) (7.339E-3) (7.339E-3) (7.6377E-3) (7.6377E-3) (8.46E-3) (8.46E-3) (8.546E-3) (8.546E-3) TABLE V. UMERICAL SOLUTIOS FOR CASE = 4 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (3.69E-4) (3.69E-4) (6.77E-4) (6.77E-4) (9.35E-4) (9.35E-4) (.5433E-3) (.5433E-3) (.384E-3) (.384E-3) (.5848E-3) (.5848E-3) (.75754E-3) (.75754E-3) (.986E-3) (.986E-3) (.3467E-3) (.3467E-3) (.357E-3) (.357E-3) TABLE VI. UMERICAL SOLUTIOS FOR CASE = 48 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (8.5E-5) (8.5E-5) (.78E-4) (.78E-4) (.579E-4) (.579E-4) (.8855E-4) (.8855E-4) (3.4E-4) (3.4E-4) (3.9533E-4) (3.9533E-4) (4.3934E-4) (4.3934E-4) (4.77E-4) (4.77E-4) (5.86E-4) (5.86E-4) (5.3387E-4) (5.3387E-4) ISS:

6 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 TABLE VII. UMERICAL SOLUTIOS FOR CASE = 96 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (.38E-5) (.38E-5) (3.99E-5) (3.99E-5) (5.645E-5) (5.645E-5) (7.4E-5) (7.4E-5) (8.66E-5) (8.66E-5) (9.883E-5) (9.883E-5) (.983E-4) (.983E-4) (.97E-4) (.97E-4) (.75E-4) (.75E-4) (.3346E-4) (.3346E-4) TABLE VIII. UMERICAL SOLUTIOS FOR CASE = 9 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (5.9E-6) (5.9E-6) (9.8E-6) (9.8E-6) (.4E-5) (.4E-5) (.83E-5) (.83E-5) (.57E-5) (.57E-5) (.47E-5) (.47E-5) (.746E-5) (.746E-5) (.98E-5) (.98E-5) (3.79E-5) (3.79E-5) (3.337E-5) (3.337E-5) TABLE IX. UMERICAL SOLUTIOS FOR CASE = 384 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (.7E-6) (.7E-6) (.45E-6) (.45E-6) (3.53E-6) (3.53E-6) (4.5E-6) (4.5E-6) (5.39E-6) (5.39E-6) (6.8E-6) (6.8E-6) (6.86E-6) (6.86E-6) (7.45E-6) (7.45E-6) (7.95E-6) (7.95E-6) (8.34E-6) (8.34E-6) TABLE X. UMERICAL SOLUTIOS FOR CASE = 768 OF TEST PROBLEM Ect Soluton EGGS EGSOR.... (.E+) (.E+) (3.E-7) (3.E-7) (6.E-7) (6.E-7) (8.8E-7) (8.8E-7) (.3E-6) (.3E-6) (.35E-6) (.35E-6) (.54E-6) (.54E-6) (.7E-6) (.7E-6) (.86E-6) (.86E-6) (.99E-6) (.99E-6) (.8E-6) (.9E-6) ISS:

7 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 TABLE XI. UMERICAL SOLUTIOS FOR CASE = 6 OF TEST PROBLEM Ect Soluton EGGS EGSOR (.3988E-) (.3988E-) (.3784E-) (.3784E-) (.3887E-) (.3887E-) (.49749E-) (.49749E-) (.64978E-) (.64978E-) ( E-) ( E-) (.83E-) (.83E-) (.36897E-) (.36897E-) (.69596E-) (.69596E-) (3.6643E-) (3.6643E-) ( E-) ( E-) TABLE XII. UMERICAL SOLUTIOS FOR CASE = OF TEST PROBLEM Ect Soluton EGGS EGSOR (3.53E-3) (3.53E-3) (3.355E-3) (3.355E-3) ( E-3) ( E-3) (3.74E-3) (3.74E-3) (4.46E-3) (4.46E-3) (4.664E-3) (4.664E-3) (5.76E-3) (5.76E-3) (5.9979E-3) (5.9979E-3) ( E-3) ( E-3) (7.6666E-3) (7.6666E-3) ( E-3) ( E-3) TABLE XIII. UMERICAL SOLUTIOS FOR CASE = 4 OF TEST PROBLEM Ect Soluton EGGS EGSOR (8.67E-4) (8.67E-4) (8.69E-4) (8.69E-4) (8.673E-4) (8.673E-4) (9.35E-4) (9.35E-4) (.35E-3) (.35E-3) (.533E-3) (.533E-3) (.38E-3) (.38E-3) (.47978E-3) (.47978E-3) (.684E-3) (.684E-3) (.9545E-3) (.9545E-3) (.743E-3) (.743E-3) TABLE XIV. UMERICAL SOLUTIOS FOR CASE = 48 OF TEST PROBLEM Ect Soluton EGGS EGSOR (.36E-4) (.36E-4) (.657E-4) (.657E-4) (.678E-4) (.678E-4) (.3379E-4) (.3379E-4) (.576E-4) (.576E-4) (.885E-4) (.885E-4) (3.9E-4) (3.9E-4) (3.6993E-4) (3.6993E-4) (4.99E-4) (4.99E-4) (4.7885E-4) (4.7885E-4) (5.435E-4) (5.435E-4) ISS:

8 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 TABLE XV. UMERICAL SOLUTIOS FOR CASE = 96 OF TEST PROBLEM Ect Soluton EGGS EGSOR (5.79E-5) (5.79E-5) (5.64E-5) (5.64E-5) (5.49E-5) (5.49E-5) (5.845E-5) (5.845E-5) (6.44E-5) (6.44E-5) (7.6E-5) (7.6E-5) (8.43E-5) (8.43E-5) (9.48E-5) (9.48E-5) (.55E-4) (.55E-4) (.97E-4) (.97E-4) (.3588E-4) (.3588E-4) TABLE XVI. UMERICAL SOLUTIOS FOR CASE = 9 OF TEST PROBLEM Ect Soluton EGGS EGSOR (.7E-5) (.7E-5) (.9E-5) (.9E-5) (.355E-5) (.355E-5) (.46E-5) (.46E-5) (.6E-5) (.6E-5) (.8E-5) (.8E-5) (.36E-5) (.36E-5) (.3E-5) (.3E-5) (.63E-5) (.63E-5) (.993E-5) (.993E-5) (3.397E-5) (3.397E-5) TABLE XVII. UMERICAL SOLUTIOS FOR CASE = 384 OF TEST PROBLEM Ect Soluton EGGS EGSOR (3.7E-6) (3.7E-6) (3.3E-6) (3.3E-6) (3.39E-6) (3.39E-6) (3.65E-6) (3.65E-6) (4.E-6) (4.E-6) (4.5E-6) (4.5E-6) (5.9E-6) (5.9E-6) (5.78E-6) (5.78E-6) (6.58E-6) (6.58E-6) (7.48E-6) (7.48E-6) (8.49E-6) (8.49E-6) TABLE XVIII. UMERICAL SOLUTIOS FOR CASE = 768 OF TEST PROBLEM Ect Soluton EGGS EGSOR (7.9E-7) (7.9E-7) (8.E-7) (8.E-7) (8.5E-7) (8.5E-7) (9.E-7) (9.E-7) (.E-6) (.E-6) (.3E-6) (.3E-6) (.8E-6) (.8E-6) (.44E-6) (.44E-6) (.65E-6) (.65E-6) (.87E-6) (.87E-6) (.E-6) (.E-6) ISS:

9 ITERATIOAL JOURAL OF APPLIED MATHEMATICS AD IFORMATICS Volume 9, 5 VI. COCLUSIO In ths pper, method hs been successfully ppled n solvng lner Fredholm ntegrl equtons of the second knd. By referrng Tbles I nd II, the numercl results show tht mplementton of the method solved the both test problems wth mnmum number of tertons nd fstest CPU tme. In terms of ccurcy, numercl solutons obtned v method re n good greement compred to the method. Through the observton n Tbles III to XVIII, ncrement of mproved the ccurcy of numercl solutons nd mmum error of the numercl soluton occurred t pont =. for both test problems. Fnlly, t cn be summrzed tht the method s better thn method, especlly n the spect of number of tertons nd CPU tme. REFERECES [] W. Wng, A new mechncl lgorthm for solvng the second knd of Fredholm ntegrl equton, Appl. Mth. Comput., vol. 7, pp , 6. [] L. Hc nd K. Domke, Integrl modelng nd smulton n some therml problems, Proceedngs of the 5 th IASME/WSEAS Interntonl Conference on Het Trnsfer, Therml Engneerng nd Envronment, WSEAS Press, pp. 4 47, 7. [3] L. Hc, K. Bednrek nd A. Tomczewsk, Computtonl results for ntegrl modelng n some problems of electrcl engneerng, Proceedngs of the th WSEAS Interntonl Conference on Computers, WSEAS Press, pp.4 9, 7. [4] M. Buke, A. Buks nd R. Vlums, One-dmensonl ntensve steel quenchng models, Recent Advnces n Mechncl Engneerng, WSEAS Press, pp. 9 7, 4. [5] K. E. Atknson, The numercl soluton of ntegrl equtons of the second knd. Cmbrdge Unversty Press, 997. [6] M. H. Rehn nd Z. Abd, Rtonlzed Hr functons method for solvng Fredholm nd Volterr ntegrl equtons, J. Comput. Appl. Mth., vol., pp., 7. [7] C. H. Hso, Hybrd functon method for solvng Fredholm nd Volterr ntegrl equtons of the second knd, J. Comput. Appl. Mth., vol. 3, pp , 9. [8] M. S. Muthuvlu nd J. Sulmn, Hlf-Sweep Arthmetc Men method wth composte trpezodl scheme for solvng lner Fredholm ntegrl equtons, Appl. Mth. Comput., vol. 7, pp ,. [9] J. Rshdn nd Z. Mhmood, Collocton method for Fredholm nd Volterr ntegrl equtons, Kybernetes, vol. 4, pp. 4 4, 3. [] X. C. Zhong, A new yström-type method for Fredholm ntegrl equtons of the second knd, Appl. Mth. Comput., vol. 9, pp , 3. [] M. S. Muthuvlu nd J. Sulmn, Performnce nlyss of the frst order lner sttonry tertve methods n solvng thrd order ewton- Cotes qudrture system, Int. J. Mth. Comput. Smul, vol. 7, pp , 3. [] M. S. Muthuvlu nd J. Sulmn, The Qurter-Sweep Geometrc Men method for solvng second knd lner Fredholm ntegrl equtons, Bull. Mlys. Mth. Sc. Soc., vol. 36, pp. 9 6, 3. [3] A. Golbb nd S. Sefollh, An tertve soluton for the second knd ntegrl equtons usng rdl bss functons, Appl. Mth. Comput., vol. 8, pp , 6. ISS: