Americn Journl of Applied Mtemtics 04; (5: 55-6 Publised online eptember 30, 04 (ttp://www.sciencepublisinggroup.com/j/jm doi: 0.648/j.jm.04005. IN: 330-0043 (Print; IN: 330-006X (Online Modified midpoint metod for solving system of liner Fredolm integrl equtions of te second kind lm Jsim Mjeed Deprtment of Pysics, College of cience, University of i-qr, i-qr, Irq Emil ddress: sm.slmmjeed@yoo.com o cite tis rticle: lm Jsim Mjeed. Modified Midpoint Metod for olving ystem of Liner Fredolm Integrl Equtions of te econd Kind. Americn Journl of Applied Mtemtics. Vol., No. 5, 04, pp. 55-6. doi: 0.648/j.jm.04005. Abstrct: In tis pper numericl solution to system of liner Fredolm integrl equtions by modified midpoint metod is considered. is metod trnsforms te system of liner Fredolm integrl equtions into system of liner lgebric equtions tt cn be solved esily wit ny of te usul metods. Finlly, some illustrtive emples re presented to test tis metod nd te results revel tt tis metod is very effective nd convenient by comprison wit ect solution nd wit oter numericl metods suc s midpoint metod, trpezoidl metod, impson's metod nd modified trpezoidl metod. All results re computed by using progrms written in Mtlb R0b. Keywords: ystem of Fredolm Integrl Equtions, Modified Midpoint Metod. Introduction Mtemticl modeling for mny problems in different disciplines, suc s engineering, cemistry, pysics nd biology leds to integrl eqution, or system of integrl equtions. Most differentil equtions cn be epressed s integrl equtions. Also, system of differentil equtions cn be written s system of integrl equtions, [, ]. For tese resons te gret interest for solving tese equtions. ince tese equtions usully cnnot be solved eplicitly, so te numericl solutions ve been igly studied by mny utors, [-4]. Now we consider te following systems of liner Fredolm integrl eqution of te second kind: were b U( = F( + K(, yu(ydy, [,b]. (. [ ] m U( = u (,u (,,u(, [ ] m F( = f (,f (,,f(, K(,y = k ij(,y, i,j = ((m. In system (. te known kernel K(, y is continuous, te function F( is given, nd U( is te solution to be determined. ere re severl numericl metods for solving tis system. For emple, Rtionlized Hr functions metod [5], Block Pulse functions metod[6], Epnsion metods [7, 8], Decompostion metod [9] nd Ortogonl ringulr functions metod [0]. In recent yers, new clss of qudrture formuls re introduced clled s modified (corrected Newton-Cotes formul wic bsed on derivtives of te function. e uses of modified qudrture formul for solving integrl equtions nd teir systems ve been considered by mny utors: modified trpezoidl [-3 ], modified impson s [3-6 ] nd modified midpoint metod [7]. In tis pper, we wnt to find te numericl solution for te system given by eq (. by using te repeted modified midpoint formul for definite integrl: b n 7 (b (4 f(d = f( j + [ f ( f (b] + f ( η. (. j= 5760 4 were n is te number of subintervl of [, b], = b n, j = + (j /, j = (n nd η (, b. k ij(,y k ij(,y o do tis, we ssume te functions, y nd f( i eist for ll i,j = (m. For furter informtion on formul (. nd oter modified qudrture formuls, see [8-5]. 4
56 lm Jsim Mjeed: Modified Midpoint Metod for olving ystem of Liner Fredolm Integrl Equtions of te econd Kind. Modified Midpoint Metod Consider te it eqution of system (. m b i i ij j j= u ( = f ( + k (,yu (ydy, i = (m (. o solve eq. (., we pproimte te integrl prt tt ppered in te rigt nd side by te repeted modified midpoint formul to get, + k ij(, 0u j( 0 J ij(, n+ u j( n+ k ij(, n+ u j( n+, were, 0 =, n+ = b nd k ij(,y J ij(,y =, y i,j = (m. Hence for = r,r = 0((n +, we get te following system of equtions: were u m n i ( f i ( = + j= s k ij (, s u j ( s + = 4 J ij (, 0 u j ( 0 m n ir = ir + j= s= ijrs js + 4 ijr0 j0 u f k u J u + kijr0u j0 Jijrn+ ujn+ kijrn+ u jn+ u u (, f f(, ir = i r ir = i r u = u(,u = u(, i0 i 0 in+ i n+ ( ( (. (.3. (.4 m b i i j= ij j u ( = f ( + H (,yu (ydy,i = (m We note tt if u solution of eq.(. ten it is solution of eq.(.4 too. Now, for solving eq.(.4, we must consider two cses, nd for simplify let L (,y ij = k ij(,y y Cse : e prtil derivtives Lij (, y eists for ec { } i, j,,..., m. In tis cse, we pproimte te integrl prt tt ppered in te rigt nd side of eq.(.4 by te repeted modified midpoint formul to get, m n u i ( = f i ( + H j s ij(, su j( s L 4 ij(, 0u j( 0 = + = + H ij(, 0u j( 0 L ij(, n+ u j( n+ H ij(, n+ u j( n+ By setting = get were ( (.5 r, r = 0((n + in eq.(.5, one cn u m n ir f = ir + j= 4 L ijr0 u j0 + s= H ijrs u js ( L 4 ijrn+ ujn+ + H 4 ijr0uj0 Hijrn+ ujn+ (.6 f = f(, Hijrs = H ij( r, s nd Lijrs = L ij( r, s. ir i r. Jijrs J ij( r, s, = ijrs ij r s k = k (,. If we differentil bot sides of eqution (. wit respect k to nd setting ij(,y H ij (,y = one cn obtin: From eq.(.6 nd eq.(.3 one cn get te following system wic consist of m(n+4 equtions nd m(n+4 unknowns:{ u,u,...,u,u,u },i = (m. i0 i in+ i0 in+ ( ( ( m n uir = fir + Jijr0uj0 + kijrsujs J j 4 s 4 ijrn + ujn + + k 4 ijr0uj0 kijrn + u jn+, = = m n u i0 = f i0 + L j 4 ij00uj0 + H s ij0sujs L 4 ij0n+ ujn+ + H 4 ij00u j0 Hij0n+ u jn+, = = u = in f + m n L u H u L u H u H u + in+ j 4 ijn+ 0 j0 + s ijn+ s js 4 ijn + n+ jn+ + 4 ijn + 0 j0 = = ijn+ n + jn+ By solving te bove system te numericl solutions of eq.(. re obtined. Cse : e prtil derivtives Lij(, y does not eist. In tis cse, we pproimte te integrl prt tt ppered in te rigt nd side of eq. (.4 by te repeted midpoint formul to get, (.8 m n u i ( = f i ( + H ij(, su j( s,i = ( m. j= s= =,,,, in eq.(.8, one cn get: By setting 0 n+ m ir ir j= s= ijrs js n (.7 u = f + H u, r = 0((n +. (.9 From eq.(.9 nd eq.(.3 one cn get te following system wic consists of m(n+4 equtions nd m(n+4 unknowns:
Americn Journl of Applied Mtemtics 04; (5: 55-6 57 ( m n uir = fir + Jijr0uj0 + kijrsujs J j 4 s 4 ijrn + ujn+ + k 4 ijr0uj0 kijrn u + jn+, = = m n ui0 = fi0 + H j s ijr0u j0, = = m n u in + = f in + + H j s ijrn+ u jn+, = = i = ( m,r = 0( (n +. (.0 By solving te system given by eq. (.0, te numericl solutions of eq.(. re obtined. 3. Numericl Emples In tis section we give tree numericl emples to test te Modified Midpoint metod for solving system of liner Fredolm integrl equtions of te second kind. All results re computed by using progrms written in Mtlb R0b. In order to sow te efficiency nd ig ccurcy of te present metod we compred te results wit te ect solutions numericlly in te tbles (-4. Also, we compred te error functions wic obtined by our metod nd oter numericl metods suc s Midpoint, rpezoidl, impson s nd Modified rpezoidl grpiclly in te figures (-4. Finlly, te following nottions re used in te tbles nd figures. ( Ect olution ( Approimtion solution Modified Midpoint metod cse, eq.(.7 Modified Midpoint metod cse, eq.(.0 MP n ble. e numericl results for emple wit n=0 nd n=30. Modified rpezoidl metod Midpoint metod rpezoidl metod impson s metod Number of subintervl nd te error function is given by : (= ( (. Emple : Consider te following system of liner Fredolm integrl equtions of te second kind, [] 5 5 u ( = + + 6 ( + y u (ydy + yu (ydy 0 0 4 7 u ( = + + yu 5 (ydy + ( y u (ydy 0 0 were te ect solution ( u (,u ( (, 4 ( ( ( ( (3. = + nd te prtil derivtive L ij(,y eists. erefore, tis system cn be solved by using nd te results presented in tble nd fig.. n=0 n=30 n=0 n=30 0 0 0 0 0.05.0050000.005090.005000 65 649 65 0.5.050000.050586.050007 0.0005065 0.00050709 0.0005066 0.5.0650000.065004.06500 0.0039065 0.00390785 0.0039067 0.35.50000.5443.5008 0.050065 0.0500876 0.050068 0.45.050000.05904.05004 0.040065 0.040098 0.040069 0.55.3050000.305386.305009 0.095065 0.09504 0.095063 0.65.450000.45889.450036 0.785065 0.7854 0.7850633 0.75.5650000.565344.565004 0.364065 0.364394 0.3640634 0.85.750000.753960.750049 0.50065 0.5056 0.500637 0.95.9050000.905457.9050056 0.845065 0.845746 0.8450639.0000489.00000059.00008.0000005
58 lm Jsim Mjeed: Modified Midpoint Metod for olving ystem of Liner Fredolm Integrl Equtions of te econd Kind 0.0004 0.0008 0.00 0.075 MP 0.000 e ( 0.050 e ( 0.05 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.000 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.036 0.00004 0.04 MP e ( e ( 0.0000 0.0 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.000 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 Fig.. e error function vi, MP,, nd for emple wit n=0. Emple : Consider te following system of liner Fredolm integrl equtions of te second kind: y e+ (+ y = + + 0 0 y e + (+ y + 0 0 u ( e e u (ydy e u (ydy u ( = e + e + e u (ydy e u (ydy (3. is system of integrl equtions s been solved by Block-Plus Function (BPF in [6] nd te ect solution is u (,u ( = e,e. Also te prtil derivtive L ij(,y ( ( eists. erefore, similr to emple tis system cn be solved by using nd te results presented in fig. nd tbles nd 3. ble. e numericl results for emple wit n=0 nd n=30. ( ( ( ( n=0 n=30 n=0 n=30 0.00000375.00000005 0.9999998 0.99999999 0.05.0570.05749.0574 0.9594 0.9586 0.9594 0.5.68344.68380.68349 0.86070798 0.8607079 0.86070797 0.5.84054.840948.840547 0.77880078 0.77879998 0.77880077 0.35.4906755.490766.4906760 0.70468809 0.7046873 0.70468808 0.45.56839.568368.56834 0.637685 0.637677 0.637684 0.55.733530.7335700.7335307 0.5769498 0.57694863 0.5769498 0.65.9554083.9554458.9554088 0.504578 0.50448 0.504576 0.75.70000.700338.700006 0.4736655 0.473646 0.4736653 0.85.33964685.33964963.33964689 0.474493 0.47434 0.47449 0.95.58570966.58576.58570968 0.386740 0.38673757 0.38674098.78883.78835.78885 0.36787944 0.36787545 0.36787939 In tble 3 we list te results obtined by wit n=5 nd compred wit BPF results given in [6] t m=36. As we see from tis tble, it is cler tt te result obtined by te present metod is better tn te results obtined by BPF metod. Moreover te Results of emple nd by nd togeter re plotted in fig. 3 nd sow tt solves system (. more ccurtely tn, becuse in we use modified midpoint metod for solving eq.(.4 insted midpoint metod.
Americn Journl of Applied Mtemtics 04; (5: 55-6 59 ble 3. e numericl solutions for emple obtined by nd BPF, [6] wit ect solutions. Ect wit (n=5 BPF wit (m=36, [6] ( ( ( ( ( ( 0.00005 e ( 5 0.00006 0.99999.0047 0.98470 0..057 0.90484.053 0.9048.64 0.89657 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.3.34986 0.7408.3499 0.7408.34547 0.7435 0.5.6487 0.60653.64879 0.6065.630 0.66 0.7.0375 0.49659.038 0.49659.098 0.4950 0.9.45960 0.40657.45964 0.4065.4365 0.4070.788 0.36788.783 0.3678.676 0.3740 Fig.. e error function vi, nd for emple wit n=0. 0.00005 e ( 5 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.00005 0.00008 0.00005 e ( Emple 0.00005 0.0000 9 e ( 6 Emple 5 3 0.0 0. 0.4 0.6 0.8.0 e ( 0.00005 0.00004 0.00003 0.0000 Emple 0.0 0. 0.4 0.6 0.8.0 0.0000 9 e ( 6 Emple 0.0000 3 0.0 0. 0.4 0.6 0.8.0 0.0 0. 0.4 0.6 0.8.0 Fig. 3. Comprsin error functions vi, nd for emple nd wit n=0. Emple 3: Consider te following system of liner Fredolm integrl equtions of te second kind: 3/ u ( = f ( ( + y u (ydy ( y u (ydy 0 0 4 3 u ( = f ( ( y u (ydy ( y u (ydy 0 0 were 7 7 6 f ( = + + + 60 30 35 9/ (3.3 4 9 7 5 9/ 7/ 5/ ( + + ( + ( +, 4 3 3 f ( 30 60 0 3 3 4 = + +. nd te ect solution ( u (,u ( (, 3 = + +. / In tis system, we note tt L 3 (,y = ( + y 4 not eists t (,y = (0,0. erefore, we used to solve tis system nd te results presented in tble 4 nd fig. 4.
60 lm Jsim Mjeed: Modified Midpoint Metod for olving ystem of Liner Fredolm Integrl Equtions of te econd Kind ble 4. e numericl results for emple 3 wit n=0 nd n=30. ( ( ( ( n=0 n=30 n=0 n=30 0 0 78 0 0-6 -3 0.05 0.005 0.0049936 0.0049999-0.047375-0.0473847-0.047375 0.5 0.05 0.049644 0.049996-0.45-0.4384-0.4508 0.5 0.065 0.0649333 0.064999-0.7875-0.788006-0.787506 0.35 0.5 0.49003 0.49988-0.8465-0.846864-0.846505 0.45 0.05 0.048653 0.049983-0.56375-0.5637746-0.5637503 0.55 0.305 0.304883 0.3049979-0.085-0.0864-0.0850 0.65 0.45 0.447895 0.449974 0.0475 0.047456 0.047499 0.75 0.565 0.5647488 0.5649969 0.34375 0.3437545 0.343750 0.85 0.75 0.747064 0.749965 0.48665 0.48666 0.486650 0.95 0.905 0.90466 0.9049958 0.809875 0.80987678 0.8098750 0.99996393 0.99999955.0000096.0000000 e ( e ( 0.00004 0.0000 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 0.0004 0.0008 0.000 Fig. 4. e error function vi, nd for emple 3 wit n=0. 4. Conclusions Modified Midpoint metod is pplied to te numericl solution for solving system of liner Fredolm integrl equtions. Numericl results, compred wit oter metods suc s te midpoint, rpezoidl, impson, Modified rpezoidl nd Block Plus Function metod, sow tt te presented metod is of iger precision nd from te illustrtive tbles, we conclude tt wen te number of subintervls n is incresed we cn obtin very good ccurcy. Also s cn be seen from ble 3, is better tn for solving system (.. References MMP 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 [] Rmp.P. Knwl, Liner Integrl Equtions, eory nd ecnique, Acdemic press, INC, 97. []. A. Burton, "Volterr Integrl nd Differentil Equtions", second ed., Elsevier, Neterlnds, 005. [3] K. E. Atkinson, e Numericl olution of Integrl Equtions of te econd Kind, Cmbridge University Press, 997. MMP [4] A. M. Wzwz, Liner nd nonliner integrl equtions: metods nd pplictions, Higer eduction, pringer, 0.. [5] K. Mleknejd, F. Mirzee, Numericl solution of liner Fredolm integrl equtions system by rtionlized Hr functions metod, Int. J. Comput. Mt. 80 ( (003 397 405. [6] K. Mleknejd, M. rezee, H. Ktmi, Numericl solution of integrl equtions system of te second kind by Block Pulse functions, J. Applied Mtemtics nd Computtion, 66 (005 5 4. [7] K. Mleknejd, N. Agzde, M. Rbbni, Numericl solution of second kind Fredolm integrl equtions system by using ylor-series epnsion metod, J. Appl. Mt. nd Comupt., Vol. 75, No., 006, pp 9-34. [8] M. Rbbni, K. Mleknejd, N. Agzde, Numericl computtionl solution of te Volterr integrl equtions system of te second kind by using n epnsion metod, J. Appl. Mt. nd Comupt., Vol. 87, No., 007, pp 43-46. [9] A. R. Vidi, M. Moktri, A. R. Vidi, On te decomposition metod for system of liner Fredolm integrl equtions of te econd Kind, J. Appl. Mt. cie., Vol., No., 008, pp 57-6. [0] E. Bbolin, Z. Msouri,. Htmzde-Vrmzyr, A direct metod for numericlly solving integrl equtions system using ortogonl tringulr functions, Int. J. Ind. Mt. 0, No., 009, pp 35-45. [] J.. Ndjfi, M. Heidri, olving Liner Integrl Equtions of te econd kind wit Composed Modified rpzoid Qudrture Metod", J. Appl. Mt. nd Comupt., 89(4(007, 980-985. []. J. Mjeed, Modified rpezoidl Metod for olving ystem of Liner Integrl Equtions of te econd Kind", J. Al-Nrin University, (4(009, pp.3-34. [3]. J. Mjeed, Numericl Metods for olving Liner Fredolm-Volterr Integro differentil Equtions of te econd Kind ", J. Al-Nrin University, 3((00, pp. 94-04.
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