Ameica Joual of Computatioal Mathematics, 05, 5, 393 404 Published Olie Septembe 05 i SciRes. http://www.scip.og/joual/ajcm http://d.doi.og/0.436/ajcm.05.53034 A Compaative Stud o Numeical Solutios of Iitial Value Poblems (IVP) fo Odia Diffeetial Equatios (ODE) with Eule ad Ruge Kutta Methods Md. Amiul Islam Depatmet of Mathematics, Uttaa Uivesit, Dhaka, Bagladesh Email: amiul.math@gmail.com Received 7 August 05; accepted Septembe 05; published 5 Septembe 05 Copight 05 b autho ad Scietific Reseach Publishig Ic. This wok is licesed ude the Ceative Commos Attibutio Iteatioal Licese (CC BY). http://ceativecommos.og/liceses/b/4.0/ Abstact This pape mail pesets Eule method ad fouth ode Ruge Kutta Method (RK4) fo solvig iitial value poblems (IVP) fo odia diffeetial equatios (ODE). The two poposed methods ae quite efficiet ad pacticall well suited fo solvig these poblems. I ode to veif the accuac, we compae umeical solutios with the eact solutios. The umeical solutios ae i good ageemet with the eact solutios. Numeical compaisos betwee Eule method ad Ruge Kutta method have bee peseted. Also we compae the pefomace ad the computatioal effot of such methods. I ode to achieve highe accuac i the solutio, the step size eeds to be ve small. Fiall we ivestigate ad compute the eos of the two poposed methods fo diffeet step sizes to eamie supeioit. Seveal umeical eamples ae give to demostate the eliabilit ad efficiec. Kewods Iitial Value Poblem (IVP), Eule Method, Ruge Kutta Method, Eo Aalsis. Itoductio Diffeetial equatios ae commol used fo mathematical modelig i sciece ad egieeig. Ma poblems of mathematical phsics ca be stated i the fom of diffeetial equatios. These equatios also occu as efomulatios of othe mathematical poblems such as odia diffeetial equatios ad patial diffeetial equatios. I most eal life situatios, the diffeetial equatio that models the poblem is too complicated to How to cite this pape: Islam, Md.A. (05) A Compaative Stud o Numeical Solutios of Iitial Value Poblems (IVP) fo Odia Diffeetial Equatios (ODE) with Eule ad Ruge Kutta Methods. Ameica Joual of Computatioal Mathematics, 5, 393 404. http://d.doi.og/0.436/ajcm.05.53034
solve eactl, ad oe of two appoaches is take to appoimate the solutio. The fist appoach is to simplif the diffeetial equatio to oe that ca be solved eactl ad the use the solutio of the simplified equatio to appoimate the solutio to the oigial equatio. The othe appoach, which we will eamie i this pape, uses methods fo appoimatig the solutio of oigial poblem. This is the appoach that is most commol take sice the appoimatio methods give moe accuate esults ad ealistic eo ifomatio. Numeical methods ae geeall used fo solvig mathematical poblems that ae fomulated i sciece ad egieeig whee it is difficult o eve impossible to obtai eact solutios. Ol a limited umbe of diffeetial equatios ca be solved aalticall. Thee ae ma aaltical methods fo fidig the solutio of odia diffeetial equatios. Eve the thee eist a lage umbe of odia diffeetial equatios whose solutios caot be obtaied i closed fom b usig well-kow aaltical methods, whee we have to use the umeical methods to get the appoimate solutio of a diffeetial equatio ude the pescibed iitial coditio o coditios. Thee ae ma tpes of pactical umeical methods fo solvig iitial value poblems fo odia diffeetial equatios. I this pape we peset two stadad umeical methods Eule ad Ruge Kutta fo solvig iitial value poblems of odia diffeetial equatios. Fom the liteatue eview we ma ealize that seveal woks i umeical solutios of iitial value poblems usig Eule method ad Ruge Kutta method have bee caied out. Ma authos have attempted to solve iitial value poblems (IVP) to obtai high accuac apidl b usig umeous methods, such as Eule method ad Ruge Kutta method, ad also some othe methods. I [] the autho discussed accuac aalsis of umeical solutios of iitial value poblems (IVP) fo odia diffeetial equatios (ODE), ad also i [] the autho discussed accuate solutios of iitial value poblems fo odia diffeetial equatios with fouth-ode Ruge kutta method. [3] studied o some umeical methods fo solvig iitial value poblems i odia diffeetial equatios. [4]-[6] also studied umeical solutios of iitial value poblems fo odia diffeetial equatios usig vaious umeical methods. I this pape Eule method ad Ruge Kutta method ae applied without a discetizatio, tasfomatio o estictive assumptios fo solvig odia diffeetial equatios i iitial value poblems. The Eule method is taditioall the fist umeical techique. It is ve simple to udestad ad geometicall eas to aticulate but ot ve pactical; the method has limited accuac fo moe complicated fuctios. A moe obust ad iticate umeical techique is the Ruge Kutta method. This method is the most widel used oe sice it gives eliable statig values ad is paticulal suitable whe the computatio of highe deivatives is complicated. The umeical esults ae ve ecouagig. Fiall, two eamples of diffeet kids of odia diffeetial equatios ae give to veif the poposed fomulae. The esults of each umeical eample idicate that the covegece ad eo aalsis which ae discussed illustate the efficiec of the methods. The use of Eule method to solve the diffeetial equatio umeicall is less efficiet sice it equies h to be small fo obtaiig easoable accuac. It is oe of the oldest umeical methods used fo solvig a odia iitial value diffeetial equatio, whee the solutio will be obtaied as a set of tabulated values of vaiables ad. It is a simple ad sigle step but a cude umeical method of solvig fist-ode ODE, paticulal suitable fo quick pogammig because of thei geat simplicit, although thei accuac is ot high. But i Ruge Kutta method, the deivatives of highe ode ae ot equied ad the ae desiged to give geate accuac with the advatage of equiig ol the fuctioal values at some selected poits o the sub-iteval. Ruge Kutta method is a moe geeal ad impovised method as compaed to that of the Eule method. We obseve that i the Eule method ecessivel small step size coveges to aaltical solutio. So, lage umbe of computatio is eeded. I cotast, Ruge Kutta method gives bette esults ad it coveges faste to aaltical solutio ad has less iteatio to get accuac solutio. This pape is ogaized as follows: Sectio : poblem fomulatios; Sectio 3: eo aalsis; Sectio 4: umeical eamples; Sectio 5: discussio of esults; ad the last sectio: the coclusio of the pape.. Poblem Fomulatio I this sectio we coside two umeical methods fo fidig the appoimate solutios of the iitial value poblem (IVP) of the fist-ode odia diffeetial equatio has the fom f,, 0, () 0 0 whee d d, is the solutio of the Equatio (). I this ad f is a give fuctio ad 394
pape we detemie the solutio of this equatio o a fiite iteval, 0, statig with the iitial poit 0. A cotiuous appoimatio to the solutio will ot be obtaied; istead, appoimatios to will be geeated at vaious values, called mesh poits, i the iteval, 0. Numeical methods emplo the Equatio () to obtai appoimatios to the values of the solutio coespodig to vaious selected values of 0 h,,,3,. The paamete h is called the step size. The umeical solutios of () is give b a set of poits, : 0,,,, ad each poit, is a appoimatio to the coespodig, o the solutio cuve. poit.. Eule Method Eule s method is the simplest oe-step method. It is basic eplicit method fo umeical itegatio of odia diffeetial equatios. Eule poposed his method fo iitial value poblems (IVP) i 768. It is fist umeical method fo solvig IVP ad seves to illustate the cocepts ivolved i the advaced methods. It is impotat to stud because the eo aalsis is easie to udestad. The geeal fomula fo Eule appoimatio is hf,, 0,,, 3,... Ruge Kutta Method This method was devised b two Gema mathematicias, Ruge about 894 ad eteded b Kutta a few eas late. The Ruge Kutta method is most popula because it is quite accuate, stable ad eas to pogam. This method is distiguished b thei ode i the sese that the agee with Talo s seies solutio up to tems of h whee is the ode of the method. It do ot demad pio computatioal of highe deivatives of as i Talo s seies method. The fouth ode Ruge Kutta method (RK4) is widel used fo solvig iitial value poblems (IVP) fo odia diffeetial equatio (ODE). The geeal fomula fo Ruge Kutta appoimatio is k k k3 k4, 0,,,3, 6 h k whee h k k hf,, k hf,, k3 hf,, k4 hf h, k3. 3. Eo Aalsis Thee ae two tpes of eos i umeical solutio of odia diffeetial equatios. Roud-off eos ad Tucatio eos occu whe odia diffeetial equatios ae solved umeicall. Roudig eos oigiate fom the fact that computes ca ol epeset umbes usig a fied ad limited umbe of sigificat figues. Thus, such umbes o caot be epeseted eactl i compute memo. The discepac itoduced b this limitatio is call Roud-off eo. Tucatio eos i umeical aalsis aise whe appoimatios ae used to estimate some quatit. The accuac of the solutio will deped o how small we make the step size, h. A ulim ma 0 deotes the appoimate meical method is said to be coveget if. Whee h0 N solutio ad deotes the eact solutio. I this pape we coside two iitial value poblems to veif accuac of the poposed methods. The Appoimated solutio is evaluated b usig Mathematica softwae fo two e ma. poposed umeical methods at diffeet step size. The maimum eo is defied b 4. Numeical Eamples steps I this sectio we coside two umeical eamples to pove which umeical methods covege faste to aaltical solutio. Numeical esults ad eos ae computed ad the outcomes ae epeseted b gaphicall. Eample : we coside the iitial value poblem, 0 o the iteval 0. The π eact solutio of the give poblem is give b e ef e. The appoimate esults ad maimum eos ae obtaied ad show i Tables (a)-(d) ad the gaphs of the umeical solutios ae displaed i Figues -7. 395
Table. (a) Numeical appoimatios ad maimum eos fo step size h 0. ; (b) Numeical appoimatios ad maimum eos fo step size h 0.05 ; (c) Numeical appoimatios ad maimum eos fo step size h 0.05 ; (d) Numeical appoimatios ad maimum eos fo step size h 0.05. Eule Method h 0. Ruge Kutta Method h 0. (a) e e Eact Solutio 0..0000000000000000 5.3465E 03.005346480083334 4.6045E 08.00534658840 0..00000000000000.8895E 0.08893798037348 8.676E 08.08894647599 0.3.03599999999998.9970E 0.05598407370900.309E 07.05599637660336 0.4.075765999999998 3.0044E 0.05387896458685.6335E 07.0538959706604 0.5.34876640000000 4.6873E 0.76974766744460.05805E 07.7697497589769 0.6.600047000000 5.86769E 0.746787363539485.5564E 07.746789997767 0.7.349670030000 7.906E 0.403987995370888 3.3030E 07.403988384007750 0.8.4667095934400.05078E 0.57787347344033 4.694E 07.57787769675660 0.9.648046836889793.3860E 0.7866655850639 6.00769E 07.786665853690383.0.87737044909877.8037E 0.0594065035735 9.085E 07.05940740534576 Eule Method h 0.05 Ruge Kutta Method h 0.05 (b) e e Eact Solutio 0..0065000000000.75E 03.0053465953968.59745E 09.0053465884 0..068460937500 6.06530E 03.0889457303 5.549E 09.08894647599 0.3.0449798075787.0E 0.0559956678 7.6495E 09.05599637660336 0.4.08999708545087.5399E 0.0538948609584.0097E 08.0538959706604 0.5.550367600056364.938E 0.7697495985963.6597E 08.7697497589769 0.6.4443907678436 3.0400E 0.7467897637353.56044E 08.746789997767 0.7.36339794960348 4.08485E 0.4039889883075.95705E 08.40398838400775 0.8.5730030075834 5.44875E 0.577877439369459.57387E 08.57787769675660 0.9.745498646493 7.39E 0.786665873944439 3.646E 08.786665853690383.0.9643507036668488 9.50567E 0.05940735064544 5.4697E 08.05940740534576 Eule Method h 0.05 Ruge Kutta Method h 0.05 (c) e e Eact Solutio 0..0039734390899.3733E 03.00534656505684.680E 0.0053465884 0..0985464503 3.06405E 03.088946535374 3.760E 0.08894647599 0.3.0500685934876 5.650E 03.055996389464 4.76790E 0.05599637660336 0.4.097503874045 7.79857E 03.05389534068 6.8600E 0.0538959706604 0.5.6584975698874.5E 0.7697497734603 7.84350E 0.7697497589769 0.6.59343793899.53576E 0.746789905588 9.650E 0.746789997767 0.7.383068996306.07776E 0.403988379999.066E 09.40398838400775 0.8.5440607986967.7766E 0.5778776800385.57534E 09.57787769675660 0.9.7498544658 3.6834E 0.786665854067.636E 09.786665853690383.0.0079538470343 4.863E 0.059407409860655 3.3565E 09.05940740534576 396
(d) Eule Method h 0.05 Ruge Kutta Method h 0.05 e e Eact Solutio 0..0046566697375803 6.8985E 04.00534658070.0399E.0053465884 0..03494875897.54003E 03.08894645595.00999E.08894647599 0.3.055943397385.59763E 03.0559963736754.97600E.05599637660336 0.4.03943547977403 3.9460E 03.053895934784 3.9900E.0538959706604 0.5.7373539445 5.606E 03.7697497470777 4.8800E.7697497589769 0.6.66939048903 7.73979E 03.74678999793 5.97400E.746789997767 0.7.3935085607509.04798E 0.4039883836378 7.44000E.40398838400775 0.8.557773406756774.4044E 0.57787769578305 9.73599E.57787769675660 0.9.76806478579363.860E 0.7866658534807.36930E 0.786665853690383.0.034808463635.4587E 0.05940740534904.07670E 0.05940740534576.4 Eact value. Eact values.8.6.4. diffeet values of Figue. Eact umeical solutios..4. RK4 appoimatio Eule appoimatio Appoimate values.8.6.4. diffeet values of Figue. Numeical appoimatio fo step size h = 0.. 397
.4. RK4 appoimatio Eule appoimatio Appoimate values.8.6.4. diffeet values of Figue 3. Numeical appoimatio fo step size h = 0.05..4. RK4 appoimatio Eule appoimatio Appoimate values.8.6.4. diffeet values of Figue 4. Numeical appoimatio fo step size h = 0.05..4. RK4 appoimatio Eule appoimatio Appoimate values.8.6.4. diffeet values of Figue 5. Numeical appoimatio fo step size h = 0.05. 398
0. 0.8 0.6 h=0. h=0.05 h=0.05 h=0.05 0.4 maimum eos 0. 0. 0.08 0.06 0.04 0.0 0 diffeet values of Figue 6. Eo fo diffeet step size usig Eule method. 0-6 Figue.6:Eos fo diffet step size usig RK4 method h=0. 0.9 h=0.05 h=0.05 0.8 h=0.05 0.7 maimums eos 0.6 0.5 0.4 0.3 0. 0. 0 diffeet values of Figue 7. Eo fo diffeet step size usig RK4 method. Eample : we coside the iitial value poblem, 0 o the iteval 0. The e eact solutio of the give poblem is give b. The appoimate esults ad mai- πefi mum eos ae obtaied ad show i Tables (a)-(d) ad the gaphs of the umeical values ae displaed i Figues 8-4. 5. Discussio of Results The obtaied esults ae show i Tables (a)-(d) ad Tables (a)-(d) ad gaphicall epesetatios ae show i Figues -7 ad Figues 8-4. The appoimated solutio is calculated with step sizes 0., 0.05, 0.05 ad 0.05 ad maimum eos also ae calculated at specified step size. Fom the tables fo each method we sa that a umeical solutio coveges to the eact solutio if the step size leads to deceased eos such that i the limit whe the step size to zeo the eos go to zeo. We see that the Eule appoimatios usig the step size 0. ad 0.05 does ot covege to eact solutio but fo step size 0.05 ad 0.05 covege slowl to eact 399
Table. (a) Numeical appoimatios ad maimum eos fo step; size h 0. ; (b) Numeical appoimatios ad maimum eos fo step size h 0.05 ; (c) Numeical appoimatios ad maimum eos fo step size h 0.05 ; (d) Numeical appoimatios ad maimum eos fo step size h 0.05. Eule Method h 0. Rugee Kutta Method h 0. (a) e e Eact Solutio 0. 0.9000000000000000.3509E 0 0.935089300458.95878E 07 0.93509789878 0. 0.88000000000000.85E 0 0.849870604544 3.4764E 07 0.84985870443 0.3 0.77600600000000.588E 0 0.8089448683 4.53096E 07 0.8083397957603 0.4 0.739063799679744.8798E 0 0.767783066058 5.4033E 07 0.76778358659507 0.5 0.74004866778 3.06849E 0 0.74468984640 5.73E 07 0.7446897004786337 0.6 0.69874774484 3.636E 0 0.73088779650559 6.0668E 07 0.730888407785085 0.7 0.698669656969 3.3447E 0 0.755066587579 6.33400E 07 0.7559970983 0.8 0.693908587047 3.46350E 0 0.77064958635 6.56635E 07 0.770708676577 0.9 0.69984770349557 3.59008E 0 0.73574909580043 6.78965E 07 0.735743588544958.0 0.7385063096446 3.7897E 0 0.753964993897 7.005E 07 0.7540359579868 Eule Method h 0.05 Ruge Kutta Method h 0.05 (b) e e Eact Solutio 0. 0.907500000000000 6.593E 03 0.93509376563 6.583E 09 0.93509789878 0. 0.8396097770965 9.95759E 03 0.84985048488676.38536E 08 0.84985870443 0.3 0.789588457598809.349E 0 0.8083378795.97359E 08 0.8083397957603 0.4 0.75407436706578.37093E 0 0.7677835603639.469E 08 0.76778358659507 0.5 0.7995707544795.4736E 0 0.74468967309678.7385E 08 0.7446897004786337 0.6 0.753744093633583.5540E 0 0.730888378944706.98840E 08 0.730888407785085 0.7 0.70906950338550.688E 0 0.755673367656 3.9353E 08 0.7559970983 0.8 0.700989998883.6873E 0 0.7707054669 3.37550E 08 0.770708676577 0.9 0.78708868437886.7477E 0 0.73574355305839 3.5493E 08 0.735743588544958.0 0.73958735770343.886E 0 0.754034709988 3.7487E 08 0.7540359579868 Eule Method h 0.05 Ruge Kutta Method h 0.05 (c) e e Eact Solutio 0. 0.9048759053907 3.060E 03 0.935097639753.5909E 0 0.93509789878 0. 0.844384774477937 4.83374E 03 0.8498580509749 6.5469E 0 0.84985870443 0.3 0.795858785640485 5.96467E 03 0.808339695988 9.97784E 0 0.8083397957603 0.4 0.76077773078 6.70587E 03 0.767783584889969.6955E 09 0.76778358659507 0.5 0.737463994496 7.57E 03 0.7446896989999765.47866E 09 0.7446897004786337 0.6 0.7363490608 7.655E 03 0.73088840344937.6440E 09 0.730888407785085 0.7 0.77840690076906 7.9673E 03 0.755974898684.783E 09 0.7559970983 0.8 0.78735476438908 8.96E 03 0.77070843788.90588E 09 0.770708676577 0.9 0.77934975998 8.647E 03 0.73574358650894.0387E 09 0.735743588544958.0 0.74585358368 8.9884E 03 0.754034985703.48E 09 0.7540359579868 400
Eule Method h 0.05 Ruge Kutta Method h 0.05 (d) e e Eact Solutio 0. 0.903644337686.48548E 03 0.93509788699.790E 0.93509789878 0. 0.84683597686434.3854E 03 0.8498586679586 3.4485E 0.84985870443 0.3 0.79887748685904.9459E 03 0.80833979055 5.5477E 0.8083397957603 0.4 0.76446694883 3.379E 03 0.767783586087474 7.3600E 0.76778358659507 0.5 0.74067096053 3.57903E 03 0.7446897003930565 8.55770E 0.7446897004786337 0.6 0.77075635837666 3.78084E 03 0.7308884068344 9.6640E 0.730888407785085 0.7 0.79759738653 3.9537E 03 0.7559967009.05089E 0 0.7559970983 0.8 0.7909634547887 4.745E 03 0.770708604554.303E 0 0.770708676577 0.9 0.7345864856067 4.8494E 03 0.7357435884407.075E 0 0.735743588544958.0 0.74667590788 4.46444E 03 0.7540358958.8405E 0 0.7540359579868 0.9 0.9 Eact value Eact values 0.88 0.86 0.84 0.8 0.8 0.78 0.76 0.74 0.7 diffeet values of Figue 8. Eact umeical solutios. 0.95 RK4 appoimatio Eule appoimatio 0.9 Appoimate values 0.85 0.8 0.75 0.7 0.65 diffeet values of Figue 9. Numeical appoimatio fo step size h = 0.. 40
0.95 RK4 appoimatio Eule appoimatio 0.9 Appoimate values 0.85 0.8 0.75 0.7 diffeet values of Figue 0. Numeical appoimatio fo step size h = 0.05. 0.95 RK4 appoimatio Eule appoimatio 0.9 Appoimate values 0.85 0.8 0.75 0.7 diffeet values of Figue. Numeical appoimatio fo step size h = 0.05. 0.9 0.9 RK4 appoimatio Eule appoimatio 0.88 0.86 Appoimate values 0.84 0.8 0.8 0.78 0.76 0.74 0.7 diffeet values of Figue. Numeical appoimatio fo step size h = 0.05. 40
0.04 0.035 0.03 h=0. h=0.05 h=0.05 h=0.05 maimum eos 0.05 0.0 0.05 0.0 0.005 0 diffeet values of Figue 3. Eo fo diffeet step size usig Eule method. 8 0-7 Figue.6:Eos fo diffet step size usig RK4 method h=0. h=0.05 7 h=0.05 h=0.05 6 maimums eos 5 4 3 0 diffeet values of Figue 4. Eo fo diffeet step size usig RK4 method. solutio. Also we see that the Ruge Kutta appoimatios fo same step size covege fistl to eact solutio. This shows that the small step size povides the bette appoimatio. The Ruge Kutta method of ode fou equies fou evaluatios pe step, so it should give moe accuate esults tha Eule method with oe-fouth the step size if it is to be supeio. Fiall we obseve that the fouth ode Ruge Kutta method is covegig faste tha the Eule method ad it is the most effective method fo solvig iitial value poblems fo odia diffeetial equatios. 6. Coclusio I this pape, Eule method ad Ruge Kutta method ae used fo solvig odia diffeetial equatio (ODE) i iitial value poblems (IVP). Fidig moe accuate esults eeds the step size smalle fo all methods. Fom the figues we ca see the accuac of the methods fo deceasig the step size h ad the gaph of the appoimate solutio appoaches to the gaph of the eact solutio. The umeical solutios obtaied b the two poposed methods ae i good ageemet with eact solutios. Compaig the esults of the two methods ude ivestigatio, we obseved that the ate of covegece of Eule s method is Oh ad the ate of covegece 4 of fouth-ode Ruge Kutta method is Oh. The Eule method was foud to be less accuate due to the iaccuate umeical esults that wee obtaied fom the appoimate solutio i compaiso to the eact solutio. 403
Fom the stud the Ruge Kutta method was foud to be geeall moe accuate ad also the appoimate solutio coveged faste to the eact solutio whe compaed to the Eule method. It ma be cocluded that the Ruge Kutta method is poweful ad moe efficiet i fidig umeical solutios of iitial value poblems (IVP). Refeeces [] Islam, Md.A. (05) Accuac Aalsis of Numeical solutios of Iitial Value Poblems (IVP) fo Odia Diffeetial Equatios (ODE). IOSR Joual of Mathematics,, 8-3. [] Islam, Md.A. (05) Accuate Solutios of Iitial Value Poblems fo Odia Diffeetial Equatios with Fouth Ode Ruge Kutta Method. Joual of Mathematics Reseach, 7, 4-45. http://d.doi.og/0.5539/jm.v73p4 [3] Oguide, R.B., Fadugba, S.E. ad Okulola, J.T. (0) O Some Numeical Methods fo Solvig Iitial Value Poblems i Odia Diffeetial Equatios. IOSR Joual of Mathematics,, 5-3. http://d.doi.og/0.9790/578-0353 [4] Shampie, L.F. ad Watts, H.A. (97) Compaig Eo Estimatos fo Ruge-Kutta Methods. Mathematics of Computatio, 5, 445-455. http://d.doi.og/0.090/s005-578-97-09738-9 [5] Eaqub Ali, S.M. (006) A Tet Book of Numeical Methods with Compute Pogammig. Beaut Publicatio, Khula. [6] Akabi, M.A. (00) Popagatio of Eos i Eule Method, Scholas Reseach Liba. Achives of Applied Sciece Reseach,, 457-469. [7] Kockle, N. (994) Numeical Method fo Odia Sstems of Iitial value Poblems. Joh Wile ad Sos, New Yok. [8] Lambet, J.D. (973) Computatioal Methods i Odia Diffeetial Equatios. Wile, New Yok. [9] Gea, C.W. (97) Numeical Iitial Value Poblems i Odia Diffeetial Equatios. Petice-Hall, Uppe Saddle Rive. [0] Hall, G. ad Watt, J.M. (976) Mode Numeical Methods fo Odia Diffeetial Equatios. Ofod Uivesit Pess, Ofod. [] Hossai, Md.S., Bhattachajee, P.K. ad Hossai, Md.E. (03) Numeical Aalsis. Titas Publicatios, Dhaka. [] Balaguusam, E. (006) Numeical Methods. Tata McGaw-Hill, New Delhi. [3] Sast, S.S. (000) Itoducto Methods of Numeical Aalsis. Petice-Hall, Idia. [4] Bude, R.L. ad Faies, J.D. (00) Numeical Aalsis. Bagaloe, Idia. [5] Geald, C.F. ad Wheatle, P.O. (00) Applied Numeical Aalsis. Peaso Educatio, Idia. [6] Mathews, J.H. (005) Numeical Methods fo Mathematics, Sciece ad Egieeig. Petice-Hall, Idia. 404