U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 7 ISSN -77 A NUMERICAL METHOD OF SOLVING CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS BASED ON A LINEAR APPROXIMATION Cristia ŞERBĂNESCU, Marius BREBENEL A alterate method o umerical itegratio o irst order ordiar dieretial equatios (i.e. solvig Cauch problem) is proposed, based o the liearizatio o the epressio which does t cotai the derivative o the ukow uctio. It is proved that the approimatio is o order ad, b give eamples, the accurac o method is illustrated. The method is eteded or d order dieretial equatios ad oe shows that it is also o order o accurac, but or some classes o such equatios, the methods becomes o order. The advatage is that the proposed method applies directl to d order equatios, without the eed o solvig sstems o irst order. Kewords: Talor series; liear approimatio; order o approimatio; Cauch problem.. Itroductio The classical umerical methods or solvig irst-order ordiar dieretial equatios (ODE) with iitial values are based o approimatios derived rom the Talor epasios o the ukow uctio. The simplest (ad oldest) method is amed ater Euler ad is based o the irst order Talor approimatio o the uctio we are lookig or. However, the Euler method is ot as accurate as egieerig problems require, so that more precise methods are eeded to solve Cauch problems or ODE s ad sstems o ODE s. Toda, the most used such a method, speciall o solvig large sstems o -st order ODE s with costat coeiciets, is kow as Ruge-Kutta method (RK), which i act is a class o methods. The most applied versio belogig to this class is the 4-th order RK method ([], [4]). Other methods are also kow, which belog to amilies o liear multistep methods, like the eplicit Adams Bashorth methods or implicit Adams-Moulto methods, havig various orders o accurac ([], [5]). Liear multi-step methods usig higher-order derivatives have bee developed (such as Störmer method) as well ([6]). Assist. Pro., Departmet o Mathematical Methods ad Models, Uiversit POLITEHNICA o Bucharest, Romaia; email: mserbaescuc@ahoo.com Assist. Pro., Departmet o Aerospace Scieces Elie Caraoli, Uiversit POLITEHNICA o Bucharest,, Romaia, e-mail: mariusbreb@ahoo.com
6 Cristia Şerbăescu, Marius Brebeel Other methods use adaptative step sizes i order to improve the accurac o approimatio, b chagig the step size durig the itegratio. The implicit methods are ot metioed i this paper, sice the proposed method is a eplicit oe ad it will be compared with methods o the same categor. The method proposed i this paper combies the versatilit o Talor approimatios with the accurac give b the eact aaltic solutios o liear ODE o st order. Oe cosequece o this eature is that or some classes o ODE s, the order o the approimatio gets higher, usig the same method. This could be a advatage over the classical methods.. First order ODE irst method Oe cosiders the Cauch problem stated as ollows: id the uctio satisig the dieretial equatio ad the iitial coditio give below:, where C D, D R,,. () The irst proposed method cosists o takig the liear term rom the, with respect to the variable : Talor epasio o,,,. (), Oe ca observe that Euler method cosidered ol the irst term o (), was take i. B perormig the uctio chage where, i additio, the value o u () the equatio () becomes a liear ihomogeeous ODE o orm: u u,, u. (4) The geeral solutio o above equatio ca be epressed as u (, e ) d. (5) Accordig to (), it ollows that
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 7, e d. (6), Sice the iterval is i geeral small (it represets the step o the umerical algorithm), a mea value theorem ca be applied o the itegral i (6): e d, where, M M. (7) B computig the itegral, oe gets iall the approimate solutio o, : e,, where. (8) Remarks: ) Sice the uctio is supposed to be o class C, its partial derivatives ca be easil oud ad the above epressio ca be implemeted without diiculties i the computig codes. ) I the case whe, oe ca observe that the last ractio i (8) has a iite limit: (i other words, becomes: lim e is a apparet sigularit). The approimate solutio, The trivial case whe (9). () ca be easil treated usig the above ormula, which is similar but ot idetical to the ormulas give b other kow methods, like improved Euler, secod order Ruge-Kutta or Heu s method. ) For the liear ODE o irst order, the rule applies as ollows: p q, (), p q, p ()
8 Cristia Şerbăescu, Marius Brebeel p e p q. () p This epressio approimates the closed orm solutio p q e d (4) wherei the itegral caot be computed aalticall ad umerical approaches were eeded ivolvig thus a higher volume o computatios. The accurac o approimatio I the sequel, the Talor epasios o the eact solutio ad the approimate oe about will be aalzed. The Talor series o uctio about has the orm: (5).... 6 Usig (), the above epasio ca be rewritte as: where d, d d,... 6 d, (6). (7) B usig the basic rules o Calculus, the total derivatives o, respect to will get the ollowig eplicit orms: d d with (8) d. (9) d
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 9 It ollows that the Talor epasio o about gets the orm: ().... 6 A similar aalsis will be perormed o the approimate solutio (8). The epoetial uctio has the epasio: e... 6 () so that the correspodig term i (8) reduces to e 6... () The other term which has to be epaded is,,,...., (), 4.... (4) Itroducig i (8), oe ields:.... 6 (5) B comparig to (8), oe ca observe that the diereces occur at the term o order, which meas that the accurac o the method is o order. The et eample will illustrate the accurac o the proposed method, b comparig the results with those give b other kow methods (like Euler ad Ruge-Kutta o order 4). The ollowig Cauch problem is cosidered:
4 Cristia Şerbăescu, Marius Brebeel e, 5. (6) The equatio is o separable variables ad admits a closed aaltical solutio, such that it is eas to compare the approimate solutios with the eact oe: For the give eample, oe derives: e 5. (7), e,, e (8) 6 e. (9), The recurret ormula derived rom (8) is, e. () A costat step o h.5 was chose or the give eample. I the Table ad Fig., the umerical results ad the graphical represetatio are show. Comparative results o umerical solutios o Cauch problem (6) usig the -st proposed method ad other kow methods - eact proposed () RK - Euler.4474.4474.4474.4474..48465.484667.484649.48765..5568.5565.5568.558..5749.57497.5749.56674.4.69446.6957.69446.665.5.6788.6777.6787.6645.6.778.756.778.7956.7.7966.79696.7965.77954.8.8669.86684.8669.84.9.945.9496.944.9..744.84.74.98689..9587.957.9585.6549..499.7.497.49888 Table
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 4..994.447.99.944.4.45995.45557.4599.4.5.58456.588844.5845.48 Fig.. Graphical represetatio o the solutio o problem (6) usig the -st proposed method ad other kow methods. First order ODE secod method Ulike the irst proposed method, the secod oe uses the epasio o the ukow solutio i Talor series o two variables, takig ito cosideratio ol the liear terms:,,, (), or, i shorter otatios:,. () A ew liear ihomogeeous ODE o irst order is obtaied: () whose geeral aaltical solutio ca be writte as e C e d. (4) B computig the itegral, oe ields:
4 Cristia Şerbăescu, Marius Brebeel C e. (5) The costat C will be determied b applig the iitial value ad, iall, a ew approimate solutio o problem () is oud: e,. (6) Oe ca see agai that is a apparet sigularit, whe the approimate solutio becomes:, whe which is the well-kow Euler approimatio. (7) The accurac o approimatio Usig the epasio o the epoetial term (), the Talor epasio o the approimate solutio (6) gets the orm: (8).... 6 B comparig with the eact epasio (), the terms up to order are idetical ad, i additio, part o the term o order is the same. However, the accurac o this method is the same as the previous oe, amel is o order. From the poit o view o the volume o computatioal work, this method ivolves the additioal calculatio o the partial derivative. For the eample give i (8), the ollowig epressios are used:, e (9) is: e, (4) 6. e, The recurret ormula derived rom (6) used or umerical calculatios
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 4 (4) e,. The results obtaied b applig the secod proposed method o approimatio are preseted i the Table ad i Fig., i compariso with the eact solutio ad 4-th order Ruge-Kutta approimatio. As i the previous eample, a costat step o h. 5 was chose. Sice the method has the same order o accurac like the previous oe, amel order, the results are comparable, as epected. The 4-th order Ruge- Kutta method remais however the most precise oe, havig the disadvatage o a higher volume o computatioal work. 4. Secod order ODE I geeral, the -d order ODE are itegrated umericall b trasormig them irst ito a sstem o two -st order ODE s o which a appropriate method o itegratig is applied. I the et, a approimate method which applies directl o the -d order ODE is proposed, usig the same idea used i the case o the -st order ODE s. Table Comparative results o umerical solutios o Cauch problem (6) usig the -d proposed method ad other kow methods - eact proposed () RK.4474.447.4474..48465.48467.484649..5568.5566.5568..5749.5757.5749.4.69446.696.69446.5.6788.67.6787.6.778.7.778.7.7966.796488.7965.8.8669.866959.8669.9.945.9449.944.744.865.74..9587.54.9585..499.7.497
44 Cristia Şerbăescu, Marius Brebeel..994.965.99.4.45995.45594.4599.5.58456.58974.5845 Fig.. Graphical represetatio o the solutio o problem (6) usig the -d proposed method ad other kow methods orm: Let us cosider a -d order Cauch problem writte i the orm:,,,,. (4) The Talor epasio o,, with respect to ad has the,,,,....,,,, (4) Cosiderig ol the liear terms ad usig short otatios, oe ca write:,,,,. (44) The secod term i the right-had side ca be also approimated b usig Euler s rule: d. (45) d
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 45 Itroducig i (44), oe ields:,,,,. (46) With the uctio chage: z (47) the -d order dieretial equatio (4) gets the ollowig approimate orm, which is a ihomogeeous liear -st order ODE: z z,,, z. (48) The solutio o the above equatio ca be writte as z,, e d (49) ad applig agai a mea value theorem i the itegral, oe ields: where M becomes: becomes: z e d,, M e d (5) is give b (7). Ater computig the itegrals, the epressio o z,, z M B replacig e e. M (5), the approimate epressio o
46 Cristia Şerbăescu, Marius Brebeel,,. e e (5) A secod itegratio ields:,,. e d e d (5) Followig the same judgmet ad computig the itegrals, the ial epressio o the approimate solutio o problem (4) is oud:,, 4. e e (54) Remark: Like i the case o the -st order ODE aalzed, all the epressios cotaiig at deomiator have iite limits as (apparet sigularit). Thus: lim e, lim e 6 lim e. (55)
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 47 The accurac o approimatio The Talor epasio o the eact solutio about :... (56) 6 will be compared with the similar epasio o the approimate solutio (54). The series (56) ca be re-writte as:\,,.... 6 (57) The third derivative o has the epressio: d,, d,, so that the epasio (57) becomes: where... 6 (58) (59),,. (6) For the approimate solutio (54), the terms cotaiig the epoetial uctio have the kow epasios:... 6 e (6) e.... (6) 6 O the other had, the ollowig irst order approimatio applies:
48 Cristia Şerbăescu, Marius Brebeel,,,, 4 4 (6). 4 Itroducig all these results i (54), the ollowig Talor epasio o the approimate solutio is oud:.... 6 4 Comparig (64) with (59), oe ca see that the epasios dier b a quatit o order : eact appro (64) 4 O. (65) 4 I the particular case whe, the proposed approimatio becomes completel o order. Eample: 4 ;,. The eact solutio is cos. For the approimate solutio, oe writes: 4,, 4, 4,,. Sice, the ormulas (55) appl. The recurret epressio or umerical computatio, derived rom (54) ad wherei the above epressios are itroduced, will read: 4 4 (66) 6 where 4 4. (67) For computatio, a costat step o h. was chose. The results are preseted i the Table ad Fig., i compariso with the eact solutio. Sice, accordig to the above aalsis, the approimatio is o order ad thus, the Ruge-Kutta method o 4-th order is more precise. O the other had, the proposed method is much easier to be implemeted tha RK
A umerical method o solvig Cauch problem or dieretial eqs. [ ] liear approimatio 49 algorithm, ivolvig ewer eplicit ormulas or computatio (the dieretial equatio does t eed to be trasormed ito a sstem o two -st order ODE s). I the Table, the -st order derivatives o the uctio we are lookig or are also preseted, i compariso with the eact values, alog with the relative errors regardig the approimate solutio. Comparative results o eact ad approimate umerical solutios o Cauch problem (4) - eact proposed Error - eact Table proposed......4.69677.694988.79 -.447 -.4465.8 -.9 -.458.59 -.99947 -.6575. -.7794 -.748.5944 -.596 -.4474.6 -.99895 -.998684.89.6748.856 -.65644 -.6444.9.565.5798.4.87499.5.6.999.9995.8.775566.78869.5.65.997..9985.9965.58 -.98 -.76688.6.685.59657.7695 -.5876 -.654 4 -.455 -.76.666 -.97876 -.9885 Fig.. Graphical represetatio o the solutio o problem (4), showig the eact ad the approimate solutio
5 Cristia Şerbăescu, Marius Brebeel 5. Coclusios A alterate method o umerical itegratio o irst order ODEs (Cauch problem) was preseted i two variats, based o the liearizatio o the epressio which does t cotai the derivative o the ukow uctio. Both variats have the order o approimatio ad, b takig a smaller step o advacig, the accurac o solutio ca be improved. The advatage versus Ruge-Kutta o 4-th order (which is more precise at the same size o the step) cosists i the simplicit o implemetig o computer. The method was adapted also or d order dieretial equatios, obtaiig the order o accurac, but it was show that or some classes o equatios, the approimatio becomes o order. The advatage is that the proposed method applies directl to d order equatios, without the eed o trasormig them ito sstems o irst order. R E F E R E N C E S []. D.W. Zigg, T.T. Chisholm, Ruge Kutta methods or liear ordiar dieretial equatios, Applied Numerical Mathematics, pp. 7 8, 999. []. J.C. Butcher, Numerical Methods or Ordiar Dieretial Equatios, Wile, New York,. []. J.D. Lambert, Computatioal Methods i Ordiar Dieretial Equatios, Wile, New York, 97. [4]. D.H. Griiths, D.J. Higham, Numerical Methods or Ordiar Dieretial Equatios, Spriger-Verlag Lodo Limited,. [5]. K. Atkiso, W. Ha, D. Stewart, Numerical Solutio o Ordiar Dieretial Equatios, Joh Wile & Sos, Ic., Hoboke, New Jerse, 9. [6]. N.S. Bakhvalov, Numerical methods: aalsis, algebra, ordiar dieretial equatios, MIR, 977 (traslated rom Russia).