Numerical Solutions of Second Order Initial Value Problems of Bratu-Type equation Using Higher Ordered Rungu-Kutta Method

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1 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 87 ISSN Numercal Solutos of Secod Order Ital Value Problems of Bratu-Tpe equato Usg Hgher Ordered Rugu-Kutta Method Habtamu Garoma Debela * Habtamu Bassa Yadeta ** Solomo Bat Kejela *** Departmet of Mathematcs Jmma Uverst Ethopa Abstract Ths paper presets the ffth order Ruge-Kutta method RK5 to fd the umercal soluto of the secod order tal value problems of Bratu-tpe ordar dfferetal equatos. I order to justf the valdt ad effectveess of the method we solve three model eamples ad compare the eact solutos to umercal solutos. The umercal results terms of pot wse absolute errors preseted tables ad the plotted graphs show that the preset method appromates the eact soluto ver well. Besdes the stablt of the method had bee checked ad verfed. Ide Terms: Ital-Value Problem; Bratu-tpe Equato; Numercal Soluto; Ruge-Kutta method M I. INTRODUCTION a problems scece ad egeerg ca be formulated terms of dfferetal equatos. A dfferetal equato s a equato volvg a relato betwee a ukow fucto ad oe or more of ts dervatves. Equatos volvg dervatves of ol oe depedet varable are called ordar dfferetal equatos ad ma be classfed as ether tal-value problems IVP or boudar-value problems BVP. Ma authors have attempted to solve tal value problems IVP to obta hgh accurac rapdl b usg umerous methods such as Talor s method Ruge-Kutta method ad also some other methods. A more robust ad trcate umercal techque s the Ruge Kutta method. Ths method s the most wdel used oe sce t gves relable startg values ad s partcularl sutable whe the computato of hgher dervatves s complcated. O the other had accordg to Abukhaled M. et al. the stadard Bratu problem s used a large varet of applcatos such as the fuel gto model of the theor of thermal combusto the thermal reacto process model the Chadrasekhar model of the epaso of the uverse radatve heat trasfer aotecholog ad theor of chemcal reacto. The Bratu tal value problems have bee studed etesvel because of ts mathematcal ad phscal propertes. Batha B studed a umercal soluto of Bratu-tpe equatos b the varatoal terato method; Feg et al.8 cosdered Bratu s problems b meas of modfed homotop perturbato method; Rashda J. et al. 3 appled Sc-Galerk method for umercal soluto of the Bratu s problems; Sam ad Hamda6 used varatoal terato method for umercal solutos of the Bratu-tpe problems; Wazwaz A5 appled Adoma decomposto method to stud the Bratu-tpe equatos. L.J appled modfed

2 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 88 ISSN varatoal terato method to Bratu- tpe problems. J-Hua He et al. 4 cosdered varatoal terato method for Bratu-lke equatos arsg electrospg. Sarav M. et al. 3 studed soluto of Bratu s equato b He s varato terato method. Motvated b the above vestgatos the objectve of the preset stud s to vestgate umercal solutos of secod order tal value problems of Bratu-tpe equato usg hgher ordered Rugu-kutta method RK5. II. FORMULATION OF THE METHOD Cosder the secod order tal value problem of Bratu-Tpe equato of the form: Subject to the tal codtos + = < λe g l = α = β where λα ad β are gve costat umbers for s ukow fucto. To reduce the order of Eq. let we use the substtutos z = ad z = so that the gve secod order tal value problem of Eq. wth Eq. ca be re-wrtte as: = z = F z = α z = g λe = Gz z = = β Dvdg the terval [ l ] to N equal subterval of mesh legth h 3 ad the mesh pot s gve b = + h for =... N. For the sake of smplct let use the deotato: z = z g = g etc. Thus at the odal pot Eq. 3 s wrtte as: = = F z = α z = G z z = β 4 where G z = g λe To solve each tal value problems wrtte Eq. 4 we appl the sgle step methods that requre formato about the soluto at to calculate at + Grewal Ja et al 7. From oe of the sgle step methods ad the faml of Ruge Kutta methods the geeral umercal soluto of Eq. 4 usg the ffth Ruge Kutta method gve as:

3 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 89 ISSN z = + + = = z + + = 5 5 wk wk 5 where 4 4 k = hf + ch + ajk j z + ajmj j= j= 4 4 m = hg + ch + ajkj z + ajmj j= j= Chrstodoulou 9 was preset the ffth order Ruge Kutta method to solve a frst order tal value problem of d the form = f t t = whch s gve b the followg equato: dt 7k + 3k + k + 3k + 7k = + 6 where k = hf h k k = f t h k k f t = + + k + 4k + 6k k + 8k k6 = ft+ h h 3k + k k3 = f t h 3k + 6k + 9k k5 = f t Thus to solve the sstem of tal value problem of Eq. 3 the ffth order Ruge Kutta method ca be re-wrtte as: z + + 7k+ 3k3 + k4 + 3k5 + 7k6 = + 9 7m + 3m + m + 3m + 7m = z where: k = F z m = G z h k m k = hf + + z + h k m m = hg + + z + h 3k+ k 3m+ m h 3k+ k 3m+ m k3 = hf + + z + m3 = hg + + z

4 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 9 ISSN h k3 m3 k4 = hf + + z + h k m m hg z = h 3k + 6k + 9k 3m + 6m + 9m k5 = hf + + z h 3k + 6k + 9k 3m + 6m + 9m m5 = hg + + z k + 4k + 6k k + 8k m + 4m + 6m m + 8m k6 = hf + h + z k + 4k + 6k k + 8k m + 4m + 6m m + 8m m6 = hg + h + z I the determato of the parameters sce the terms are up to 5 h 5 ad the order of method soh. Grewal ad Ja et al 7. be compared the trucato error s 6 Oh III. STABILITY ANALYSIS Here the secod order tal value problem Bratu-Tpe equato of the form of Eq. s reduced to frst order sstem of equatos of the form of Eq. 4 ad let take the secod equato from Eq. 4 the we have: z = G z z = β 8 where G z = g λe The olear fucto Eq. 8 ca be learzed b epadg the fucto G Talor seres about the pot z ad trucatg t after the frst term as: G G z = G z + z + z G + z z z z 9 B the dfferetato rules of fucto of several varables Eq. 9 ca be wrtte as:

5 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 9 ISSN z = g λe + g λe + λ e = C ' where C= g λe + g λe + λ e ' ' z = C whch s lear the fucto of z. Let the test fucto s ' = λ whch s called the lear test equato for the o-lear Eq. 8. The soluto of the test equato Eq. s: = = e = e λ h λh where s costat. Now b applg Eq. 6 o Eq. we have: m h = λ m = λh + λh 3 3 h λ h λ m3 = hλ λh h λ h λ m4 = λh m5 = λh + λ h + λ h + h λ + λ h m6 = λh + λ h + λ h + h λ + h λ + λ h B substtutg the values of mad m3 m6 to obta: 7m + 3m + m + 3m + 7m = + we = λh λ h λ h λ h λ h λ h E λh + = 3

6 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 9 ISSN where: E λh = + λh+ λ h + λ h + λ h + λ h + λ h The errors umercal computato do t grow f the propagato error teds to zero or f at least bouded Ja et al 7. Now from Eq. t s easl observed the eact value of creases for the costat λ > ad decreases for λ < wth the factor of h decreases wth the factor of E λ h. Ifλ h > the stable. If λ h <.e. λ < 5.64 < λh <. h e λ. Whle from Eq. the appromate value of creases or e λ ; so the ffth order Ruge Kutta method s relatvel the the ffth order Ruge Kutta method s absolutel stable the terval of IV. NUMERICAL EXAMPLES AND RESULTS As dscussed above we are mplemetg the ffth order Ruge Kutta method o three model eample of the secod order tal value problems of Bratu s - tpe equato as follows: Eample : Cosder the Bratu-tpe tal value problem " e = ; < < = ' = whose eact soluto s = lcos Table. Comparso of absolute errors for eample Absolute errors at h =. h =. Darwsh ad Eslam Morad 5 Sa ad Necdet Our Method Our Method Kashkar 4 6 RK5 RK e e e e e e-6 3.e e-5.7e e e-5.3e-6.754e e-8.454e e-4 e e-6.333e-8.956e e-4.9e e e-8.43e e-4 4.e e e-8.938e e-4 6.5e e e-8 3.4e e-3 7.5e-6.8e e e e e e-4.578e e e e e e e-4

7 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 93 ISSN Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure : Plot of eact ad appromated solutos of Bratu tpe- Equato usg RK5 for h =..4. Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure : Plot of eact ad appromated soluto of Bratu tpe- Equato usg RK5 for h =.

8 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 94 ISSN Eample : Cosder the Bratu-tpe tal value problem d = π e ; = ' = π d Whose eact soluto s = l + s π Table : Comparso of absolute errors for eample Absolute errors at h =. h =. Eslam Morad 5 Our Method Our Method RK5 RK e e-8.668e e e e e-5.638e e e e e e e e-3.66e-4.99e e-3.7.e e e e-4.494e e e-4.794e e e e e-3.7 Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure 3: Plot of eact ad appromated solutos of Bratu tpe- Equato usg RK5 for h =.

9 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 95 ISSN Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure 4: Plot of eact ad appromated soluto of Bratu tpe- Equato usg RK5 for h =. Eample 3: Cosder the Bratu-tpe tal value problem " e = ; < < = ' = whose eact soluto s = lsec : Table 3: Comparso of absolute errors for eample 3 Absolute errors at h =. h =. Eslam Morad 5 Our Method Our Method RK5 RK e-6.454e e e e e e e e-5.4.e-6.66e e e e-8.e-4.9e-6.3e e e e-8.74e e e e e e-8.8e-4..4e-5.377e-7.444e-4

10 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 96 ISSN Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure 5: Plot of eact ad appromated solutos of Bratu tpe- Equato usg RK5 for h =..7 Numercal soluto Eact Soluto Eact ad Appromate solutos Fgure 6: Plot of eact ad appromated soluto b Bratu tpe- Equato usg RK for h =.

11 Iteratoal Joural of Scetfc ad Research Publcatos Volume 7 Issue October 7 97 ISSN V. DISCUSSION AND CONCLUSION I ths paper we preseted hgher order Rag Kutta RK5 method to vestgate umercal solutos of secod order tal value problems of Bratu-tpe equato. To further justf the applcablt of the proposed method; tables of pot wse absolute errors ad graphs have bee plotted for the three model eamples to compare the eact solutos ad umercal solutos at dfferet mesh sze h. Tables ad 3 depcted that the ffth order Rug Kutta method mproves the fdgs of Darwsh ad Kashkar 4 Eslam Morad 5 Sa ad Necdet 6. Moreover t s evdet that all the absolute errors decrease rapdl as the mesh sze h decreases whch tur shows that the smaller mesh sze provdes the better appromate value. Fgures 6 show that the preset method appromates the eact soluto a ecellet maer. VI. REFERENCES [] Abukhaled M. Khur S. ad Saf A. Sple-Based Numercal Treatmets of Bratu-Tpe Equatos. Pales- te Joural of Mathematcs [] Wazwaz A. A Relable Stud for Etesos of the Bratu Problem wth Boudar Codtos. MathematcalMethods the Appled Sceces [3] Batha B. Numercal Soluto of Bratu-Tpe Equatos b the Varatoal Iterato Method. Hacettepe Joural of Mathematcs ad Statstcs [4] Feg X. He Y. ad Meg J. 8 Applcato of Homotop Perturbato Method to the Bratu-Tpe Equatos.Topologcal Methods Nolear Aalss [5] Rashda J. Malekejad K. ad Taher N. 3 Sc-Galerk Method for Numercal Soluto of the Bratu s Problems. Numercal Algorthms [6] Sam M.I. ad Hamda A. 6 A Effcet Method for Solvg Bratu Equatos. Appled Mathematcs adcomputato [7] Wazwaz A. 5 Adoma Decomposto Method for a Relable Treatmet of the Bratu-Tpe Equatos. Appled Mathematcs ad Computato [8] Grewal B S Numercal method egeerg ad scece wth programs FORTRAN 77 C ad C++ khaa publsher sth edto [9] Ja M. K. Iegar S.R.K ad Ja R.K. 7. Numercal methods for scetfc ad Egeerg Computatos; equatos 3rd edto. [] A. Islam.5. Accurate Solutos of Ital Value Problems for Ordar Dfferetal Equatos wth Fourth Order Ruge Kutta Method. Joural of Mathematcs Research [] Nkolaos S. Chrstodoulou. 9. A Algorthm Usg Ruge-Kutta Methods OfOrders 4 Ad 5 For Sstems Of Odes Iteratoal Joural of Numercal Methods ad Applcatos [] L. J Applcato of Modfed Varatoal Iterato Me- thod to the Bratu-Tpe Problems Iteratoal Jour- al of Egeerg Cotemporar Mathematcs ad Sc- eces Vol. 4 pp [3] Sarav M. Herma M. & Kaser D. 3. Soluto of Bratu s equato b He s varatoal terato method. Amerca Joural of Computatoal ad Appled Mathematcs VII. AUTHORS Frst Author- Habtamu Garoma Debela Jmma Uverst Ethopa E-mal: habte@gmal.com Secod Author - Habtamu Bassa Yadeta Jmma Uverst Ethopa E mal: habtamubassa@gmal.com Thrd Author- Solomo Bat Kejela Jmma Uverst Ethopa E mal: solomobat@ahoo.com