SOLVING INITIAL VALUE PROBLEM USING RUNGE-KUTTA 6 th ORDER METHOD

Transcription

1 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. SOLVING INITIAL VALUE PROBLEM USING RUNGEUTTA th ORDER METHOD As Fdhl As AlShmmr Deprtmet o Eletrl Egeerg College o Egeerg Uverst o AlMustsrh Bgdd Irq EMl: husse_l8@hoo.om ABSTRACT The tl vlue prolems IVPs ordr deretl equtos re umerll solved oe step explt methods or deret order the ehvor o Ruge utt o thrd order method s oted Heu [] utt oud the omplete lssto d dervto o ourth d th order methods [] the dervto o sxth order method ws oud Hut [] ut eght stges Buther [] [] [] preseted the relto etwee the order odtos d the rooted trees up to sxth order. the m m o ths pper s to exht ew more smplest represetto or the trees d the dervto o Ruge utt method o order sx wth seve stges ludg rooted trees order odto d stlt rego smolll omputtos re used the stud to smpl the method ll exmple llustrte the method re preseted. ewords: tl vlue prolems ordr deretl equto rugeutt methods order o methods.. INTRODUCTION Ordr deretl equtos rse requetl severl models o mthemtl phss ologl sees egeerg d ppled mthemts. Uortutel m ot e solved extl. Ths s wh umerl tretmet s ver mportt d provdes powerul ltertve tool or solvg the deretl equtos. Whh re modeled s tl vlue prolems IVPs?There re vrous lsses o methods ths dreto. Tlor seres method hs mor dsdvtge t requres evluto o prtl dervtves o hgher orders mull d ths s ot possle prtl pplto thereore we see or mproved methods whh do ot eed evluto or repeted deretto o the deretl equto Although the mous pper Ruge d susequetl developed Heu d utt stll the explt Rugeutt o the th order method hve ee wdel used d the most populr verso s the lssl th order the Ruge pper s ow reogzed s the strtg pot or moder oestep methods wth multvlued d multstge ostruto o ths method eeds the dervto d soluto o m oler lger sstem o equtos lled the order odtos thus the lulto proess were perormed usg Mthemt progrm ths sstem o oler lger equtos hs m solutos thereore we must guess the tegrto step sze usg trl d error to estmte the truto error ths s the m drw o Rugeutt methods. smpled dervto o ourth order Ruge utt method gve [] the dervto o th order method were trodue utt [] d orreted Nstrom [] d the sxth order wth eght stges ouded Hut [] ths s short re hstor o the method lso we d tht t s mpossle to preset geerl ormul to the order odtos or ll mles o Rugeutt methods ut there s oeto e preseted ths pper etwee the severl sequees orders. Compltos rse spell t the odtos or order ve d more. Now we outle the orgzto o ths pper: set. oused o mthemtl ormulto d dervto o the explt Ruge utt Buther o the sx order method set. we preset the smplest ew tree represetto relted to deretl elemetr or the order odtos o sxth orders wth seve stges. set. exhts the ler omto etwee the order odtos. The stlt rego o the method s preseted se. ; ll we gve umerl test exmple set... OUTLINE AND MATHEMATICAL DERIVATION OF THE METHOD I ths seto we wll trodue the s tools requred the dervto to ot the order odto or Ruge utt o sx orders wth seve stges method the tpll geerl orm o the slr rst order o ordr deretl equto s x x x Whh s ow wth tl odto the tl vlue prolem IVB the m m ths seto s to ot pproxmte soluto o x the pproh o the sxth order wth seve stges Rugeutt method s proeed to evlute s pproxmto to x x h mportt spel se wthout loss o geerlt the uto does ot deped o x ut ol o settg x the equto redues to so lled utoomous d wrtte the orm x x x the proposed explt Rugeutt method o order sx wth seve stges deoted or oe step ordg to ths method the soluto o equto t the rst ed step te the orm 9

2 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. h d the ext soluto o equto hs the orm Y x h Oh d ordg to Tlor seres expso equto te the orm x h Y where re slops dee s elow or dg requred the ext seve termedte lultos: x x x x x x x h h h h h h h h h h h h h Where the vetor C dted to the odes [] weghts wth the step tpll the mtrx A dted to the depedee o the stges o dervtves oud t prevous stges d the row sum odto hold... the vetor B T dted to the qudrture weghts whh the l result deped o the dervtves the pproh o R methods s expded equto Tlor s seres expso ter some lger smplto ths expso s equted to the ext soluto x h tht s lso gve the Tlor s seres x h x h x h x! h x O h! the rst step s to lulte the suessve dervtves o up to order sx usg h rule or equto s the ollowg: x x x x x 8 9 x x 8 Susttute equtos 8 d ter some smpltos up to order sx d seve stges we ot the equto 9

3 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. h h h h... O h se equto s slr utoomous prolem so we m use smpled pproh or ste see [] expressos le... re ow s elemetr deretls the re relted to rooted trees whh s deed the ext seto.. ROOTED TREE THEORY Frstl we wll lss the odes or trees to ew three represetto tpes the rst d lled the tl ode root whh s usull exst eh tree d deoted t the smol the seod tpe s lled the termedte odes ole odes tht te the smol the thrd tpe o odes s med the les terml odes tht te the smol. I ths seto we eed to gve some mportt detos A rooted tree s set o p umer o the deret tpes o odes pequl to the order o the tree s the mesure o how muh odes the tree hs oeted edges oreted w rom the spe ode lled root so t s ew smplest omtorl grphs tht ot o les o oeto etwee tl d terml odes. Grphll we represet tree tht orrespodg to elemetr deretl or root... ode lelled we me the th prtl dervtves o the uto w.r.t depedet vrle tthed to su trees orrespodg to those su lsts lsts wth o su lsts re leves. Eh tree s med egg rom the tl ode root pssg through the termedte odes or ruhes to the terml odes leves d the dex strt rom dow ode root suessvel to upper odes depedg o the umer o rhes tht growths exept the leves odes d rom let rh to rght or ths purpose we preset gure elow smple exmple o treet o order umer o totl deret tpes o odes s lws the root ode lelled the ole odes lelled d l d eglet the leves wthout lel F t s the elemetr deretl uto gve terms o expresso o seve ompoets umer o ompoets s equl to the order tree t s the weghted uto tht orrespod to eh elemetr deretl ormul Ft d t s the dest o the tree whh equl to the produt o the umers odes ove. t t where t re the dest o tt the su trees ove the smmetr o the tree t s the set o permutto geerted ll memers o smmetr or the tree t the mppg t : T T tht preserves the root d the struture whh equl to how the tree smmetr hs Fgure. Smple tree o order seve wth elemetr deretl d weght utos d ts dest. elow we llustrte the orrespodee etwee elemetr deretl d rooted trees usg oth ormulto Tle or trees up to order sx wth ts dest. Tle. Illustrte the relto etwee elemetr deretl d rooted trees up to order. Order Ft F tree t 8 9

4 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. Order Ft tree t Order Ft Tree t Order Ft Tree t 8 Order Ft Tree t We wrte equtos d terms o the p t h weghted utos s ollows: x h x F t t p! p 9

5 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. p t t h F t p! Where p p! t t t d t p! t Rugeutt o order sxth wth seve stges re overges t t omprg equto wth see [].the results re sstem o oler lger equtos whh s deoted the order odtos whh s rrged ths pper regulr rrgemet d sequetl up to sxth order wth seve stges the ollowg twet two odtos must e stsed equtos e ddto to the ostrts gve equto [ [ [ ] ] ]!!!!!! l l l [ [ l [ m m l l l l ] l l l!!! ]! lm m ]!! d e d solve m o them tht sts the sxth order Ruge utt method d ths proess lulto requred usg Mthemt progrm.. REDUCTION THE ORDER CONDITIONS Se the umer o rooted trees eg rese or order greter th our the umer o odtos rese d omplext or order sx or ths reso t s eessr to d reltoshps ler omto etwee the results order odtos orrespodg to deret trees or exmple the two prtulr trees t d t wth the sme sx order whh der ol smll respet Fgure. Illustrte two equvlet prtulr trees. Hee t t re equvlet d the ler omto etwee the two odtos d t t gve ths result stte tht two trees re equvlet tht orrespods t e expressed terms o tht orrespods t d trees o order. Smlrl we lsted ll odtos whh lower e wrtte s ler omto d sts the Rugeutt method o order sx d seve stges elow.we ssume the moded odtos CD C l l or... d l... D or

6 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved ppled to ll stges the ol odto tht rem t t or the trees Fgure. Illustrte the trees tht ot e wrtte s ler omtos. The ler omtos pper equtos 9e9h elow; we lst ll odtos whh re eeded to spe the method d...9e g...9h L...9m...9 We solve or the ompoets T rom equtos 99d the solve or d rom the osstet equtos 9e9h we the solve or rom 9 d the or d rom equtos 9 wth = d rom 9L9 we ompute the rst olum o the mtrx A rom 9 d the lst row rom 9 ll we gve the ollowg exmple or Buther tleu rom the ove equtos tht sts the sx order method wth seve stges.

7 8 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved STABILITY REGION I ths seto we wll stud ddto to the umerl method overges to the ext soluto over ouded tervl the qulttve or stlt lss tht ssg to the tess o the method wth the used umerl soluto euse the omplext o ths tpe o lss we restre to the ler prolem wth ostt oeets so lled the st prolem or test equto where s ostt d lso omplex ture. h / h / h / h / h / h / h / h / h / h / h / h / h / h / h / 8 h / h / 8 h /8 h / 8 h / h / h / 8 h / h / h /8 h / h / h / h / h / h / h / h / h/8 h/ h/ h / h / h / h /8 h / h / h / h / h / h / h / h / h / h / h / 8h / h / h / 8 h / h / h /8 h /h / h / h / h / h / h / h / h / 8h / 9 h /9 8h / 9 h / h / h / h / h / h / 8h / 9 h / h / h / h / h / h / h /9 h / 8 h / h / h / 8 h / h / h / 8 h / h / h / h / h / h / 8h / 9 h / h / h / h / 8 h / h / h / h / h / h / h / h / h / h / h / 8 h / h / h / 8 h / h / h / 8 h / h / h / h / h / h / Susttutg z h d the vlue o ove the th order seve stges method elow h / 8 99

8 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. we get h / z z z z 8 z z 9 smpled d dvde oth sde the the stlt poloml Q z / gve 8 Q z z z z z z z z Fgure show the stlt rego o the sx order seve stges o Ruge utt method ths stlt regothe rge or the rel prt o s. Re z.. Tle. Illustrte the umerl soluto o exmple. usg R th order wth seve stges. X usg order sx ext Error CONCLUSIONS I ths pper we hve smpled the trees represetto whh relted to elemetr deretls tht t uses dervto Rugeutt methods d to redue the omplext tht rse dexg the trees t should e metoed tht the lssl Rugeutt proess o order our m e stll more eooml the dervto th the sx order. ACNOWLEDGEMENTS The uthor would le to th AlMustsrh Uverst Bghdd Irq or ts support the preset wor. REFERENCES Fgure. the stlt rego o the Ruge utt method o sx order seve stges.. NUMERICAL PROBLEM I ths seto we tsted the umerl soluto o tl vlue prolem where the method o Rugeutt o order sx wth seve stges s ppled to show the ee d the pproxmte o the method d ompred the result wth the ext soluto. The prolem re solved or x [..] Prolem. Solve the tl vlue prolem s usg Ruge utt method o order sx wth seve stges where the ext soluto gve x ls ot.8see [] J. C. Buther. 8. Numerl Methods or Ordr Deretl Equtos d Ed.Wle. [] J. C. Buther. 9. O th d Sxth OrderExplt Rugeutt MethodsOrderCodto d Order Brrers. Cd Appled Mthemts Qurterl.. [] J.C. Buther.. Numerl methods or ordr deretl equtos the th etur. J. o Comput. d Appled Mth. : 9. []. Heu. 9. Neue Methode zur pproxmtve Itegrto der Deretl glehuge eer uägge Veräderlhe Z. Mth. Phs. : 8. [] A. Hut. 9. Ue mélorto de l méthode de RugeuttNstrom pour l résoluto umérque des équtos déretelles du premer ordere At F. Nt. Uv. Come. Mth. :. 9

9 VOL. NO. JULY ISSN 898 ARPN Jourl o Egeerg d Appled Sees As Reserh Pulshg Networ ARPN. All rghts reserved. [] W. utt. 9. Betrg zur äherugswese Itegrto totler Deretl glehuge Z. Mth. Phs. :. [] Ppgeorgou G. d Tstours Ch.. Ruge utt Prs or slrutoomous tl Vlue Prolems It. J. Comput. Mth.8: 9. [8] C. Ruge. 89. Üer de umershe Aulösug vo Deretl glehuge Mth. A.: 8. [9] E.J. Nström. 9. Üer de umershe tegrto vo deretl glehuge At. So. S. Fee. :. [] Del I. OuorRoert D. Seel. 99. Col Rugeutt Nström methods o order ve d sx. o Comp. d Appled Mth. : 8. [] l Bshr M.Isml Mohmmed M. 8. Ruge utt methods o hgh order or solvg st prolems. AlRde J. o omputer sees d Mthemts. p.. [] Mohmmed M. Isml.. GoeeJohso sxorder Rugeutt method J. Edu. & S. p.. [] J usls.. Numerl methods egeerg wth Mth L d Ed. Cmrdge. [] Mus H. Irhm S. M. Wzr.. A smpled Dervto d Alss o Fourth Order Rugeutt Method It. J. o Computer pplto. 9:. 9