Application of Maple on the Differential Problem

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1 Uivrsl Jourl o Applid Si : - DOI:./ujs.. Applitio o Mpl o th Dirtil Prol Chii-Hui Yu Dprtt o Mt d Iortio N Jo Uivrsit o Si d Tholo Ti Cit Tiw *Corrspodi Author: hiihui@il.ju.du.tw Copriht Horizo Rsrh Pulishi All rihts rsrvd. Astrt This ppr uss th thtil sotwr Mpl or th uilir tool to stud th dirtil prol o two tps o utios. W oti th iiit sris ors o ordr drivtivs o ths two tps o utios usi Liiz dirtil rul dirtitio tr tr d itrtio tr tr. Ad h rtl rdu th diiult o lulti th hihr ordr drivtiv vlus o ths utios. O th othr hd w propos two pls to do lultio prtill. Th rsrh thods doptd i this stud ivolvd idi solutios throuh ul lultios d vrii ths solutios usi Mpl. This tp o rsrh thod ot ol llows th disovr o lultio rrors ut lso hlps odi th oriil dirtios o thii ro ul d Mpl lultios. For this rso Mpl provids isihts d uid rrdi prol-solvi thods. Kwords Drivtivs Iiit Sris Fors Liiz Dirtil Rul Dirtitio Tr B Tr Itrtio Tr B Tr Mpl. Itrodutio As iortio tholo dvs whthr oputrs o oprl with hu ris to pror strt tss suh s strt rt siilr to th pitis o Pisso d usil opositios siilr to thos o Mozrt is turl qustio. Currtl this pprs uttil. I dditio whthr oputrs solv strt d diiult thtil prols d dvlop strt thtil thoris suh s thos o thtiis lso pprs usil. Nvrthlss i si or ltrtivs w stud wht ssist thtil sotwr provid. This stud itrodus how to odut thtil rsrh usi th thtil sotwr Mpl. Th i rsos o usi Mpl i this stud r its sipl istrutios d s o us whih l irs to lr th oprti thiqus i short priod. B ploi th powrul oputi pilitis o Mpl diiult prols sil solvd. Ev wh Mpl ot dtri th solutio prol-solvi hits idtiid d irrd ro th pproit vlus lultd d solutios to siilr prols s dtrid Mpl. For this rso Mpl provid isihts ito sitii rsrh. Iquiri throuh oli support sst providd Mpl or rowsi th Mpl wsit ilitt urthr udrstdi o Mpl d iht provid uptd isihts. For th istrutios d oprtios o Mpl [-] doptd s rrs. I lulus d iri thtis urriul vluti th -th ordr drivtiv vlu o utio t i rl ds to o throuh two produrs: irstl dtrii th -th ordr drivtiv o d th ti ito. Ths two produrs will us with irsil opl lultios wh lulti hihr ordr drivtiv vlus o this utio i.. is lr d h to oti th swrs ul lultios is ot s. I this ppr w il stud th dirtil prol o th ollowi two tps o utios t ot Whr r rl urs. W oti th iiit sris ors o ordr drivtivs o ths two tps o utios usi Liiz dirtil rul dirtitio tr tr d itrtio tr tr ; ths r th jor rsults o this stud i.. Thors d d h rtl rdu th diiult o lulti thir hihr ordr drivtiv vlus. For th stud o rltd dirtil prols rr to [-]. I dditio w provid two pls to do lultio prtill. Th rsrh thods doptd i this stud ivolvd idi solutios throuh ul lultios d vrii ths solutios usi Mpl. This tp o rsrh thod ot ol llows th disovr o lultio rrors ut lso hlps odi th oriil dirtios o thii ro ul d Mpl lultios. Thror Mpl provids isihts d uid rrdi prol-solvi thods.. Mi Rsults

2 Applitio o Mpl o th Dirtil Prol Firstl w itrodu ottio d orul usd i this ppr... Nottio Suppos r is rl ur is positiv itr. Di r r r r d r... Gotri sris I u is rl ur u. Th u u Nt w itrodu thr iportt thors usd i this stud... Dirtitio tr tr [ p] I or ll o-tiv itr th utios : R stis th ollowi thr oditios:i thr ists poit suh tht is ovrt ii ll utios r dirtil o op itrvl iii d is uiorl d ovrt o. Th is uiorl ovrt d dirtil o. Morovr its d drivtiv d. d d.. Itrtio tr tr [ p] Suppos { } is squ o Lsu itrl utios did o itvl I. I is I ovrt th. I I.. Liiz dirtil rul [ p] Suppos is o-tiv itr d r -tis dirtil utios. Th th -th ordr drivtiv o th produt utio! whr.!! Bor drivi th irst jor rsults i this stud w d l... L Suppos is rl ur. Th th ivrs tt utio t... Proo Bus th drivtiv o I th i t d t d i i B otri sris Usi itrtio tr tr w oti t. O th othr hd wh ± holds. Thus I us t / or. / B otri sris

3 Uivrsl Jourl o Applid Si : - Thror itrtio tr tr w hv t C Whr C is ostt. Ti w oti / C. Thror t or. Th s o ti ito w oti / C. Thus t or Th ollowi is th irst jor rsult i this stud w dtri th iiit sris ors o ordr drivtivs o th utio... Thor Lt rl urs. Suppos th doi o th utio t is { } ist R. Cs : I d. Th th -th ordr drivtiv o Cs : I th Cs : I th... Proo Cs : I th o L w hv t Dirtiti -tis with rspt to o oth sids o w oti B Liiz dirtil rul B dirtitio tr tr Cs : I th o L w hv t Dirtiti -tis with rspt to o oth sids o w oti B Liiz dirtil rul B dirtitio tr tr Cs : I o L w oti

4 Applitio o Mpl o th Dirtil Prol t Dirtiti -tis with rspt to o oth sids o w hv B Liiz dirtil rul B dirtitio tr tr Also to oti th sod jor rsults i this stud w d l... L Suppos is rl ur. Th th ivrs ott utio ot i i i... Proo Bus ot t th L w iditl oti this rsult. Th ollowi is th sod jor rsult i this ppr w oti th iiit sris ors o ordr drivtivs o th utio... Thor Lt th ssuptios th s s Thor. Suppos th doi o th utio ot is { } ist R. Cs : I d. Th th -th ordr drivtiv o Cs : I th Cs : I th... Proo Cs : I d. B o L w oti ot Dirtiti -tis with rspt to o oth sids o w t B Liiz dirtil rul

5 Uivrsl Jourl o Applid Si : - B dirtitio tr tr Cs : I th o L w hv ot Dirtiti -tis with rspt to o oth sids o w oti B Liiz dirtil rul B dirtitio tr tr Cs : I th o L w oti ot Dirtiti -tis with rspt to o oth sids o w hv B Liiz dirtil rul B dirtitio tr tr. Epls I th ollowi or th dirtil prol o th two tps o utios i this stud w provid two pls d us Thors to dtri th iiit sris ors o ordr drivtivs d so hihr ordr drivtiv vlus o ths utios. O th othr hd w plo Mpl to lult th pproitios o ths hihr ordr drivtiv vlus d thir solutios or vrii our swrs... Epl Suppos th doi o th utio / t is { } R th s o i Thor.... I d. B Cs o Thor w oti -th ordr drivtiv o Thus w vlut th -th ordr drivtiv vlu o t / W us Mpl to vri th orrtss o s ollows: :-p**^^/*rt^; vld@@/; vlp*susu!/!*-!*^-*-^* produt*/-jj..-/**/^* /-..iiit..;

6 Applitio o Mpl o th Dirtil Prol... I. B Cs o Thor w oti -th ordr drivtiv o H w dtri th -th ordr drivtiv vlu o t / Also w us Mpl to vri th orrtss o. vld@@/; vlpi/*p*su!/!*-!*^-*produt /-jj..-*/^/-..-p*susu! /!*-!*^-*-^*produt-*-/-pp..- /**/^-*-/-..iiit..;... I. B Cs o Thor w oti -th ordr drivtiv o Thus th -th ordr drivtiv vlu o t Usi Mpl to vri th orrtss o s ollows: vld@@-;. vl-pi/*p-*su!/!*-!*^-* produt/-jj..-*^/*-/^..-p- *susu!/!*-!*^-*-^*produt- *-/-pp..-/**/^*-/^/ *-/^..iiit..;... Epl Lt th doi o th utio / ot { } R th s o i Thor.... I d. B Cs o Thor w oti -th ordr drivtiv o Thror th -th ordr drivtiv vlu o t / W plo Mpl to vri th orrtss o s ollows: :-p**^^/*rot^;

7 Uivrsl Jourl o Applid Si : - vld@@/; vlpi/*p/*su!/!*-!*^-*produt /-jj..-*/^/-..-p/*susu!/!*-!*^-*-^*produt*/-pp.. -/**/^*/-..iiit.. ;... I. B Cs o Thor w dtri -th ordr drivtiv o Thus th -th ordr drivtiv vlu o t Vrii th orrtss o s ollows: vld@@; vlp*susu!/!*-!*^-*-^ *produt-*-/-pp..-/**^-*-/-..iiit..; Usi Mpl to vri th orrtss o. vld@@-; vlpi*p-*su!/!*-!*^-*produt /-jj..-*^^/*-/^..p- *susu!/!*-!*^-*-^*produt-* -/-pp..-/**/^^*-/^/* -/^..iiit..;. Colusio As tiod th Liiz dirtil rul th dirtitio tr tr d th itrtio tr tr pl siiit rols i th thortil irs o this stud. I t th pplitios o ths thors r tsiv d usd to sil solv diiult prols; w dvor to odut urthr studis o rltd pplitios. O th othr hd Mpl lso pls vitl ssistiv rol i prol-solvi. I th utur w will td th rsrh topi to othr lulus d iri thtis prols d solv ths prols usi Mpl. Ths rsults will usd s thi trils or Mpl o dutio d rsrh to h th oottios o lulus d iri thtis. REFERENCES... I. B Cs o Thor w oti -th ordr drivtiv o H w dtri th -th ordr drivtiv vlu o t [] F. Grv Th Mpl Boo Lodo: Chp & Hll/CRC. [] C. Toi d S. G. Ads Applid Mpl or Eirs d Sitists Bosto: Arth Hous. [] M. L. All d J. P. Brslto Mpl Epl rd d. Nw Yor: Elsvir Adi Prss. [] R. J. Stror d J. F. Ksho Disovri Mthtis with Mpl : A Itrtiv Eplortio or Mthtiis Eirs d Eootriis Bsl: Birhusr Vrl. [] D. Rihrds Advd Mthtil Mthods with Mpl Nw Yor: Crid Uivrsit Prss. [] J. S. Rortso Eiri Mthtis with Mpl Nw Yor: MGrw-Hill. [] C. T. J. Dodso d E. A. Gozlz Eprits i

8 Applitio o Mpl o th Dirtil Prol Mthtis Usi Mpl Nw Yor: Sprir-Vrl. [] A. Griw d A. Wlthr Evluti Drivtivs: Priipls d Thiqus o Alorithi Dirtitio d d. SIAM Phildlphi. [] M. Wr A. Wlthr d B. J. Shr O th iit oputtio o hih-ordr drivtivs or ipliitl did utios Coputr Phsis Couitios Vol. Issu. pp.. [] M. A. Pttrso M. Wisti d A. V. Ro A iit ovrlodd thod or oputi drivtivs o thtil utios i MATLAB ACM Trstios o Mthtil Sotwr TOMS Vol. Issu.. [] C. -H. Yu A stud o so dirtil prols with Mpl Prodis o th th IEEE/Itrtiol Cor o Advd Ioo Tholo No.. [] C. -H. Yu Usi Mpl to vlut th drivtivs o so utios Itrtiol Jourl o Rsrh i Coputr Applitios d Rootis Vol. Issu. pp. -. [] C. -H. Yu Applitio o Mpl o solvi so dirtil prols Prodis o IIE Asi Cor Vol. pp. -. [] C. -H. Yu Th dirtil prol o so utios Itrtiol Jourl o Coputr Si d Moil Applitios Vol. Issu. pp. -. [] C. -H. Yu Th drivtivs o so utios Itrtiol Jourl o Rsrh i Iortio Tholo Vol. Issu. pp. -. [] C. -H. Yu Evluti th drivtivs o triootri utios with Mpl Itrtiol Jourl o Rsrh i Coputr Applitios d Rootis Vol. Issu. pp. -. [] C. -H. Yu A Stud o th dirtil prol Itrtiol Jourl o Rsrh i Aroutil d Mhil Eiri Vol. Issu. pp. -. [] C. -H. Yu Th dirtil prol o two tps o utios Itrtiol Jourl o Coputr Si d Moil Coputi Vol. Issu. pp. -. [] C. -H. Yu Evluti th drivtivs o two tps o utios Itrtiol Jourl o Coputr Si d Moil Coputi Vol. Issu. pp. -. [] C. -H. Yu Th dirtil prol o our tps o utios Jourl o K-Ni vol. pp. -. [] C. -H. Yu A stud o th dirtil prols usi Mpl Itrtiol Jourl o Coputr Si d Moil Coputi Vol. Issu. pp. -. [] T. M. Apostol Mthtil Alsis d d. Bosto: Addiso-Wsl..