
samavaakyangalude deferential-algibraic sisttam

electrical enchineeyaringil, deferential-algibraic samavaakyangalude samvidaanam (day) onnukil deferential samavaakyangalum beejaganita samavaakyangalum adangiyirikkunna allengil atharam oru samvidaanathinu thulyamaaya samavaakyangalude oru samvidaanamaanu.
ganithashaasthrathil iva "deferential beejaganita enangalude" udaaharanangalaan, koodaathe deferential polynomial ringukalile idealukalumaayi poruthappedunnu (beejaganita sajjeekaranathinulla deferential beejaganithathekkurichulla lekhanam kaanuka). 
x" enna veriyabilukalude ashrita vektarinaayi namukk eee deferential samavaakyangal oru swatantra variable "t" ll ezhuthaam,
eee chinnangale oru yadhaartha veriyabilinte pravarthanangalaayi kanakkaakkumbol (electrical enchineeyaringilo niyanthrana sidhaanthathilo ulla prayogangalilennapole) nammal formula _ 2 neey dependent variable formula _ 3-nte oru vektaraayi kaanunnu, koodaathe systethinu niravadhi samavaakyangalund, avaa function formula _ 4 aayi kanakkakkunnu.
avaa saadhaarana deferential samavaakyathil (ooo. dii. iii) ninnu vyathyasthamaan, kaaranam "x" functionte ellah gadakangaludeyum derivetteevukalkk oru day poornnamaayum pariharikkanaavilla, kaaranam ivayellaam prathyakshappedanamennilla (i.iii. chila samavaakyangal beejaganithamaanu); saankethikamaayi oru avyakthamaaya oodu sisttavum (athu vyakthamaayi vivarikkaam) oru day sisttavum thammilulla vyathyasam jacobian matricks formula _ 5 oru day sisttathinte eaka matricks aanu ennathaanu. deysukalkku vyathyastha swabhaavasavisheshathakalundennathu saadhaaranayaayi pariharikkan kooduthal budhimuttullathaayathum kondaanu oodsum dais thammilulla eee vyathyasam.
praayogikamaayi paranjaal, dais oodukalum thammilulla vyathyasam palappozhum oru day sisttathinte pariharam inputt signalinte derivetteevukale aasrayichirikkunnu ennathaanu, oodsilennapoole signal mathramalla; ssmitt trigger polulla histereesulla nonleeniyar sisttangalil eee prashnam saadhaaranayaayi abhimukeekarikkunnu.
sisttam maattiyezhuthukayaanenkil eee vyathyasam kooduthal vyakthamaayi drishyamaakum, angane "x" nuu pakaram dependent veriyabilukalude vektarukalude oru jody formula _ 6 nammal pariganikkunnu, koodaathe dake roopamundu.
eee roopathilulla oru day samvidaanathe "semi-eksplisit" ennu vilikkunnu. samavaakyathinte randaam pakuthiyile "g" enna oro pariharavum samavaakyangalude aadya pakuthiyiloode "x" ennathinu oru savishesha disha nirvachikkunnu, athesamayam "y" ennathinte disha ekapaksheeyamaanu. ennaal ellah pooyintukalum (x, y, t) "g" yude parihaaramalla. samavaakyangalude "x", aadya pakuthi "f" ennivayile variabilukalkku "deferential" enna attribute labikkum. samavaakyangalude "y", randaam pakuthi "g" ennivayude gadakangale "beejaganita" variableukal allengil sisttathinte samavaakyangal ennu vilikkunnu. [deysinte pashchaathalatthil "beejaganita" enna padatthinte ardham "derivateevukal illaathathu" ennaanu, ithu (amoortha) beejaganithavumaayi bandhappettathalla.]
oru deyude parihaarathinu rund bhaagangalund, aadyam sthiramaaya prarambha moolyangalkkaayulla thirayal, randaamathethu oru sanjaarapathathinte kanakkukoottal. sthiramaaya prarambha moolyangal kandethunnathinu deyude chila ghadaka pravartthanangalude derivateevukal pariganikkendathu palappozhum aavasyamaanu. eee prakriyaykku aavashyamaaya oru derivetteevinte ettavum uyarnna kramatthe "deferentiation indux" ennu vilikkunnu. soochika kanakkukoottunnathil ninnu uruthirinja samavaakyangalum sthiramaaya prarambha moolyangalum sanjaarapathathinte kanakkukoottalil upayogapradamaayekkaam. deferentiation soochika onnu kurachukondu oru ardha-vyakthamaaya day systethe avyakthamaayi parivartthanam cheyyaan kazhiyum, thirichum.
mattu tharatthilulla days.
chila ashrita variableukal avayude derivateevukal illaathe sambhavikkukayaanenkil deysukaludeyum odesukaludeyum vyathyasam vyakthamaakum. dependent veriyabilukalude vekttar pinneedu jody formula _ 6 aayi ezhuthaam, koodaathe deyude deferential samavaakyangalude samvidaanam roopathil prathyakshappedunnu.
evide
mothatthil, deysukalude koottam oru pravarthanamaanu.
prarambha vyavasthakal roopathinte samavaakyangalude samvidaanathinte pariharamaayirikkanam.
udaaharanagal.
cartician kordinettukalil ("x", "y") (0,0) madhyathilulla "l" neelamulla oru pendulatthinte swabhaavam euler-lagranch samavaakyangal vivarikkunnu.
ivide soothravaakyam _ 26 oru lagranch gunakamaanu. u", "v" ennee momentam variableukal oorjja samrakshana niyamathaal parimithappedutthukayum avayude disha vrithathiloode choondikkaanikkukayum venam. aaa samavakyangalil rund vyavasthakalum vyakthamalla. avasaanathe samavaakyathinte vyathyasam nayikkunnath
chalanatthinte disha vrithathinte sparshanathilekku parimithappedutthuka. eee samavaakyathinte adutha derivative soochippikkunnathu
aaa avasana aidantittiyude derivative formula 29 lake lalithamaakkunnu, ithu oorjja samrakshanathe soochippikkunnu, kaaranam samyojanathinu sesham sthiramaaya formula 30 gathikorjjathinteyum pottensial oorjjathinteyum aakethukayaan.
ellah ashrita veriyabilukalkkum advitheeya derivative moolyangal labhikkunnathin avasaanathe samavaakyam moonniratti vyathyasthamaayirunnu. ithu 3 enna deferentiation soochika nalkunnu, ithu parimithamaaya mechanical systengalkku saadhaaranamaanu.
prarambha moolyangalaaya formula _ 31 umm "y" ennathinulla chihnavum nalkiyittundenkil, mattu variableukal formula _ 32 vazhiyum formula _ 33 aanenkil formula _ 34 umm formula _ 35 umm vazhiyum nirnnayikkunnu. adutha poyintilekku pookaan "x", "u" ennivayude derivateevukal labhikkaan mathiyakum, athaayath, ippol pariharikkenda sisttam
ithu soochika 1-nte ardha-vyakthamaaya day aanu. samanamaaya samavaakyangalude mattoru koottam formula _ 37-ll ninnu aarambhich "x" ennathinu oru chihnamaayi labikkum.
noonn-leanier upakaranangalulla sarkyoottukalude modelingilum swaabhaavikamaayum days sambhavikkunnu. udaaharanathinu, numeric surcuted simulettarukalude sarvvyaapiyaaya sugandhavyanjana kudumbatthil deysukal upayogikkunna parishkkaricha nodel vishakalanam upayogikkunnu. athupole, oru nettu listtil ninnu deysukal nedunnathinum chila sandarbhangalil samavaakyangal lalithamaakkaano pratheekaathmakamaayi pariharikkaano polum franhoferinte analog insides maathamaattikkal paakkej upayogikkaam. positteev feedbaakkukalulla kappaasittarukal vazhi pravartthana amplifierukal vazhi cascading/coupling vazhi oru deyude (oru sarkyoottinte) soochika ekapaksheeyamaayi uyarthaan kazhiyumennathu sradhikkendathaanu.
soochika 1-nte ardha-vyakthamaaya day.
roopathinte
avaye ardha-vyakthamennu vilikkunnu. soochika-1 propertic "y" ennathinu "g" pariharikkaavunnathaanenna aavasyamaanu. mattoru vidhathil paranjaal, "t" ennathinu beejaganita samavaakyangalude vyathyaasathiloode oru antharleenamaaya oodu sisttam phalangal undengil, deferentiation soochika 1 aanu.
formula _ 41 aanenkil formula _ 40 nuu ithu pariharikkaavunnathaan
mathiyaya minusamarnna oro daye eee ardha-vyakthamaaya soochika-1 roopathilekku mikkavarum ellaayidathum kuraykkaan kazhiyum.
deyudeyum prayogangaludeyum samkhyaaparamaaya chikitsa.
soochika kuraykkal", "sthiramaaya prarambha vyavasthakal" ennivayaanu deysukal pariharikkunnathile rund pradhana prasnangal. mikka samkhyaa pariharangalkkum saadhaarana deferential samavaakyangalum roopathinte beejaganita samavaakyangalum aavasyamaanu.
shudhamaaya oodu solverukal vazhi ekapaksheeyamaaya day sisttangale parihaaratthinaayi oodu aakki maattunnath nissaaramallaattha oru joliyaanu. pantalides algoritam", "dummy derivative indux reduction reethi" enniva upayogikkaan kazhiyunna saankethikavidyakalil ulppedunnu. pakaramaayi, asthiramaaya prarambha avasthakalulla uyarnna indux deysukalude nerittulla pariharavum saadhyamaan. eee parihara sameepanathil "parimithamaaya moolakangalile oorthoogoonal kolokkeshan" allengil "direct transcription" vazhi derivative moolakangale beejaganita padaprayogangalaayi parivartthanam cheiyunnathu ulppedunnu. ithu eathu soochikayudeyum deysukal ooppan ekweshan roopathil punakrameekarikkathe pariharikkan anuvadikkunnu.
maathruka beejaganita samavaakya roopathilekku parivartthanam cheythukazhinjaal, valia thothilulla nonleeniyar prograaming solvarukalkku ithu pariharikkanaakum (apponittar kaanuka).
tractability.
deferentiation indux", "perterbation indux", "tractability indux", "jyaamitheeya soochika", "chronecker indux" thudangiya samkhyaa reethikalude adisthaanathil days tractabilityude niravadhi alavukal vikasichittundu.
deysinaayulla gadanaaparamaaya vishakalanam.
oru day vishakalanam cheyyaan nammal formula _ 44-reethi upayogikkunnu. oro variyum oro samavaakya soothravaakyavumaayum 46-nuu yojikkunna oro nirayum oro variable soothravaakyavumaayum poruthappedunna oru signature matricks formula nammal dakeyi nirmmikkunnu. position formula entry formula 49 aanu, ithu formula 46-ll eathu soothravaakyavumaayi 47 sambhavikkunnu ennathinte ettavum uyarnna kramatthe soochippikkunnu, allengil formula 46-ll formula 47 sambhavikkunnillenkil formula 52.
mukalilulla pendulam dake, variableukal formula _ 55 aanu. anubandha signature matricks ithaanu

