ganiteeya model

kisi bhautik tantr (physical system) ya prakram (process) ya amoort tantr (abstract system) ke vibhinn avayavon ke antarsambandhon ka ganit ki bhaasha mein varnan us tantr ka ganiteeya pratiroop ya ganiteeya model (mathematical model) kahalaata hai.

ganiteeya model praay: sangat tantr ke saraleekrut roop hote hain. isse us tantr ki kaaryapranaali ko aasaani se samajhne mein suvidha hoti hai. iski sahaayata se yeh ganana ki ja sakti hai ki kis sthiti mein kya hoga. ganiteeya model ki sahaayata se hi us bhautik tantr ka niyntran bhi kiya ja sakta hai. kisi tantr ko computer dvaara simulet (simulate) karne ke liye us tantr ka ganiteeya model banaana pehli jaroorat hai.

ganiteeya model ka praakrutik vijnyaaanon evam praudyogiki mein bahutaayat se upayog hota hai. iske atirikt iska saamaajik vijnyaaanon, jaise arthashaastr, samaaj shaastr evam raajaneeti shaastr mein bhi upayog hota hai.

kisi tantr ya yukti ke ganiteeya model ko jab kisi vidyut paripth ke roop mein nirupit kiya jaata hai to is vidyut paripth ko tulya paripth (equivalent circuit) kehte hain. udaaharan ke liye kisi baitari ko ek aadarsh voltej sors evam ek pratirodh ke shreneekram (sireej) sanyojan ke roop mein pradarshit kiya ja sakta hai.

jaane anajaane har koi anekon modelon ka upayog karta hai. ye kisi vastu ya tantr ki maanasik model hote hain. ham apne anubhav evam samajh se banaaye hue vibhinn maanasik modelon ke aadhaar par hi nirnaya lete hain ya kaam karte hain. jaroori naheen hai ki hamaare mastishk mein rachi-basi kisi tantr ki maanasik chhavi (mental imej) us tantr ke vaastavik model se mel khaati ho.

maalthas ka janasankhya model

maalthas ne vichaara ki janasankhya samay ke saath iksponensiyl roop se (charaghaataanki roop se) badhegi jabki jeevika ke saadhan (anaaj aadi ka utpaadan) rekhiiya gati se hi badhega. in maanyataaon (esampsans) ke aadhaar par maalthas ne bhavishya ki sambhaavit tasveer pesh ki.

bhautiki mein ganiteeya modelon ka upayog

ganiteeya model bhautiki mein bahut mahatva rakhate hain. bhautik siddhaant praay: ganiteeya model ke roop mein hi prastut kiye jaate hain. udaaharan ke liye gaison ka vyavahaar aadarsh gas sameekaran ke roop mein (PV=nRT) vyakt kiya jaata hai. yeh ek ganiteeya model hi hai. isi prakaar kisi potenshiyl fild mein kisi kan ka vyavahaar ke liye difreinshiyl sameekaran ke roop mein ek ganiteeya model prastut kiya ja sakta hai.

mukhyat: do vidhiyaaain hai:

• siddhaanton evam niyamon ka upayog karte hue tantr ka ganiteeya varnan (jaise difrenshiyl sameekaran ya difreins sameekaran ke roop mein)

iske liye tantr ki banaavat evam kaaryapaddhati ka sampoorn gyaan aavashyak hai.

• tantr ki pehchaan (system aaidentifikeshan) : ismein tantr ko samyak prakaar ka input dekar outaput rikaard kar liya jaata hai. aur in aakadon se gyaat kiya ja sakta hai ki tantr kis order ka hai va uske vibhinn pairaameetar ka maan kya hoga. iska mukhya laabh yeh haiki iske liye system ki poorn jaankaari jaroori naheen hai; kintu iske liye vaastavik tantr ka vidyamaan hona jaroori hai jis par prayog karke aankade liye ja sakein.

system aaidentifikeshan ki mukhyat: do vidhiya hain:

• • samay-domen mein, praay: step-input ke sangat outaput rekaard karke

• • aavrutti-domen mein, alag-alag aavrutti ke saainasvaayadal (sign aakaar vaale) sanket dekar unke sangat outaput ka aayaam va kala (phase) naapakar

• ek baar ganiteeya model ban jaane ke baad bhautik model jaroori naheen rah jaata. iske naye design ya teknaaloji ko ganiteeya model ki sahaayata se hi viksit kiya ja sakta hai. at: vartamaan tantr ka kaamadhaam roke bina hi kaam ban jaata hai.

• ganiteeya model, bhautik model ki tulana mein bahut adhik lacheela aur upayogi hota hai:

• • samay ka koi bandhan naheen, adhik samay lene wala prakram bhi kam samay mein simulet karke dekha ja sakta hai.

• • vibhinn pairaameetaron ke maan jaaainche ja sakte hain. unhein adal-badal kar unke prabhaav ka adhyayan kiya ja sakta hai.

• • ve bhi raashiyaaain maapi/ganana ki ja sakti hain jinhein bhautik tantr mein dekh paana kathin/asambhav hota hai.

• ganiteeya model par koi bhi prayog 100% surakshit hote hain.

• • bhautik tantron par prayog praay: khatarnaak hote hain.
  • bhautik tantron par logon ko paise kharch karke prashikshit karna padta hai.

• ganiteeya model antardrushti pradaan karne mein madad karte hain. isse unke baare mein achhi samajh paida hoti hai.

• traansafar fankshan ke roop mein
• state-space roop mein
• block ke roop mein (jaise simoolink ya saaikaus ki sahaayata se)

• rekheeya (liniyr) tantr
• arekheeya tantr (naun-liniyr)

• bhautik model
• simuleshan (Simulation)
• saankhyikeeya model (Statistical model)

• McLaughlin, Michael P. (1999) 'A Tutorial on Mathematical Modeling' peedeeऍf (264 KiB)
• Patrone, F. Introduction to modeling via differential equations, with critical remarks.