Chii chinonzi Partial Differential Equation?
Maequations ePartial differential (PDEs) inyaya inokosha mumasvomhu anoshandiswa, anoshandiswa zvakanyanya kutevedzera zviitiko zvakasiyana-siyana zvechisikigo uye maitiro einjiniya. Kana tichida kunzwisisa kuti tembiricha inopararira sei kuburikidza nechinhu, kuti mafungu anopararira sei patambo, kana kuti mvura inoyerera sei mupombi, tingangosangana nemaequations ePartial differential. Chinyorwa chino chinokurukura tsananguro yadzo, chimiro chakajairika, mhando, mienzaniso, uye mashandisirwo chaiwo.
Kunzwisisa Zviyero Zvishoma Zvinosiyanisa
Zvichitaurwa zviri nyore, chikamu che "partial differential equation" i "equation" inosanganisira "derivative" yebasa maererano ne "independent variable" dzinopfuura imwe chete. Kusiyana ne "ordinary differential equations" (ODEs), iyo inosanganisira "derivatives" maererano ne "variable" imwe chete (semuenzaniso, nguva), "PDI" inomuka kana mamiriro achitsamira pane "variables" mbiri kana kupfuura, dzakadai se "space" nenguva, panguva imwe chete.
Semuenzaniso, ngatitii tine basa rekupisa \(u(x,t)\) padanda resimbi: kupisa kunochinja maererano nenzvimbo \(x\) nenguva \(t\). Kana tichida kutsanangura hukama hwekuchinja kwekupisa maererano nenzvimbo nenguva, taizoshandisa zvikamu zvema "partial derivatives" zvakaita se:
\[
\frac{\partial u}{\partial t}, \quad \frac{\partial u}{\partial x}, \quad \frac{\partial^2 u}{\partial x^2}
\]
Nekuti inosanganisira zvikamu zvechikamu (partial derivatives), equation iyi inonzi "partial differential".
Sei Zvikamu Zvisina Kukwana Zvichidiwa?
Zvikamu zvema "partial derivatives" zvinoshandiswa kana basa richienderana ne "variable" dzinopfuura imwe chete, uye tinoda kuziva mwero wekuchinja kwebasa maererano nerimwe rema "variables" uku tichibata zvimwe "variables" zvakafanana. Semuenzaniso, mu "\(u(x,y)\), "partial derivative" maererano ne "\(x\)" inoratidza shanduko mu "\(u\)" kana "\(x\)" ikachinja, asi "\(y\)" inoramba yakadaro.
Munyaya yefizikisi neinjiniya, izvi zvakakosha nekuti masisitimu mazhinji epasi rese anokanganiswa nezvinhu zvakawanda panguva imwe chete. Kupararira kwekupisa kunoenderana nenzvimbo nenguva; mafambiro emvura anoenderana nemakonekita matatu enzvimbo nenguva; uye masimba emagetsi nemagineti anoenderana nenzvimbo nenguva.
Fomu Rakajairika reMaenzaniso Ekusasiyana Kwechikamu
Chimiro chePDP chinosiyana zvikuru, asi kazhinji chinogona kunyorwa seizvi:
\[
F\left(x_1, x_2, \madotsi, x_n, u, \frac{\partial u}{\partial x_1}, \madotsi, \frac{\partial u}{\partial x_n},
\frac{\partial^2 u}{\partial x_i \partial x_j}, \dots \right)=0
\]
Pano, \(u\) ibasa risingazivikanwe (basa rinofanira kugadziriswa), nepo \(x_1, x_2, \dots, x_n\) ari mavariable akazvimiririra (semuenzaniso, nzvimbo nenguva). Equation inogona kusanganisira first-, second-, kana higher-order partial derivatives.
Pamusoro pezvo, PDP inogona kukamurwa kuita:
– Mutsetse: kana \(u\) uye zvinobuda mazviri zvichionekwa zvakatevedzana (zvisina kukwidziridzwa kumasimba, zvisingawanziridzwe nezvimwe zvinobuda mazviri, uye zvisingapindi mumabasa asiri emutsara).
– Zvisiri zvemutsara: kana paine zvinhu zvisiri zvemutsara zvakaita se \((\partial u/\partial x)^2\), \(u^2\), kana \(\sin(u)\).
Kurongeka uku kwakakosha nekuti maPDP akarongana anowanzo kuve nyore kuongorora uye ane matekiniki ekugadzirisa akasimbiswa.
Kurongeka kweZviyero zvePartial Differential
Kurongeka kwePDP kunoonekwa ne "highest-order partial derivative" inoonekwa mu "equation".
– Odha yekutanga: ine chikamu chekutanga chinobva pane chimwe chete, semuenzaniso:
\[
\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x}=0
\]
– Chechipiri: chine chikamu chechipiri chinobva pane chimwe chinhu, semuenzaniso:
\[
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
\]
Maequation mazhinji akakosha efizikisi maPDE echipiri.
Makirasi matatu ekare ePDP: Elliptic, Parabolic, uye Hyperbolic
MuPDP dzemutsara wechipiri, kune marudzi anozivikanwa zvikuru ekupatsanura, anoti elliptic, parabolic, uye hyperbolic. Magadzirirwo aya anokanganisa mhando yemhinduro uye nzira dzekudzigadzirisa.
1. Kutenderera kwedenderedzwa resimbi
Muenzaniso unonyanya kuzivikanwa iequation yaLaplace:
\[
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0
\]
MaPDP eelliptic anowanzoonekwa mumamiriro "asina kumira" kana akaenzana, semuenzaniso kugoverwa kwesimba remagetsi muchadenga pasina kuchinja nekufamba kwenguva.
2. Kufananidzira
Muenzaniso mukuru iequation yekupisa:
\[
\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}
\]
Parabolic PDP inotsanangura maitiro ekupararira kana kupararira, zvakaita sekupisa, makemikari, kana huwandu hwevanhu.
3. Hyperbolic
Muenzaniso unonyanya kufarirwa iwave equation:
\[
\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}
\]
Hyperbolic PDP inotevedzera kupararira kwemafungu, akadai semafungu patambo, ruzha, kana mafungu emagetsi.
Mienzaniso yeZviyero Zvishoma zvekusiyanisa muhupenyu chaihwo
Kuti zvinyatsobatsira, heano mimwe mienzaniso yemapurogiramu ePDP anowanzo sangana nawo:
1. Kupararira kwekupisa muzvinhu
Mainjiniya anoshandisa maequation ekupisa kufanotaura kuti tembiricha inopararira sei kuburikidza nemichina, zvikamu zvemagetsi, kana zvinhu zvekuvaka. Izvi zvakakosha pakugadzira kutonhora uye kudzivirira kukuvara kunokonzerwa nekupisa zvakanyanya.
2. Mafungu nekudengenyeka
Maequation emafungu anoshandiswa muinjiniya yezvekuvaka (semuenzaniso, kuongorora kutenderera kwemabhiriji), maacoustics (kuparadzirwa kweruzha), uye seismology (mafungu ekudengenyeka kwenyika).
3. Mvura uye mamiriro ekunze
Kuenzanisa kuyerera kwemvura kunosanganisira masisitimu akaomarara ePDP akadai seNavier-Stokes equations. Kufanotaura kwemamiriro ekunze, mafungu egungwa, uye mhepo inovhuvhuta zvinoenderana zvakanyanya nenzira dzekuenzanisa idzi.
4. Mari yehuwandu
Mumasvomhu ezvemari, equation yeBlack-Scholes yemitengo yesarudzo iPDP ine chekuita nenguva, mutengo wezvinhu, kusagadzikana, nezvimwe zvinhu.
5. Biology nemishonga
Kupararira kwezvirwere, kukura kwebundu, uye kupararira kwemishonga munyama zvinogona kuenzaniswa nemaitiro ePDP ekupararira kwemasero.
Kupedzisa PDP: Mamiriro Ekutanga uye Mamiriro Emuganhu
Kusiyana nemaequation akajairwa earbegian, anogona kuva nemhinduro imwe chete, maPDE anowanzova nemhinduro dzakawanda dzinogoneka. Kuti tiwane mhinduro inoenderana nemamiriro ezvinhu chaiwo, tinowanzoda:
– Mamiriro ekutanga: kukosha kwebasa panguva yekutanga, semuenzaniso \(u(x,0)=f(x)\).
– Muganho: maitiro ebasa riri pamuganhu wenzvimbo, semuenzaniso \(u(0,t)=0\) kana \(\frac{\partial u}{\partial x}(L,t)=0\).
Muenzaniso uri nyore: kune tsvimbo refu \(0 \le x \le L\), tinogona kuva netembiricha yekutanga uye magumo etsvimbo anochengetwa patembiricha yakafanana. Musanganiswa we equation yekupisa + mamiriro ekutanga + mamiriro emuganhu unoumba dambudziko guru.
Nzira dzekugadzirisa Partial Differential Equations
Haasi ese maPDP ane mhinduro ye "closed formula" inogona kunyorwa zviri nyore. Kazhinji, kune nzira dzakasiyana siyana:
1. Nzira yekuongorora
Semuenzaniso, kupatsanurwa kwezvimiro, Fourier transform, Laplace transform, uye characteristic method (yekutanga kurongeka).
2. Nzira dzekuverenga nhamba
Kana mhinduro dzekuongorora dzakaoma kana dzisingaite, nzira dzekuverenga, dzakadai se finite difference, finite element, uye finite volume methods, dzinoshandiswa. Nzira dzekuverenga nhamba dzakakosha zvikuru muinjiniya yemazuva ano uye simulations yesainzi.
3. Maitiro emhando yepamusoro
Dzimwe nguva chinotsvakwa hachisi chimiro chaicho chemhinduro, asi hunhu hwayo: kana mhinduro yacho yakagadzikana, kana paine mafungu ekuvhunduka, kana mhinduro yacho yakatsetseka kana kuti ine zvinhu zvakasiyana-siyana, nezvimwewo.
Mhedziso
Maequation ePartial differential (PDEs) zvishandiso zvine simba zvemasvomhu zvekutsanangura shanduko mumasisitimu anoenderana nezvinhu zvakawanda, kunyanya nzvimbo nenguva. Maequation ePartial differential (PDEs) anogona kushandiswa kuratidza kupisa, mafungu, kuyerera kwemvura, maitiro ekupararira, uye kunyange mafambiro ehurongwa hwehupfumi nehwezvipenyu. Kunyange zvazvo kazhinji zvakaoma uye zvichinetsa, maPDE ihwaro hwakakosha hwesainzi yemazuva ano, mainjiniya, uye tekinoroji, sezvo zviitiko zvakawanda zvepanyika chaiyo zvichitsanangurwa zviri nani kuburikidza nehukama huripo pakati pekuchinja kwenzvimbo nenguva.
Kana muchida, ndinogonawo kuwedzera mienzaniso iri nyore yematambudziko ePDP pamwe chete nematanho avo ekugadzirisa (semuenzaniso, 1D heat equations ine mamiriro chaiwo emiganhu), kana kugadzira vhezheni yakakurumbira yechinyorwa ichi kune vaverengi vechikoro chesekondari.
