Linear vs Nonlinear Differential Equations
Ib kab zauv uas muaj tsawg kawg yog ib qho sib txawv coefficient lossis derivative ntawm qhov tsis paub sib txawv yog hu ua qhov sib npaug sib txawv. Ib qho kev sib npaug sib txawv tuaj yeem yog linear lossis non-linear. Lub Scope ntawm tsab xov xwm no yog los piav txog dab tsi yog linear differential equation, dab tsi yog nonlinear differential equation, thiab dab tsi yog qhov txawv ntawm linear thiab nonlinear sib npaug sib npaug.
Txij thaum kev txhim kho ntawm kev suav lej hauv xyoo pua 18th los ntawm cov lej zoo li Newton thiab Leibnitz, qhov sib npaug sib txawv tau ua lub luag haujlwm tseem ceeb hauv zaj dab neeg ntawm kev ua lej. Kev sib npaug sib txawv yog qhov tseem ceeb hauv kev ua lej vim lawv cov kev siv ntau yam. Kev sib npaug sib txawv yog lub hauv paus ntawm txhua tus qauv uas peb tsim los piav qhia txhua qhov xwm txheej lossis kev tshwm sim hauv ntiaj teb txawm tias nws yog hauv physics, engineering, chemistry, txheeb cais, kev txheeb xyuas nyiaj txiag, lossis biology (cov npe tsis muaj qhov kawg). Qhov tseeb, txog thaum suav suav tau los ua ib qho kev tshawb xav, cov cuab yeej ua lej raug tsis muaj los txheeb xyuas cov teeb meem nthuav dav hauv qhov xwm txheej.
Cov txiaj ntsig tau los ntawm ib daim ntawv thov tshwj xeeb ntawm kev suav lej yuav nyuaj heev thiab qee zaum tsis tuaj yeem daws tau. Txawm li cas los xij, muaj qee qhov uas peb tuaj yeem daws tau, tab sis yuav zoo ib yam thiab tsis meej pem. Yog li ntawd, kom yooj yim dua kev txheeb xyuas qhov sib npaug sib npaug yog categorized los ntawm lawv tus cwj pwm lej. Linear thiab nonlinear yog ib qho xws li categorization. Nws yog ib qho tseem ceeb los txheeb xyuas qhov sib txawv ntawm cov kab sib npaug thiab cov kab sib txawv tsis sib txawv.
Dab tsi yog Linear Differential Equation?
Xav tias f: X → Y thiab f (x)=y, qhov sib npaug sib txawv yam tsis muaj cov ntsiab lus tsis sib xws ntawm qhov tsis paub ua haujlwm y thiab nws cov derivatives yog hu ua linear differential equation.
Nws imposes tus mob uas koj tsis tuaj yeem muaj cov ntsiab lus siab dua xws li y2, y3, … as
Nws kuj tsis tuaj yeem muaj cov lus tsis sib xws xws li Sin y, e y ^-2, lossis ln y. Nws siv daim ntawv,
qhov twg y thiab g yog cov haujlwm ntawm x. Qhov sib npaug yog qhov sib txawv ntawm qhov kev txiav txim n, uas yog qhov ntsuas ntawm qhov kev txiav txim siab tshaj plaws.
Hauv kab zauv sib txawv, tus neeg teb xov tooj sib txawv yog tus neeg teb xov tooj thiab cov kev daws teeb meem tsim qhov chaw vector. Raws li qhov tshwm sim ntawm qhov xwm txheej ntawm qhov kev daws teeb meem, ib qho kev sib xyaw ua ke ntawm cov kev daws teeb meem kuj yog ib qho kev daws rau qhov sib npaug sib txawv. Qhov ntawd yog, yog tias y1 thiab y2 yog cov kev daws teeb meem ntawm qhov sib txawv, ces C1 y 1+ C2 y2 kuj yog kev daws.
Lub linearity ntawm qhov sib npaug tsuas yog ib qho parameter ntawm kev faib tawm, thiab nws tuaj yeem muab faib ua homogenous lossis non-homogenous thiab zoo tib yam lossis ib nrab qhov sib txawv. Yog tias qhov ua haujlwm yog g=0 ces qhov sib npaug yog linear homogeneous sib txawv sib npaug. Yog tias f yog ib qho kev ua haujlwm ntawm ob lossis ntau qhov sib txawv ntawm tus kheej (f: X, T → Y) thiab f (x, t)=y, ces qhov sib npaug yog qhov sib npaug ntawm qhov sib txawv.
Txoj kev daws teeb meem rau qhov sib npaug sib txawv yog nyob ntawm hom thiab cov coefficients ntawm qhov sib npaug sib txawv. Qhov teeb meem yooj yim tshaj plaws tshwm sim thaum cov coefficients tas li. Cov piv txwv classic rau rooj plaub no yog Newton txoj cai thib ob ntawm kev txav thiab nws cov kev siv ntau yam. Newton txoj cai thib ob ua rau qhov kev txiav txim thib ob kab sib npaug sib npaug nrog cov coefficients tas li.
Dab tsi yog qhov sib txawv tsis sib luag?
Kev sib npaug uas muaj cov ntsiab lus tsis sib xws yog lub npe hu ua non-linear sib txawv.
Tag nrho cov saum toj no yog nonlinear sib npaug sib npaug. Nonlinear sib npaug sib npaug yog qhov nyuaj rau kev daws, yog li ntawd, kev kawm ze yuav tsum tau txais kev daws teeb meem. Nyob rau hauv cov ntaub ntawv ntawm ib feem sib npaug sib npaug, feem ntau ntawm cov kab zauv tsis muaj kev daws teeb meem. Yog li ntawd, txhua qhov sib npaug yuav tsum tau kho nws tus kheej.
Navier-Stokes kab zauv thiab Euler qhov sib npaug hauv cov dej dynamics, Einstein cov kev sib npaug ntawm cov kev sib raug zoo yog paub zoo tias tsis yog kab zauv sib txawv ib nrab. Qee lub sij hawm daim ntawv thov ntawm Lagrange kab zauv mus rau qhov sib txawv ntawm qhov sib txawv yuav ua rau muaj qhov sib npaug ntawm qhov sib txawv tsis sib luag.
Dab tsi yog qhov sib txawv ntawm Kab Kab thiab Tsis Sib Tw Sib Txawv?
• Ib qho kev sib npaug sib txawv, uas tsuas muaj cov kab lus sib txawv ntawm qhov tsis paub lossis qhov sib txawv thiab nws cov derivatives, yog lub npe hu ua linear differential equation. Nws tsis muaj lub sij hawm nrog qhov sib txawv ntawm qhov ntsuas siab dua 1 thiab tsis muaj ntau yam ntawm nws cov derivatives. Nws tsis tuaj yeem muaj cov haujlwm tsis zoo xws li trigonometric functions, exponential functions, thiab logarithmic functions nrog rau qhov sib txawv ntawm qhov sib txawv. Ib qho kev sib npaug sib txawv uas muaj cov lus hais saum toj no yog qhov sib npaug tsis sib luag.
• Cov kev daws teeb meem ntawm kab sib npaug sib npaug tsim vector qhov chaw thiab tus neeg teb xov tooj sib txawv kuj yog tus neeg teb xov tooj hauv qhov chaw vector.
• Cov kev daws teeb meem ntawm cov kab sib npaug sib txawv yog qhov yooj yim dua thiab cov kev daws teeb meem dav dav. Rau cov kev sib npaug nonlinear, feem ntau, cov kev daws teeb meem tsis muaj nyob thiab cov kev daws yuav yog teeb meem tshwj xeeb. Qhov no ua rau kev daws teeb meem nyuaj dua li cov kab zauv sib npaug.
