Laplace vs Fourier Transforms
Ob qho Laplace hloov pauv thiab Fourier hloov pauv yog qhov kev hloov pauv, uas feem ntau siv los ua lej los daws cov lej ua qauv ntawm lub cev. Cov txheej txheem yog yooj yim. Ib tug complex lej yog hloov dua siab tshiab rau hauv ib tug yooj yim, solvable qauv siv ib tug integral transform. Thaum tus qauv yooj yim daws tau, qhov kev hloov pauv hloov pauv hloov pauv tau siv, uas yuav muab kev daws teeb meem rau tus qauv qub.
Piv txwv li, vim tias feem ntau ntawm lub cev ua rau muaj qhov sib npaug sib txawv, lawv tuaj yeem hloov mus rau hauv cov lej sib npaug lossis qis dua qhov yooj yim daws tau qhov sib txawv sib txawv siv qhov kev hloov pauv. Ces daws qhov teeb meem yuav yooj yim dua.
Laplace transform yog dab tsi?
Muab qhov ua haujlwm f (t) ntawm qhov sib txawv tiag tiag t, nws qhov kev hloov pauv Laplace yog txhais los ntawm qhov tseem ceeb [latex] F(s)=\\int_{0}^{ \\infty} e^{- st}f(t)dt [/latex] (thaum twg nws muaj), uas yog ib qho kev ua haujlwm ntawm qhov sib txawv ntawm qhov sib txawv s. Nws yog feem ntau denoted los ntawm L {f (t)}. Lub inverse Laplace transform ntawm ib tug muaj nuj nqi F (s) yog coj mus ua lub function f (t) nyob rau hauv xws li ib tug txoj kev uas L { f (t)}=F (s), thiab nyob rau hauv ib txwm lej cim peb sau, L-1{ F (s)}=f (t). Kev hloov pauv hloov pauv tuaj yeem ua tshwj xeeb yog tias tsis tso cai ua haujlwm. Ib tug tuaj yeem txheeb xyuas ob qho no raws li tus neeg ua haujlwm tawm tau hais tseg hauv qhov chaw ua haujlwm, thiab nws kuj yooj yim pom tias, L -1{ L { f (t)}}=f (t), yog tsis pub muaj nuj nqi null.
Cov lus hauv qab no teev cov kev hloov pauv ntawm Laplace ntawm qee qhov haujlwm feem ntau.
Fourier transform yog dab tsi?
Muab qhov muaj nuj nqi f (t) ntawm qhov sib txawv tiag tiag t, nws qhov kev hloov pauv Laplace yog txhais los ntawm qhov tseem ceeb [latex] F(\ alpha)=\\ frac{1}{\ sqrt{2 \\ pi}} \int_{- \\infty}^{\infty} e^{i \\alpha t}f(t)dt [/latex] (thaum twg muaj), thiab feem ntau yog denoted los ntawm F {f (t)}. Qhov hloov pauv hloov pauv F -1{ F (α)} yog muab los ntawm qhov tseem ceeb [latex] f(t)=\\frac{1}{\sqrt{2 \\pi }}\\int_{- \\infty}^{\infty} e^{-i \\alpha t}F(\alpha)d\\alpha [/latex]. Fourier transform kuj yog linear, thiab tuaj yeem xav tias yog tus neeg teb xov tooj uas tau teev tseg hauv qhov chaw ua haujlwm.
Siv lub Fourier hloov pauv, thawj lub luag haujlwm tuaj yeem sau tau raws li hauv qab no yog tias txoj haujlwm tsuas muaj qhov txwv tsis pub dhau thiab yog qhov sib xyaw ua ke.
Qhov txawv ntawm Laplace thiab Fourier Transforms yog dab tsi?
- Fourier transform of a function f (t) txhais tau tias yog [latex] F(\alpha)=\\frac{1}{\sqrt{2 \\pi}} \int_{- / \infty}^{infty} e^{i \\alpha t}f(t)dt [/latex], whereas the laplace transform of it is definition to be [latex] F(s)=\\int_{ 0}^{ \\infty} e^{-st}f(t)dt [/latex].
- Fourier transform yog txhais tau tsuas yog rau cov haujlwm uas tau hais tseg rau txhua tus lej tiag tiag, qhov Laplace hloov pauv tsis tas yuav muaj txoj haujlwm los txiav txim siab rau cov lej tsis zoo.
- Fourier transform yog qhov tshwj xeeb ntawm Laplace transform. Nws tuaj yeem pom tau tias ob qho tib si coincide rau cov lej tsis zoo. (piv txwv li coj s hauv Laplace los ua iα + β qhov twg α thiab β yog tiag tiag li uas e β= 1/ √(2ᴫ))
- Txhua txoj haujlwm uas muaj Fourier hloov pauv yuav muaj Laplace hloov pauv tab sis tsis hloov pauv.
