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Teorema na Nau'i mai suna da dukkan cikin al'amuran Laplace transform. An bayar shi ne aiki na tarihi na French Mathematical Physicist Pierre Simon Marquis De Laplace. Ya yi aiki mai yawa a kan abubuwa masu harkokin kasa da amfani da teoriya na Gravitation na Newton. Aikinsa game da teoriya na probability da statistika ya zama babban nasara da take tabbatar da jami'a masu aiki a cikin tarihin Mathematician. Laplace ya zama daya daga 72 mutanen da suka fi sune sunan su a Eiffel Tower.
Teorema na Nau'i da Teorema na Rike suna da suka kira waɗannan ana kiran suna da suka kira Limiting Theorems. Ana kiran Teorema na Nau'i mafi tsawo a matsayin IVT. Zai iya ba muna da nau'in bayan f(t) (laplace) a lokacin t = (0+) bace ba a yi aiki mai kyau don samun f(t) wanda shi ne aiki mai karfi a cikin halin da ba.

Kunan da ke nufin da aka da Teorema na Nau'i

1. Funkshin f(t) da karamin f(t) ya kamfanon da za su iya Laplace transformable.

2. Idan lokacin t ta gama zuwa (0+) funkshin f(t) ya kamfanon da za su iya wuce.

1. Funkshin f(t) = 0 idan t > 0 da kuma bai da impulses ko singularities masu yawan adadin da suka haɗa a asalin.

Bayanin Teorema na Nau'i na Laplace

Idan f(t) da F(s) suka bi Laplace transform pairs. i.e

don haka Teorema na Nau'i ya kasance

Laplace transform na funkshin f(t) shi ne

don haka Laplace transform na karamin f ‘ (t) shi ne

Yana da integral part first

An substituta (2) a (1) muna samu

Ba da cancella f (0–) a duk biyu muna samu

Za a iya rubuta equation da ma ake so a matsayin ma ake so, amma intensiongaski a matsayin limits of integration daga (0– zuwa ∞) shine cewa idan an samun values na negative ya kamfanon da za su iya samun results na positive.

Note:
An san cewa Laplace transform ya kamfanon da za su iya applicability only for causal functions.
Idan an samun (s) ta gama zuwa infinity a duk biyu a (3)

Don haka, Teorema na Nau'i ya gama da shiga.

Ayoyinta na Teorema na Nau'i

Daga cikin irin da na ce bayanin Teorema na Nau'i shine don samun nau'in bayan funkshin f (t) idan an bayar Laplace transform.
Misali 1 :
Samun nau'in bayan funkshin f (t) = 2 u (t) + 3 cost u (t)
Sol:

By initial value theorem

Nau'in bayan shi ne 5.
Misali 2:
Samun nau'in bayan transformed function

Sol:

By initial value theorem

[as s → ∞ the values of s become more and more insignificant hence the result is obtained by simply taking the ratio of leading co-efficient]