Laplace formula

Laplace Formula Reference From Advanced Engineering Mathematics (ATUL PUBLICATION)

1. ARHAM VEERAYATAN INSTITUTE OF ENGINEERING, TECHNOLOGY & RESEARCH Presented by Kotadiya Urvish(150920106001) Kathiriya Vishal(150920106003) VYAS KEYUR (150920106004) GASHVI YOGESH (160920316017) LAPLACE FORMULA Guided by : Pro. Sachin gaikwad 30 April 2017 1

2. TOPIC Laplace formula Laplace inverse formula 30 April 2017 2

3. 1. Laplace Formula 1 . 2. L{S} = 3. L{ } = 4. L{ } = 5. L{ sinat } =   30 April 2017 3

4. Continue….. 6. L{ cosat } = 7. L{ sinhat } = 8.L{ coshat } = 9. L{ } =   30 April 2017 4

5. Continue……Formula 1 . Definition :- let f(t) be any function and t There, t = variable S = parameter L = Laplace operators   30 April 2017 5

6. Continue……. Formula 2. Prove that :- f(t) = 1 L{ f(t) } = f(t) dt L { 1 } = (1) dt L{ 1 } = dt L{ 1 } = [0-1] L{ 1 } =   30 April 2017 6

7. Continue…… Formula 3. Prove that :- f(t) = L{ f(t) } = f(t) dt L { } = ( ) dt L{ } = dt L{ } = [0-1] L{ } =   30 April 2017 7

8. Continue….. Formula 4. Prove that :- f(t) = L{ f(t) } = f(t) dt L { } = ( ) dt L{ } = dt L{ } = [0-1] L{ } =   30 April 2017 8

9. Continue….. Formula 5 & 6. Prove that :- L{ sinat } = & L{ cosat } = = cosx + isinx (1) put x = at in equation (1) = cosat + isinat L{ } = L{ cosat } + L{ isinat }   30 April 2017 9

10. Continue………. = L{ cosat } + L{ isinat } = L{ cosat } + L{ isinat } + = L{ cosat } + L{ isinat } L{ sinat } = L{ cosat } =   30 April 2017 10 Continue….. Formula 5 & 6.

11. Continue….. Formula 7. Prove that :- L{ sinhat } = L{ sinhat } = L{ sinhat } = L L{ sinhat } = L L { sinhat } = L{ sinhat } =   30 April 2017 11

12. Continue….. Formula 8. Prove that :- L{ coshat } = L { coshat } = L { coshat } = L L{ coshat } = L L{ coshat } = L{ coshat } =   30 April 2017 12

13. Continue….. Formula 9. Prove that :- L { } = y = log y = log y = y = = L{ } = L{ } L{ } =   30 April 2017 13

14. 2. Laplace I nverse Formula 1. 2. 3. = sinat 4. = cosat 5. = sinhat 6. = coshat   30 April 2017 14

15. Continue…… 7. 8. = sinbt 9. = cosbt 10. = sinhbt 11. = coshbt   30 April 2017 15

16. Thank you 30 April 2017 16