1. DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE

2. MATHEMATICS - II ● LAPLACE TRANSFORMS ● FOURIER SERIES ● FOURIER TRANSFORMS ● VECTOR DIFFERENTIAL CALCULUS ● VECTOR INTEGRAL CALCULUS ● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS ● BETA AND GAMMA FUNCTIONS FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING] DEPARTMENT OF MATHEMATICS, CVRCE TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS – ERWIN KREYSZIG [8 th EDITIONS]

3. MATHEMATICS-II Convolution Theorem of Laplace Transforms LectureS : 7 - 8 DEPARTMENT OF MATHEMATICS, CVRCE

4. CONVOLUTION THEOREM STATEMENT: Let f(t) and g(t) be two functions whose laplace transforms exists. Let F(s) = L(f(t)) and G(s) = L(g(t)). Then F(s)G(s) is the laplace transform of the convolution of f(t) and g(t), which is denoted by (f  g)(t) and defined by In other words

5. PROOF OF CONVOLUTION THEOREM PROOF: Given that and

6. PROOF OF CONVOLUTION THEOREM t   =0  =t (0,0)  t = 0  t =  t =0 Region of double integration: 0 <  < t, 0 < t < . By altering the variables  and t the region of double integration can be restated as :  < t < , 0 <  < . By altering the order of integration we have

7. PROOF OF CONVOLUTION THEOREM

8. 1. Using convolution find the value of t*t Solution : SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

9. 2. Using convolution find the value of Solution :

10. 3. Using convolution to find the value o f Solution : SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

11. 4. Using Convolution Theorem to find the inverse of Solution : SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

12. 5.Using convolution to find the inverse of SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Solution :

13. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

14. 6. Using convolution theorem to find the inverse of Solution : SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

16. 7. Using convolution find the inverse of SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Solution :

17. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

18. 8. Using convolution theorem solve the following differential equation. Solution : The given differential equation is Applying Laplace transform on (1), we get

19. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

21. 8. Using convolution theorem solve the following differential equation. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Solution: The given initial value problem is Example 9 Using convolution theorem of laplace transform solve the following differential equation. Taking Laplace transform of (1), we get

22. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Using (2) and (3), we get The required solution is

23. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM [By convolution theorem]

24. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Example 10: Using convolution theorem of laplace transform solve the following differential equation. Solution: The given initial value problem is Taking Laplace transform of (1), we get

25. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Using (2) and (3), we get The required solution is [By convolution theorem]

26. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

27. Example 11: Using convolution theorem of laplace transform solve the following differential equation. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Solution: The given initial value problem is

28. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Taking Laplace transform of (1), we get Using (2) and (3), we get

29. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is [By convolution theorem]

30. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

31. Example : 12 Using Laplace transformation solve the integral equation. Solution : The given integral equation is Taking Laplace of the above integral equation we get

32. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

33. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Example 13: Using Laplace transformation solve the integral equation. Solution : The given integral equation is Taking Laplace of the above integral equation we get

34. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM [By convolution theorem of Laplace transform]

35. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

36. Example 14: Using Laplace transformation solve the integral equation. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM Solution : The given integral equation is Taking Laplace of the above integral equation we get

37. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM [By convolution theorem of Laplace transform]

38. SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

39. i) Show that Assignment