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

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]

MATHEMATICS-II Convolution Theorem of Laplace Transforms LectureS : DEPARTMENT OF MATHEMATICS, CVRCE

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

PROOF OF CONVOLUTION THEOREM PROOF: Given that and

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

PROOF OF CONVOLUTION THEOREM

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

2. Using convolution find the value of Solution :

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

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM [By convolution theorem]

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

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

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

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

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

SOLVED PROBLEMS BASSED ON CONVOLUTION THEOREM The required solution is

i) Show that Assignment