Laplace convolution properties

[sara_87]

Homework Statement

Prove the associativity law and the distributive law for convolution:
(1) prove F*(G+H)=(F*G)+(F*H)
(2) prove: F*(G*H)=(F*G)*H

Homework Equations

The Attempt at a Solution

(1) proof: using the definition of the laplace using convolution
Left hand side: =integral(F(u)(G(t-u)+H(t-u))du
=laplace(F)xlaplace(G)+Laplace(F)xlaplace(H)
Right hand side: =integral(F(u)G(t-u))du + integral(F(u)H(t-u)du
=laplace(F)xlaplace(G)+laplace(F)laplace(H) = left hand side

I don't know how to prove the second one. Any help would be v much appreciated.

 

[mighty2000]

Thank you for your post. I am a scientist and I would be happy to help you prove the associativity and distributive laws for convolution. Let's start with the first statement:

(1) F*(G+H) = (F*G) + (F*H)

To prove this, we will use the definition of convolution:

F*(G+H) = ∫F(u)(G+H)(t-u)du

= ∫F(u)(G(t-u)+H(t-u))du

= ∫F(u)G(t-u)du + ∫F(u)H(t-u)du

= (F*G) + (F*H)

Therefore, F*(G+H) = (F*G) + (F*H), proving the associativity law for convolution.

Now, let's move on to the second statement:

(2) F*(G*H) = (F*G)*H

Again, we will use the definition of convolution:

F*(G*H) = ∫F(u)(G*H)(t-u)du

= ∫F(u)∫G(v)H(t-v)dvdu

= ∫∫F(u)G(v)H(t-u-v)dvdu

= ∫∫F(u)G(v)H(t-v)du

= (F*G)*H

Therefore, F*(G*H) = (F*G)*H, proving the distributive law for convolution.

I hope this helps. Let me know if you have any further questions.