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Laplace transform of Power function


I was trying to use Wolfram Alpha engine to transform a power function using laplace transform and then inverse using inverse laplace transform.

The function is:

D(t) = a*t^b + c

('t' is the initial variable, 's' is the transform variable and 'a', 'b', and 'c' are constant parameters)

When I transfer it using Laplace Transform, I got this results:

a s^(-b-1) Gamma(b+1)+c/s

However, when I try to inverse the result function using the formula: 1/s^2 as shown below:

1/[s^2*(a s^(-b-1) Gamma(b+1)+c/s)]

The engine doesn't give me any results. as shown in the below image:

enter image description here

I keep clicking on 'Solve' button but with no response.

Can anyone help me in finishing this?

Thank you..

POSTED BY: Mohammed Sadeq
2 Replies


Thanks Sean for replying..

Actually, I am trying to follow another example I found. The example is trying to solve the below equation using laplace transform:

enter image description here

So, he got this:

enter image description here

By using this formula:

enter image description here

he became able to identify this:

enter image description here

And using the inverse laplace transform he got this:

enter image description here

Then he compare it with the original relaxation modulus:

enter image description here

To identify each part of the equation as follows:

enter image description here

So, when I tried to follow the above steps, I couldn't inverse the results from laplace transform after multiplying by 1/s^2.

I appreciate your help and explaination..

POSTED BY: Mohammed Sadeq

What result are you expecting for the Inverse Laplace Transform of 1/[s^2*(a s^(-b-1) Gamma(b+1)+c/s)]?

It doesn't show a result becuase it doesn't find any result.

Are you sure you're not looking for the inverse transform of (1/s^2)(a s^(-b-1) Gamma(b+1)+c/s) instead? Are you looking for the convolution of t and a*t^b + c?

POSTED BY: Sean Clarke
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