Please see the attached file for the fully formatted problem.

My problem lies in manipulating the equation to one which the inverse transform can be taken, but would appreciate this example of convolution worked with some more of the blanks filled. I cannot figure out how the equation
gets manipulated into what appears to be .
The main objective here is to find the inverse Laplace transform for h(s) using convolution.

k and t are separate variables, not a single variable kt.

Side Note: Is there a mathematical method for taking the inverse transform or are there just tables from people who did the original Laplace transform?

Hello and thank you for posting your question to Brainmass!
The solution is attached below in two files. the files are identical in content, only differ in ...

Solution Summary

A Convolution Applied to an Inverse Transform Problem. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.

... The transform of a convolution in the time domain is a product of transfroms of the individual functions in the s ... And, applying the inverse transform: ...

...Applying the Laplace transform to both sides we obtain: ... Then we need to apply the inverse transform to ... a transform of a product, the result is a convolution: ...

...Applying this to the equation system (with respect to x ... The transform of multiplication is convolution, thus ... Now we need to find the inverse transforms of ϕ (k ...

...Convolution with a delta function: http://130.191.21.201/multimedia ... So applying the transform to the equation yields ... This is an inverse transform of a product of ...

... We will apply knowledge from many branches of science and ... not yet exist and even theories applied to classes ... in the Laplace domain gives a convolution in the ...

... nf F n = (− iσ ) F (σ) n ( 12) dx 8) Convolution formulae: L ... 0 and 0 for t <0) Now, we need to apply the inverse Fourier transform...

... html http://mathworld.wolfram.com/Convolution.html http ... s 2 L[ f (t )] − sf (0 ) − f ′(0 ) Applying it to ... x(t) we need to use the inverse Laplace transform...

... LaPlace transform of 1 / s2 using the integration theorem, and then apply the s ... Following is the Convolution theorem, which will be applied to solve the ...

... Now, we need to apply the inverse Fourier transform... of 2 Fourier images, the original function u(x,t) can be expressed as a convolution product, according ...

... html http://mathworld.wolfram.com/Convolution.html http ... http://www.sosmath.com/diffeq/ laplace/application/application.html. ...Applying the transform on both sides: ...