Inverse Laplace Transform of a product of exponential functions

In summary, the inverse Laplace transform is a mathematical operation that converts a function in the Laplace domain back to its original form in the time domain. The product of exponential functions is a mathematical expression where multiple functions of the form f(x) = e^(ax) are multiplied together. The inverse Laplace transform can be applied to a product of exponential functions, resulting in the convolution of the individual inverse Laplace transforms. The formula for the inverse Laplace transform of a product of exponential functions is given by the convolution theorem, with special cases such as when all the exponential functions have the same exponent.
  • #1
metdave
5
0
I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question.

Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse transform simply be a convolution?

Thanks!
David
 
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  • #2
Hi !
I fear that this may not be possible.
Try to find the Inverse Laplace transform of exp(-s²) for example.
 

What is the inverse Laplace transform?

The inverse Laplace transform is a mathematical operation that takes a function in the Laplace domain and converts it back to its original form in the time domain.

What is the product of exponential functions?

The product of exponential functions is a mathematical expression in which multiple functions of the form f(x) = e^(ax) are multiplied together.

Can the inverse Laplace transform be applied to a product of exponential functions?

Yes, the inverse Laplace transform can be applied to a product of exponential functions. The resulting function in the time domain will be the convolution of the individual inverse Laplace transforms of each exponential function.

What is the formula for the inverse Laplace transform of a product of exponential functions?

The formula for the inverse Laplace transform of a product of exponential functions is given by the convolution theorem, which states that the inverse Laplace transform of a product of two functions is equal to the convolution of the individual inverse Laplace transforms of each function.

Are there any special cases for the inverse Laplace transform of a product of exponential functions?

Yes, there are special cases for the inverse Laplace transform of a product of exponential functions. One special case is when all the exponential functions have the same exponent, in which case the inverse Laplace transform can be simplified to the inverse Laplace transform of a single exponential function.

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