Inverse Laplace Transform of a product of exponential functions

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SUMMARY

The discussion centers on the Inverse Laplace Transform of the function u(s,t) = e^((as^2 + bs)t) where a and b are both negative. David seeks to determine if the inverse transform can be expressed as a convolution. The response indicates skepticism about the feasibility of this inversion, referencing the specific case of exp(-s²) as a challenging example. This highlights the complexities involved in inverting Laplace Transforms of products of exponential functions.

PREREQUISITES
  • Understanding of Laplace Transforms
  • Knowledge of Partial Differential Equations (PDEs)
  • Familiarity with convolution operations
  • Experience with complex function analysis
NEXT STEPS
  • Research the properties of Inverse Laplace Transforms
  • Study convolution theorems in the context of Laplace Transforms
  • Explore specific examples of Inverse Laplace Transforms, such as exp(-s²)
  • Investigate techniques for solving PDEs using Laplace Transforms
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Mathematicians, engineers, and students studying differential equations, particularly those interested in the application of Laplace Transforms in solving PDEs.

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

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