Solving PDE with Laplace Transforms & Inverse Lookup

In summary, the conversation involves trying to solve a problem using Laplace Transforms and Duhamel's principle, but struggling to find the inverse Laplace transform at the end. The proposed solution involves using a table or integral, but the integral is deemed too difficult and an alternate calculator is suggested. The conversation also mentions studying residue theory as a potential solution.
  • #1
fahraynk
186
6
I am trying to solve with Laplace Transforms in an attempt to prove duhamels principle but can't find the Laplace transform inverse at the end. The book I am reading just says "from tables"...

The problem :
$$
U_t = U_{xx}\\\\
U(0,t)=0 \quad 0<t< \infty\\\\
U(1,t)=1\\\\
U(x,0)=0 \quad 0<x<1\\\\
$$

The solution attempt :
$$
SU(x,s) = U_{xx}(x,s)\\\\
U(1,s) = \frac{1}{S}\\\\
U = \frac{1}{S} \frac{e^{\sqrt{S}x}-e^{-\sqrt{S}x}}{e^{\sqrt{S}}-e^{-\sqrt{S}}} = \frac{1}{S} \frac{sinh(\sqrt(S)x)}{sinh(\sqrt{S}})\\\\
$$
The inverse transform is the convolution $$1 \ast
\mathcal{L}^{-1}(\frac{sinh(\sqrt(S)x)}{sinh(\sqrt{S}}) $$

Does anyone know of a table where I can find this... The integral to actually compute it myself is... terrifying. Do I have to use the integral... if so... can someone show me how...
 
  • #3
I don't know but I found a good inverse Laplace and inverse Fourier calculator on wolfram. Maybe I will try the integral again after I study residue theory or something.
 

1. What is a Laplace Transform?

A Laplace Transform is a mathematical operation that transforms a function from the time domain to the frequency domain. It is denoted by the symbol "L" and is often used in the study of differential equations.

2. How does the Laplace Transform help in solving PDEs?

The Laplace Transform allows us to convert a partial differential equation (PDE) into an algebraic equation, making it easier to solve. This is because the Laplace Transform has the property of converting derivatives into algebraic expressions, making it a powerful tool in solving PDEs.

3. What is the inverse lookup method for solving PDEs with Laplace Transforms?

The inverse lookup method involves using a table of Laplace Transform pairs to find the inverse Laplace Transform of an algebraic expression. This allows us to convert the solution of the PDE back to the time domain, giving us the final solution.

4. What are the advantages of using Laplace Transforms to solve PDEs?

Using Laplace Transforms to solve PDEs has several advantages. It can simplify the problem by converting it into an algebraic equation, which is often easier to solve. It also allows for the use of initial and boundary conditions, making it applicable to a wider range of PDEs. Additionally, the method is systematic and can be applied to a variety of PDEs.

5. Are there any limitations to using Laplace Transforms for solving PDEs?

While Laplace Transforms can be a powerful tool in solving PDEs, they do have some limitations. They may not be suitable for all types of PDEs, especially those with complex boundary conditions. Additionally, the inverse lookup method may not always yield a closed-form solution, which may make it difficult to interpret the final solution.

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