Using fourier and laplace transform to solve PDE

In summary, The heat equation can be solved using the Fourier transform, which involves taking the transform with respect to x for each term in the equation. The exercise then asks to solve it using both the Fourier and Laplace transforms. This involves taking the Fourier transform in space and the Laplace transform in time, resulting in a polynomial equation in two variables and a dispersion relation. It is important to note that the Fourier transform is defined in \mathbb{R} while the Laplace transform is defined in \mathbb{R}^+, and they must be done independently in their respective domains. This type of question provides information about solving the heat equation using multiple transforms and the resulting dispersion relation.
  • #1
cyw1984
4
0
PDE is type of heat equation.

Many book only gives an example of solving heat equation using Fourier transform.
An exercise asks me to solve it for using Fourier and laplace transform:

In the heat equation, we'd take the Fourier transform with respect to x for
each term in the equation. How to combine it with using Fourier and laplace transform

Can anyone suggest some example and notes to me??
 
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  • #2
umum...the question mention us to use Fourier and laplace transform. Is there mention us to use Fourier transform with respect to t (t>0) and then inverse Fourier transform to solve the solution; Separately, solve the question using laplace transform with respect to x -\infty< x < \infty and then inverse laplace transform?

Can there combines two transformation in one??
What is the result compared with two method??
 
  • #3
It's all the way around... The Fourier transform is defined in [itex]\mathbb{R}[/itex], while the Laplace transform is defined in [itex]\mathbb{R}^+[/itex], so you have to take the Fourier transform in space and the Laplace transform in time. That way, you will land with a polinomial equation in two variables, which will give you some sort of dispersion relation.
 
  • #4
AiRAVATA said:
It's all the way around... The Fourier transform is defined in [itex]\mathbb{R}[/itex], while the Laplace transform is defined in [itex]\mathbb{R}^+[/itex], so you have to take the Fourier transform in space and the Laplace transform in time. That way, you will land with a polinomial equation in two variables, which will give you some sort of dispersion relation.

O...
you mean that it must need to do it independent in two times??

What information will give in this type of question??
 

1. What is the difference between Fourier and Laplace transforms?

The Fourier transform is used to convert a function from its original domain (such as time or space) to its frequency domain. The Laplace transform, on the other hand, is used to convert a function from its original domain to a complex s-plane. While both transforms involve integration, the Fourier transform uses an integral over time or space, while the Laplace transform uses a complex integral over a certain contour in the s-plane.

2. Can Fourier and Laplace transforms be used interchangeably to solve PDEs?

No, Fourier and Laplace transforms are used for different types of PDEs. The Fourier transform is typically used for PDEs with initial value problems, while the Laplace transform is used for PDEs with boundary value problems. Additionally, the Laplace transform is more useful for transient problems, while the Fourier transform is better suited for steady-state problems.

3. How do Fourier and Laplace transforms help to solve PDEs?

Fourier and Laplace transforms can help to solve PDEs by converting them into algebraic equations that are easier to solve. The transforms essentially change the variables of the PDE, making it easier to apply boundary or initial conditions and find a solution.

4. Are Fourier and Laplace transforms limited to linear PDEs?

No, Fourier and Laplace transforms can also be used to solve non-linear PDEs. However, the resulting equations may be more complex and difficult to solve compared to linear PDEs.

5. Are there any limitations to using Fourier and Laplace transforms for solving PDEs?

While Fourier and Laplace transforms can be powerful tools for solving PDEs, they are not applicable to all types of PDEs. They are most commonly used for PDEs with constant coefficients and certain types of boundary conditions. Additionally, the transforms may not always yield an exact solution and may require further approximations or numerical methods.

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