# Laplace and Fourier Transform of a PDE

• bvic4
In summary, the problem at hand is to derive an explicit solution for Langmuir waves in a plasma using the method of Laplace and Fourier transforms. The wave equation derived in part (a) of the problem matches the solution in the book, indicating it is correct. The initial conditions are given and the solution is assumed to be a plane wave, but this assumption cannot be made in this case. The equations that can be used for this problem are the Fourier transform and the Laplace transform, which leads to an equation with a complex exponential term. The inverse Fourier transform is needed to complete the solution, but there are difficulties in obtaining it due to the presence of alpha inside the square root.
bvic4

## Homework Statement

In this problem I'm trying to derive an explicit solution for Langmuir waves in a plasma. In part (a) of the problem I derived the wave equation

$$(\partial_t_t+\omega_e^2-3v_e^2\partial_x_x) E(x,t) = 0$$

This matches the solution in the book so I believe it's correct.

The initial conditions are
$$E(x,t=0)=f(x)$$
$$\partial_tE(x,t=0)=0$$
(The second i.c. should be the time derivative, but I can't figure out how to write it correctly)

For this part of the problem I am supposed to solve this equation explicitly. The problem statement says we should use the method of Laplace and Fourier transforms to do this. Generally, for this class, we assume the solution is a plane wave (E(x,t)=exp[-iwt+ikx]), but we don't get to do that here.

## Homework Equations

I don't have a lot of experience using Fourier transforms so the equations that I think I should use are:

Fourier Transform:

$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(\alpha)e^{-i\alpha x}d\alpha$$
$$f(\alpha)=\int_{-\infty}^{\infty}f(t)e^{i\alpha x}dx$$

## The Attempt at a Solution

I'm doing this problem by taking the Laplace Transform with respect to time and the Fourier transform with respect to x, which I think is right, but I'm not 100% sure about.

First I took the Fourier Transform (using tables) which gave me:

$$\delta_t_tE(\alpha,t) + \omega_e^2 E(\alpha,t) + 3 v_e^2 \alpha^2 E(\alpha,t) = 0$$

Then I took the Laplace Transform, which after some simplification, gave me:

$$E(\alpha,s) = s f(\alpha)/(s^2 + \omega_e^2 + 3 v_e^2 \alpha^2)$$

I used contour integration to get the inverse Laplace transform:

$$E(\alpha,t) = f(\alpha) (e^{i \sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t} + e^{-i \sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t})$$

Now I need to take the inverse Fourier transform and I think I'm done. The problem is I'm not sure how to do this. The only explicit way I know to do this is using contour integration, which isn't useful here because there aren't any poles. I'm not sure how I can use the tables because of that alpha inside of the square root. Any ideas?

Thanks,

Brian

I realized that the last equation I wrote can be written as

$$E(\alpha,t) = f(\alpha) (2*\cos{\sqrt{\omega_e^2 + 3 v_e^2 \alpha^2} t})$$

Is that helpful? Does anyone know how to take the inverse Fourier transform?

Thanks,

Brian

## 1. What is a Laplace transform of a PDE?

A Laplace transform of a PDE (Partial Differential Equation) is a mathematical operation that transforms a PDE from the time domain to the frequency domain. It is named after the French mathematician Pierre-Simon Laplace and is commonly used in engineering and physics to solve differential equations.

## 2. What is a Fourier transform of a PDE?

A Fourier transform of a PDE is a mathematical operation that transforms a PDE from the time domain to the frequency domain. It is named after the French mathematician Joseph Fourier and is commonly used in engineering and physics to analyze signals and solve differential equations.

## 3. What is the difference between Laplace and Fourier transform of a PDE?

The main difference between Laplace and Fourier transform of a PDE is the type of function that is being transformed. Laplace transform is used for functions that are defined for all positive time values, while Fourier transform is used for functions that are defined for all time values. Additionally, Laplace transform is used to solve initial value problems, while Fourier transform is used to analyze signals and solve boundary value problems.

## 4. What are the applications of Laplace and Fourier transform of a PDE?

Laplace and Fourier transform of a PDE have many applications in various fields such as engineering, physics, and mathematics. They are used to solve differential equations, analyze signals and systems, and study the behavior of physical phenomena. They are also used in image and signal processing, control systems, and electronic circuit analysis.

## 5. What are the limitations of Laplace and Fourier transform of a PDE?

One limitation of Laplace and Fourier transform of a PDE is that they can only be used for linear systems. They also assume that the signals or functions being transformed are continuous and have a finite number of discontinuities. Additionally, they may not work well for functions that have a rapidly changing behavior or infinite energy.

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