Fourier/Laplace transform for PDE

In summary, the conversation discusses finding the fundamental solution for a diffusion equation with a Dirac delta function as the initial condition. The laplace and Fourier transforms are used to solve for the first and second derivatives, respectively. The usefulness of the Green function for solving more complicated problems is also mentioned.
  • #1
vladimir69
130
0
hello
i am trying to find the fundamental solution to
[tex]\frac{\partial c}{\partial t} = D\frac{\partial ^2 c}{\partial x^2}[/tex]
where c=c(x,t)
with initial condition being [tex]c(x,0)=\delta (x)[/tex]
where [tex]\delta (x)[/tex] is the dirac delta function.
i have the solution and working written out in front of me.
first off its got the laplace transform of [tex]\frac{\partial c}{\partial t}[/tex] as
[tex] u\hat c (x,u) -c(x,0)[/tex]
and the Fourier transform of [tex]\frac{\partial ^2 c}{\partial x^2}[/tex] as
[tex]-q^2 \tilde c (q,t) [/tex]
and then out of nowhere we get
[tex]\hat c (q,u) = \frac{c(q,0)}{u+Dq^2}[/tex]
once that bit of magic is done and a leap of faith is taken then i can see how the rest of it falls into place but can anyone explain the above steps to me?
 
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  • #2
To get the Fourier and laplace transform of the first derivative you put df/dx into the definitions, integrate by parts and use, in the Fourier case, that f(x) must vanish as x goes to infinity in order for the the Fourier transform of f(x) to exist. And this you easily can generalize to higher derivatives.

If you set c(q,0)=1 you get the Green function for the diffusion equation in the transformed space. The Green function is very useful for solving more complicated problems. A general solution to the diffusion equation is then integrals over the Green function G(r,r';t,t') times the source term, the boundary conditions, the initial condition, respectively.
 
  • #3


The use of Fourier and Laplace transforms in solving partial differential equations (PDEs) is a powerful technique that allows us to convert the PDE into a simpler algebraic equation. In this case, we are solving for the fundamental solution of a PDE with an initial condition.

To begin, we apply the Laplace transform to both sides of the PDE, which gives us:

\mathcal{L}\left[\frac{\partial c}{\partial t}\right] = D\mathcal{L}\left[\frac{\partial^2 c}{\partial x^2}\right]

Using the properties of Laplace transforms, we can rewrite this as:

u\hat{c}(x,u) - c(x,0) = D(-q^2\tilde{c}(q,t))

Where \hat{c}(x,u) and \tilde{c}(q,t) are the Laplace and Fourier transforms of c(x,t) respectively. The Laplace transform of the initial condition c(x,0) is simply the value of c at t=0, which is given as \delta(x).

Next, we apply the Fourier transform to both sides of the equation, which gives us:

\mathcal{F}\left[u\hat{c}(x,u) - c(x,0)\right] = \mathcal{F}\left[D(-q^2\tilde{c}(q,t))\right]

Using the properties of Fourier transforms, we can rewrite this as:

\tilde{c}(q,t) = \frac{\hat{c}(x,0)}{u+Dq^2}

Where \hat{c}(x,0) is the Fourier transform of c(x,0), which is simply the constant value 1.

Finally, we substitute the initial condition c(x,0)=\delta(x) into our equation, which gives us:

\tilde{c}(q,t) = \frac{1}{u+Dq^2}

This is the fundamental solution to the PDE, which satisfies the given initial condition. This approach may seem like a "leap of faith", but it is a well-established method for solving PDEs with initial conditions using Fourier and Laplace transforms.
 

What is a Fourier transform for PDE?

A Fourier transform for PDE (partial differential equation) is a mathematical technique used to convert a function of time or space into a function of frequency. This allows for the analysis of the behavior of a function over a continuous range of frequencies, which is useful in solving PDEs.

What is a Laplace transform for PDE?

A Laplace transform for PDE is a mathematical operation that transforms a function of time (or other independent variable) into a function of a complex variable, allowing for the solution of initial value problems for PDEs. It is particularly useful in solving linear PDEs with constant coefficients.

What is the relationship between Fourier and Laplace transforms for PDEs?

The Laplace transform is a generalization of the Fourier transform, where the Fourier transform is a special case of the Laplace transform with the imaginary part of the complex variable set to zero. Both transforms are used in solving PDEs, but the Laplace transform is particularly useful in solving initial value problems while the Fourier transform is useful in studying the behavior of a function over a continuous range of frequencies.

How are Fourier and Laplace transforms used in solving PDEs?

Fourier and Laplace transforms are used in solving PDEs by converting the PDE into an ordinary differential equation (ODE), which is typically easier to solve. This is done by applying the transform to the dependent variable in the PDE, resulting in a new equation in terms of the transform variable. After solving the ODE, the inverse transform is used to obtain the solution to the original PDE.

What are some real-world applications of Fourier and Laplace transforms in PDEs?

Fourier and Laplace transforms have numerous real-world applications in PDEs, including heat transfer, fluid dynamics, quantum mechanics, and electrical engineering. They are used to model and solve various physical phenomena, such as heat conduction in a solid, airflow over a wing, and electronic circuits. These transforms are also widely used in image and signal processing, where they are used to analyze and manipulate signals in the frequency domain.

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