The equation provided is a diffusion equation of the form:

$\frac{\partial f}{\partial t} - \frac{1}{2} \frac{\partial^2 f}{\partial z^2} = 0$

This is a parabolic partial differential equation, often representing diffusion processes in physics and engineering.

Solution Method

To solve this, we can use separation of variables or the Fourier transform method, depending on the boundary conditions. Here, I'll outline a general solution using Fourier transforms for an infinite domain:

1. Take the Fourier transform of $f(z,t)$ with respect to $z$, denoted by $\hat{f}(k,t)$, where $k$ is the wavenumber.

   $\hat{f}(k,t) = \int_{-\infty}^{\infty} f(z,t) e^{-ikz} \, dz$

   Applying the Fourier transform to both sides of the diffusion equation:

   $\frac{\partial f}{\partial t} \Rightarrow \frac{\partial \hat{f}(k,t)}{\partial t}$ and $\frac{\partial^2 f}{\partial z^2} \Rightarrow -k^2 \hat{f}(k,t)$

   So the equation becomes:

   $\frac{\partial \hat{f}(k,t)}{\partial t} + \frac{1}{2} k^2 \hat{f}(k,t) = 0$

2. Solve the resulting ordinary differential equation (ODE) in time:

   $\frac{\partial \hat{f}(k,t)}{\partial t} = -\frac{1}{2} k^2 \hat{f}(k,t)$

   This is a separable ODE, with the solution:

   $\hat{f}(k,t) = \hat{f}(k,0) e^{-\frac{1}{2} k^2 t}$

   Here, $\hat{f}(k,0)$ is the Fourier transform of the initial condition $f(z,0)$.

3. Finally, take the inverse Fourier transform to find $f(z,t)$:

   $f(z,t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(k,0) e^{-\frac{1}{2} k^2 t} e^{ikz} \, dk$

The result depends on the initial condition $f(z,0)$, but the general form involves Gaussian functions due to the exponential decay in the Fourier space.

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Would you like to explore more about this solution process or see a specific example with an initial condition?

Related Questions:

1. How is the separation of variables used to solve the diffusion equation?
2. Can we solve the diffusion equation on a finite domain?
3. How does the heat equation relate to the diffusion equation?
4. What are the boundary conditions typically used in diffusion problems?
5. How can Green's functions be applied to solve the diffusion equation?

Tip:

The Fourier transform is a powerful tool for solving linear PDEs, especially for problems with infinite or periodic domains.