Third-order Linear Parabolic PDE

What's the best method to solve analytically an equation of the form

$$f_t=f_x+af_{xx}+bf_{xxx}$$

with $a,b\in\mathbb{R}$ ?

Solution 1:

The partial differential equation specified is given by,

$$\frac{\partial f(x,t)}{\partial t}=\frac{\partial f(x,t)}{\partial x} + a \frac{\partial^2 f(x,t)}{\partial x^2}+b\frac{\partial^3 f(x,t)}{\partial x^3}$$

We approach the problem with the Fourier transform, i.e.

$$F(k,t)=\int_{-\infty}^{\infty} \mathrm{d}x \, e^{-ikx} \, f(x,t)$$

The new differential equation in terms of the function in Fourier space is given by,

$$\frac{\partial F(k,t)}{\partial t}=F(k,t)\left(ik-ak^2-ibk^3\right)$$

where we have employed the standard formula for the Fourier transform of a derivative, derived by integration by parts, c.f. Fourier Transform. Can you proceed from here? Notice as the equation does not contain any $k$ derivatives, $F=F(t)$ from the perspective of the equation.

Additional Information

It is clear a particular solution to the equation in Fourier space is simply an exponential given by,

$$F(k,t)=\exp \left[ \left(ik -ak^2-ibk^3 \right)t\right]$$

To convert back to physical space is a daunting task,$^{\dagger}$ the inverse Fourier integral required:

$$f(x,t)=\int_{-\infty}^{\infty} \frac{\mathrm{d}k }{2\pi}\, \exp \left[ \left(ik -ak^2-ibk^3 \right)t + ikx\right]$$

$\dagger$ As MIT Professor Arthur Mattuck stated, jokingly, "that's conservation of mathematical difficulty!"