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

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

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

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3 Answers 3

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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!"

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  • $\begingroup$ this is fine as long a the PDE is defined on $\mathbb R\times \mathbb R$ (space-time). What about smaller space domains? Is anything known? $\endgroup$
    – DeM
    Commented Jan 13, 2016 at 15:52
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The way I would do it would be to write the equation in reciprocal space. The equation becomes algebraic (i.e., there are no differential operators). You will get an expression $\omega = \omega(\vec{k})$. Then given an initial $f(x,t_0)$, you can decompose it as $f(x,t_0) = \int f(\vec{k},t_0) e^{i\vec{k}\cdot \vec{x}} d\vec{k}$. Then the solution for $t\ne t_0$ will be $f(x,t) = \int f(k,t_0) e^{-i\omega(\vec{k})t} e^{i\vec{k}\cdot \vec{x}} d\vec{k}$

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It seems that the best method is the Fourier substitution $$f(x,t) = \chi(x)\tau(t),$$ then $${\tau'(t)\over\tau} = {b\chi'''(x)+a\chi''(x)+\chi(x)\over\chi}=-\lambda = const,$$ $$\begin{cases} \tau'(t)+\lambda\tau = 0\\ b\chi'''(x) + a\chi''(x) + \chi'(x) + \lambda\chi(x) = 0. \end{cases}$$ That leads to generalize solution in the form of $$f(x,t) = \int\limits_{\Omega(\lambda)}\left(C_1e^{r_1(\lambda)x} + C_2e^{r_2(\lambda)x} + C_3e^{r_3(\lambda)x} + C_4e^{r_4(\lambda)x}\right)e^{-\lambda t}dt,$$ where $r_i(\lambda)$ are the roots of the algebraic equation $$br^3+ar^2+r+\lambda = 0.$$

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