Self-similar solution for parabolic equations

Let us consider the following parabolic Eq.

$$\frac{\partial u}{\partial t} = \eta \frac{\partial^2 u}{\partial x^2} \tag{1}$$

This equation can be written in a more general form

$$\frac{\partial u}{\partial t}=\frac{1}{r^{s-1}}\frac{\partial}{\partial r}\left(r^{s-1}D(u)\frac{\partial u}{\partial r}\right), \quad D(u)=Ku^{n} \tag{2}$$

where $K$ is a constant, $n>0$ is the (positive) nonlinearity exponent, and $s\in{1,2,3}$ is the number of space dimensions. For a one-dimensional equation like Eq.(1), we have $s=1$, and thus $r=x$. For the limiting case of $n\rightarrow0$, the general form reduces to Eq.(1), with $K=\eta$.

Pattle's self-similar solution [1, 2, 3] is of the form

$$u(r,t)=u_0(1+\chi t)^{-\frac{s}{sn+2}} \left(1-\frac{r^2}{R^2(t)} \right)^{\frac{1}{n}}$$

for $r < R(t)$, and 0 otherwise, where

$$R(t)= R_0(1+\chi t)^{\frac{1}{sn+2}}, \quad \chi=\frac{sn+2}{n}\cdot\frac{2D(T_0)}{R_0} $$

For the one-dimensional linear equation, the solution becomes

$$u(x,t)=u_0(1+\chi t)^{-\frac{1}{2}} \exp\left(-\frac{x^2}{2R_{\sigma}^2(t)} \right)$$

where $R_{\sigma}(t)=R_{\sigma 0}(1+\chi t)^{\frac{1}{2}}$, $\chi=\frac{2 \eta}{R_{\sigma 0}^2}$. Here where $R_{\sigma}(t)$ is the standard deviation of the Gaussian distribution. The total quantity is $\phi_0=(2\pi)^{\frac{1}{2}}u_0R_{\sigma 0}$. The 1D solution can be used to test the implementation in ex_5. Note: here, $u_0 = \max[u(t = 0)]$ is the initial representative peak value.

[1] Pattle, R. E. 1959, Q. J. Mech. Appl. Math., 12, 407

[2] Furuseth, S. V., Cherry, G., & Martínez-Sykora, J. 2024, A&A, 659, A186

[3] Moreno-Insertis, F., Nóbrega-Siverio, D., Priest, E. R., & Hood, A. W. 2022, A&A, 662, A42

[4] Furuseth, S. V., Cherry, G., Martinez-Sykora, J.