I'm wondering how to solve this 1D PDE using Matlab pdepe function:
$$ \frac{\partial u}{\partial t} = \partial_x \Big( D_0 \; (u\circ\theta)^{\alpha}\; \partial_x u \Big)$$
where $u(0,0) = 1,\; u(0,1) = 0, u(0,x) = u_0(x), x \in [0,1] $
the deviation is $\theta(x) = ax^{\beta} + b$
and $D_0, a, b, \alpha, \beta $ are constants.
I'm trying to "adjust" the pde to use Matlab pdepe function but I don't see how to choose the pde function to take into account the deviation since we must have a function of $(x,t,u,u_x)$.
Any idea to help?
Separation of variables $u=X(x)T(t)$ yields $$ X(x)\dot{T}(t) = D_0\, T(t)^{\alpha+1} \big( ( X\circ \theta )^\alpha X' \big)' (x) , $$ where the dot denotes differentiation w.r.t. time, and the prime denotes spatial differentiation. Dividing by $X(x) T(t)^{\alpha+1}$ on both sides, a separation constant $\lambda$ is introduced so as to write differential equations for $X(x)$ and $T(t)$. The differential equation $$ \dot{T}(t) = -\lambda T(t)^{\alpha+1} $$ satisfied by $T(t)$ is a Bernoulli equation, which can be solved analytically. The differential equation $$ \big(X' (X\circ\theta)^\alpha \big)'(x) = -\lambda X(x) $$ satisfied by $X(x)$ is more difficult to solve in general (it is a second-order nonlinear delay differential equation). However, it is possible to do so in some simpler particular cases -- consider for instance the case $\theta(x) = x$ where $(X^{1+\alpha})'' = -\lambda(1+\alpha) X$ may be addressed analytically. In the general case, one may rather use perturbation methods or numerical integration.