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I want to solve the following nonlinear system in 1D

\begin{cases} \dot{R} + v \frac{\partial R }{\partial x} - \frac{\partial }{\partial x}\left( D \frac{\partial R }{\partial x} \right) -\Gamma = 0 \\ \dot{h}+\Gamma = 0 \end{cases}

where

\begin{equation} \Gamma(h,R) = -R(x,t)\left[ \left( \gamma \frac{\partial h }{\partial x} \right)+ \left( k \frac{\partial^2 h }{\partial x^2 } \right) \right] \end{equation}

is called the interaction term, $\ h(x,t)$ is the height of the sandpile and $R(x,t)$ describes the behavior of the rolling grains. $\gamma, D, k, v$ are constants.

These equations describe the evolution of sandpile surface. In this system $h,R$ are linear but the system is nonlinear since the term $\Gamma$ is nonlinear.

my attempt:

let $\Omega = [a,b]$, multiplying each equation of the system by $\phi_{i}$, integrating and using integration by parts

\begin{multline} \int_{\Omega} \dot{R}\phi_{i} + v \frac{\partial R }{\partial x}\phi_{i} + R \left( \gamma \frac{\partial h }{\partial x} \right)\phi_{i} - \frac{\partial }{\partial x}\left( D \frac{\partial R }{\partial x} \right)\phi_{i} + R \left( k \frac{\partial^2 h }{\partial x^2 } \right) \phi_{i} = \\ \int_{\Omega} \left[ \dot{R}\phi_{i} + v \frac{\partial R }{\partial x}\phi_{i} + R \left( \gamma \frac{\partial h }{\partial x} \right)\phi_{i} + \left( D \frac{\partial R }{\partial x} \right) \frac{d \phi_{i} }{dx} \ - \left( k \frac{\partial h }{\partial x } \right) \left( \frac{\partial R }{\partial x}\phi_{i} + R \frac{ d\phi_{i} }{dx} \right) \right] + \\ \left[ \left( k \frac{\partial h }{\partial x } \right)\left( R \phi_{i} \right) \right]_{a}^{b} - \left[ \left( D \frac{\partial R }{\partial x} \right) \phi_{i} \right]_{a}^{b} = 0 \label{RollingXphi} \end{multline}

\begin{equation} \int_{\Omega} \dot{h} \phi_{i} - R \left( \gamma \frac{\partial h }{\partial x } \right) \phi_{i} - R \left( k \frac{\partial^2 h }{\partial x^2 } \right) \phi_{i} = \\ \int_{\Omega} \left[ \dot{h} \phi_{i} - R \left( \gamma \frac{\partial h }{\partial x } \right) \phi_{i} + \left( k \frac{\partial h }{\partial x } \right) \left( \frac{\partial R }{\partial x}\phi_{i} + R \frac{ d\phi_{i} }{dx} \right) \right] \ - \\ \left[ \left( k \frac{\partial h }{\partial x } \right)\left( R \phi_{i} \right) \right]_{a}^{b} = 0 \end{equation}

I could write $h,R$ as a linear combination of the basis functions $\{\phi_{j}\}_{j=1}^{N}$ but the expression looks more complicated. I'm stuck.

My questions are:

  1. Do you know references(books or papers) where I can learn how solve this system using the finite element method?
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    $\begingroup$ For nonlinear problems like this, you have certainly gone about doing things correctly so far. From here, you just need to use some nonlinear equation technique (like Newton's method) to solve your system of nonlinear equations. $\endgroup$ – spektr Jul 15 at 17:23
  • $\begingroup$ If you have access to matlab you can probably solve that system using the pdepe function, mathworks.com/help/matlab/ref/pdepe.html. $\endgroup$ – Bill Greene Jul 15 at 17:35

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