Ants decision model fitting with fokker planck

Consider the Fokker-Planck equation,

\begin{equation} \frac{\partial G}{\partial t}=\frac{\sigma^{2}}{2}\frac{\partial^{2}G}{\partial s^{2}}-\frac{\partial}{\partial s}\left(G\frac{\partial U(s)}{\partial s}\right),\\ \end{equation}

where \(U(s)\) is the effective potential for the decision variable:

\begin{equation} U(s)=\gamma s+ks\sum_{\alpha}\delta\left(t-t_{\alpha}\right).\\ \end{equation}

Using \(U(s)\), this becomes

\begin{equation} \label{eq:FP} \label{eq:FP}\frac{\partial G}{\partial t}=\frac{\sigma^{2}}{2}\frac{\partial^{2}G}{\partial s^{2}}-\frac{\partial G}{\partial s}\left(\gamma+k\sum_{\alpha}\delta\left(t-t_{\alpha}\right)\right).\\ \end{equation}

To solve this using the finite element method, first let \(G=\eta_{i}g_{i}\), where \(\eta_{i}\) are the shape functions and \(g_{i}\) are the nodal variables. Summation notation applies over roman indices, e.g. over \(i\) and \(j\). After writing the weak form of the equation and setting the integral of the residual to zero, we obtain the finite element matrix equation:

\begin{equation} \label{eq:FPmatrix} \label{eq:FPmatrix}M_{ij}\frac{dg_{j}}{dt}=\frac{\sigma^{2}}{2}A_{ij}g_{j}-\left(\gamma+k\sum_{\alpha}\delta\left(t-t_{\alpha}\right)\right)B_{ij}g_{j},\\ \end{equation}

where \(M_{ij}\), \(A_{ij}\), and \(B_{ij}\) are matrices: \(M_{ij}\) is the mass matrix, \(A_{ij}\) is a second-derivative operator, and \(B_{ij}\) is a derivative operator.

The mass matrix is defined by integrating the shape functions:

\begin{equation} M_{ij}=\int_{-1}^{1}\eta_{i}\eta_{j}ds.\\ \end{equation}

To define \(A_{ij}\), we will use integration by parts so that only a first derivative remains (and thus we will only need to use linear shape functions. Writing out the integral, and then integrating by parts, we have

\begin{align} \label{eq:Aijcalc} A_{ij} & =\int_{-1}^{1}\eta_{i}\frac{\partial^{2}\eta_{j}}{\partial s^{2}}ds\notag \\ & \label{eq:Aijcalc}=\left.\eta_{i}\frac{\partial\eta_{j}}{\partial s}\right|_{-1}^{1}-\int_{-1}^{1}\frac{\partial\eta_{i}}{\partial s}\frac{\partial\eta_{j}}{\partial s}ds.\@add@PDF@RDFa@triples\end{document}\\ \end{align}

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