Problem Description
Model Descriptions
\[\begin{array}{rl} {d C \over d t} \: = \: & D_x {\partial^2 C \over \partial x^2} \: + \: D_y {\partial^2 C \over \partial y^2} \: - \: v_x {d C \over d x} \: - \: v_y {d C \over d y} \: - \: R C \\ \\ & \mbox{where:} \\ & \ \ D_x \mbox{ is the hydrodynamic dispersion coefficient in the } x \mbox{ direction} \\ & \ \ D_y \mbox{ is the hydrodynamic dispersion coefficient in the } y \mbox{ direction} \\ & \ \ v_x \mbox{ is the advective transport or seepage velocity in the } x \mbox{ direction} \\ & \ \ v_y \mbox{ is the advective transport or seepage velocity in the } y \mbox{ direction} \\ & \ \ R \mbox{ is the reaction coefficient which takes into acount the linear} \\ & \ \ \ \ \mbox{equilibrium retardation factor and effective first order decay} \\ & \ \ \ \ \mbox{rate due to combined biotic and abiotic processes} \end{array}\]
A nonlinear response function is built to model the concentrations over the spatial and temporal domain, resulting from the ADR PDE
For each model, a spatial domain of \(0<x<10\) was used, with \(D_x = D_y = 0.57\), \(v_x = v_y = 2.25\), \(R = 0.01\), and the boundary condition \(C(0) = 1\).
Results
Resulting \(C(t)\) curves are generated for each model, for \(x\) values of 2, 4, 6, 8, and \(y\) values of 2, 4, 6, 8. Differential equation model values are shown as data points, and nonlinear response function model values are shown as lines.
Prediction curves are generated for an \(x\) value of 5 at \(y = 2, 4, 6, 8\).
All of the resulting plots are shown here.
Conclusions
The nonlinear response function show accurate fits to the data derived from solution of the partial differential equation.
The nonlinear response function is able to accurately predict the time curves for \(x = 5\) and \(y = 2, 4, 6, 8\).
The nonlinear response function provides an accurate model for time curves over the entire spatial domain, thereby extending its applicability in modeling nonlinear dynamics to spatially-variant nonlinear dynamics.