An important property for solutions of differential equations is stability. Stability is important in physical applications because it determines whether or not a solution may be seen in the real world. If a solution is unstable, small perturbations of the system will fore the system away from the solution. If a solution is stable, the system will return to that solution after small perturbations. To determine whether or not a solution is stable, one linearizes the differential equation and examines the spectrum of the resulting linear operator. The geometry of the spectrum reveals stability properties, but is typically very difficult to find analytically. The spectral problem may be truncated to a matrix eigenvalue problem. Once the matrix is formed, we use computers to find the eigenvalues and plot them to understand the geometry of the spectrum. In this project, students will be given a partial differential equation (PDE) and class of traveling wave solution to that equation. This class of solutions typically has various parameters that may take a continuum of values in some fixed region of space. From a given solution, students will use the above described method to compute the spectrum using a computer. If there is time, the solution parameter space can be examined to determine regions where the spectrum has qualitatively different shapes. If there is still time, students may look at transition regions by altering their program to allow for dynamic solution parameter changes. In doing this, interesting movies and an applet can be created.
An important property for solutions of differential equations is stability. Stability is important in physical applications because it determines whether or not a solution may be seen in the real world. If a solution is unstable, small perturbations of the system will fore the system away from the solution. If a solution is stable, the system will return to that solution after small perturbations. To determine whether or not a solution is stable, one linearizes the differential equation and examines the spectrum of the resulting linear operator. The geometry of the spectrum reveals stability properties, but is typically very difficult to find analytically. The spectral problem may be truncated to a matrix eigenvalue problem. Once the matrix is formed, we use computers to find the eigenvalues and plot them to understand the geometry of the spectrum. In this project, students will be given a partial differential equation (PDE) and class of traveling wave solution to that equation. This class of solutions typically has various parameters that may take a continuum of values in some fixed region of space. From a given solution, students will use the above described method to compute the spectrum using a computer. If there is time, the solution parameter space can be examined to determine regions where the spectrum has qualitatively different shapes. If there is still time, students may look at transition regions by altering their program to allow for dynamic solution parameter changes. In doing this, interesting movies and an applet can be created.