Families of Self-Similar Symmetric Laplacians on the Interval and Sierpinski Gasket

We were able to use our eigenfunction/value pairs of the laplacian to solve the heat and wave equations on both the interval and sierpinski gasket according to our nonstandard laplacians. Since the space of possible parameters and initial conditions is huge, here we present a small sample of cases which we found to be interesting. For more mathematical details, please see paper. For more computational details, and the ability to set initial conditions and parameters at will, see matlab code . It is common practice to examine the heat equation when applied to an identifyng function as the intial condition. A solution for the interval for the standard p=0.5 and an initial condition of 0 everywhere except the exact midpoint of the interval yields the expected classical solution avi . When can repeat these numerical computations for p=0.1 and p=0.9. Meaningful analysis of such complicated solutions can be difficult, but note that the rate of convergence to the uniform distribution is not independent of p. We can extend this method to the sierpinski gasket. In thise case we let the identifying function used for our initial condition be zero everywhere except for the point in level 1 connecting the inner and outter sets of triangles. We can create time series for r=1 ,r=0.3 ,r=3 . Note again that the rate of convergence to the uniform distribution varies as we vary r. By changing the spectral operator from exponential decay to trigonometric oscillations, we can use these methods to solve the wave equation. Here we use an initial condition on the first derivative; specifying an identifying function zero everywhere except for a small interval near x=0. Then in the standard p=0.5 case we obsreve a crisp travelling wave along the interval. Drastically altering the value of p to p=0.1 or p=0.9 appears to completely destroy the aforementioned wave in favor of more violent oscillations. Approaching the boundaries with p=0.01 or p=0.99 , these oscillations appear to concentrate onto specific portions of the interval; those which are being assigned smaller and smaller portions of the measure. A more subtle alteration to p=0.48 or p=0.52 produces a solution that appears structurally similar to that of the standard case. The most significant alteration is that the wave now leaves a very significant 'wake' of smaller oscillations in its trail. The sierpinski gasket produces very similar results on a more complicated structure that is more difficult to visualize. Here are the associated animations for r=1 , r=0.01 , and r=100 .