One-Dimensional Fractal Wave Equations

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition, and therefore the existing theory, introduced by Kigami and developed by many other mathematicians, cannot be appled. First, by using a weak formulation of the problem, we prove the existence, uniqueness and regularity of weak solutions of these wave equations. Second, we study numerical computations of the solutions. By using the second-order self-similar identities introduced by Strichartz et al., we discretize the equation and use the finite element method and central difference method to obtain numerical solutions. Last, we also prove that the numerical solutions converge to the weak solution, and obtain estimates for the convergence of this approximation scheme