Abstract
This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given.
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Supported by National Natural Science Foundation of China (Grant No. 10571084)
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Ma, L.T., Su, W.Y. Two-dimensional wave equations with fractal boundaries. Acta. Math. Sin.-English Ser. 29, 2321–2342 (2013). https://doi.org/10.1007/s10114-013-1764-5
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DOI: https://doi.org/10.1007/s10114-013-1764-5