Two-dimensional wave equations with fractal boundaries |
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Authors: | Lin Tao Ma Wei Yi Su |
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Institution: | 1. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China 2. College of Mathematical Sciences, Guangxi Normal University, Guilin, 541004, P. R. China 3. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China
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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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Keywords: | Von Koch type curve p-type derivative two-dimensional wave equation fractal boundary fractal dimension |
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