A speculative extension of the differential operator definition to fractal via the fundamental solution

This paper makes a speculative extension of the fundamental solution of the standard integer-order differential operators to fractal. Then, the fractal fundamental solution is used via the implicit calculus equation modeling approach to define differential operators on fractal for modeling complex mechanical behaviors of fractal materials. By employing the singular boundary method, a recent boundary discretization technique with the fundamental solution, this study also makes numerical simulation of fractal Laplace problems of multiply-connected and composite material. Results show the validity and rationality of the conjectured definition of Laplace operator on fractal. Furthermore, the fractional and the fractal Laplace operators are also compared in our numerical experiments.