Abstract
The generic nonlocal fractal calculus scheme have been formulated in
this work. A unified derivative operator which employs an interpolated
characteristic between the generic nonlocal derivative in
Riemann–Liouville and Caputo senses has also been derived. For being
generic, an arbitrary kernel function has been adopted. The condition on
fractional order has been derived so that it is not related to the
γ-dimension of the fractal set. The fractal Laplace transforms of our
operators have been derived. A simple illustrative example and practical
ones have been presented. Unlike the previous power law kernel-based
nonlocal fractal calculus operators, ours are generic, consistent with
the local fractal derivative and employ higher degree of freedom. The
inverse relationships between our derivative and integral operators can
be achieved. The results obtained from the examples are significantly
different from such previous operator-based counterparts and
significantly depended on the kernel function. The unified operator
displays an interpolated characteristic as expected.