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- W2022787284 abstract "Abstract It is shown that, if the problem is defined in the setting of fractional calculus via fractional difference on non-differentiable functions, then the solution of the functional equation f ( x + y ) = f ( x ) f ( y ) is exactly defined as the solution of a linear fractional differential equation. The dual or counterpart problem, that is the fractional solution of the equation g ( xy ) = g ( x ) + g ( y ), is also considered, and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized Mittag–Leffler function which is nowhere differentiable. This framework suggests a definition of fractional Laplace’s transform expressed in terms of generalized Mittag–Leffler function, and its main properties are outlined: mainly inverse function and convolution. One takes this opportunity to display a (new) fractional Taylor’s series for functions f ( x , y ) of two variables x and y . Many open problems are stated, which are directly related to the non-differentiability of the functions so involved, therefore the title “on the fractional solution…”." @default.
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- W2022787284 date "2012-11-01" @default.
- W2022787284 modified "2023-09-26" @default.
- W2022787284 title "On the fractional solution of the equation f(x+y)=f(x)f(y) and its application to fractional Laplace’s transform" @default.
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- W2022787284 doi "https://doi.org/10.1016/j.amc.2012.08.004" @default.
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