On the fractional solution of the equation f(x + y) = f(x)f(y) and its application to fractional Laplace's transform

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…”.

[1]  Guy Jumarie,et al.  On the representation of fractional Brownian motion as an integral with respect to (dt)alpha , 2005, Appl. Math. Lett..

[2]  Guy Jumarie,et al.  Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions , 2009, Appl. Math. Lett..

[3]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[4]  T. Osler.  Taylor’s Series Generalized for Fractional Derivatives and Applications , 1971 .

[5]  M. Al-Akaidi.  Fractal Speech Processing , 2004 .

[6]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[7]  Guy Jumarie,et al.  An approach to differential geometry of fractional order via modified Riemann-Liouville derivative , 2012 .

[8]  Guy Jumarie,et al.  On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion , 2005, Appl. Math. Lett..

[9]  G. Jumarie.  Stochastic differential equations with fractional Brownian motion input , 1993 .

[10]  Kiran M. Kolwankar,et al.  Hölder exponents of irregular signals and local fractional derivatives , 1997, chao-dyn/9711010.

[11]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[12]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[13]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[14]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[15]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[16]  Kiran M. Kolwankar,et al.  Local Fractional Fokker-Planck Equation , 1998 .

[17]  L. M. B. C. Campos,et al.  On a Concept of Derivative of Complex Order with Applications to Special Functions , 1984 .

[18]  Guy Jumarie,et al.  New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations , 2006, Math. Comput. Model..

[19]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[20]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[21]  I. Podlubny.  Fractional differential equations , 1998 .