Fractional Taylor Series Research Articles

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form f (χ + h) = E α ( h α D χ α) f(χ), 0 α ≤ 1, where E α is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

A norandom variational approach to stochastic linear quadratic Gaussian optimization involving fractional noises (FLQG)

It is shown that the problem of minimizing (maximizing) a quadratic cost functional (quadratic gain functional) given the dynamicsdx=(fx+gu)dt+hdb(t,a) whereb(t, a) is a fractional Brownian motion of ordera, 0<2a<1, can be solved completely (and meaningfully!) by using the dynamical equations of the moments ofx(t). The key is to use fractional Taylor's series to obtain a relation between differential and differential of fractional order.

On calculus of local fractional derivatives

Local fractional derivative (LFD) operators have been introduced in the recent literature (Chaos 6 (1996) 505–513). Being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD operators. Generalization of directional LFD and multivariable fractional Taylor series to higher orders have been presented.