Fractional diffusions with time-varying coefficients

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators $L$ and converting their fractional power $L^{\alpha}$ into Erd\'elyi--Kober fractional integrals. We study also probabilistic properties of the r.v.'s whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed r.v.'s

Mellin Transforms of the Generalized Fractional Integrals and Derivatives

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized $\delta_{r,m}$ operators with generalized Stirling numbers and Lah numbers. For example, we show that $\delta_{1,1}$ corresponds to the Stirling numbers of the $2^{nd}$ kind and $\delta_{2,1}$ corresponds to the unsigned Lah numbers. Further, we show that the two operators $\delta_{r,m}$ and $\delta_{m,r}$, $r,m\in\mathbb{N}$, generate the same sequence given by the recurrence relation $$S(n,k)=\sum_{i=0}^r \big(m+(m-r)(n-2)+k-i-1\big)_{r-i}\binom{r}{i} S(n-1,k-i), \;\; 0< k\leq n,$$ with $S(0,0)=1$ and $S(n,0)=S(n,k)=0$ for $n>0$ and $1+min\{r,m\}(n-1) < k$ or $k\leq 0$. Finally, we define a new class of sequences for $r \in \{\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, ...\}$ and in turn show that $\delta_{\frac{1}{2},1}$ corresponds to the generalized Laguerre polynomials.Comment: 17 pages, 1 figure, 9 tables, Accepted for publication in Applied Mathematics and Computatio

Diffusion in quantum geometry

The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is controlled by a multiscale fractional diffusion equation, and physically interpreted as a composite stochastic process. The simplest example is a fractional telegraph process, describing quantum spacetimes with a spectral dimension equal to 2 in the ultraviolet and monotonically rising to 4 towards the infrared. The general profile of the spectral dimension of the recently introduced multifractional spaces is constructed for the first time.Comment: 5 pages, 1 figure. v2: title slightly changed, discussion improve

Fractional Calculus of the Generalized Wright Function

Mathematics Subject Classification: 26A33, 33C20.The paper is devoted to the study of the fractional calculus of the generalized Wright function pΨq(z) defined for z ∈ C, complex ai, bj ∈ C and real αi, βj ∈ R (i = 1, 2, · · · p; j = 1, 2, · · · , q) by the series pΨq (z) It is proved that the Riemann-Liouville fractional integrals and derivative of the Wright function are also the Wright functions but of greater order. Special cases are considered.* The present investigation was partially supported by Belarusian Fundamental Research Fund

Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and associated solitonic hierarchies. The constructions yield horizontal/vertical pairs of fractional vector sine-Gordon equations and fractional vector mKdV equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of Schrodinger maps.Comment: latex2e, 11pt, 21 pages; the variant accepted to J. Math. Phys.; new and up--dated reference

An Analog of the Tricomi Problem for a Mixed Type Equation with a Partial Fractional Derivative

Mathematics Subject Classification 2010: 35M10, 35R11, 26A33, 33C05, 33E12, 33C20.The paper deals with an analog of Tricomi boundary value problem for a partial differential equation of mixed type involving a diffusion equation with the Riemann-Liouville partial fractional derivative and a hyperbolic equation with two degenerate lines. By using the properties of the Gauss hypergeometric function and of the generalized fractional integrals and derivatives with such a function in the kernel, the uniqueness and existence of a solution of the considered problem are proved, and its explicit solution is established in terms of the new special function

Fractional Integration of the Product of Bessel Functions of the First Kind

Dedicated to 75th birthday of Prof. A.M. Mathai, Mathematical Subject Classification 2010:26A33, 33C10, 33C20, 33C50, 33C60, 26A09Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann-Liouville and Erdélyi-Kober fractional integral operators. Formulas for compositions of such generalized fractional integrals with the product of Bessel functions of the first kind are proved. Special cases for the product of cosine and sine functions are given. The results are established in terms of generalized Lau-ricella function due to Srivastava and Daoust. Corresponding assertions for the Riemann-Liouville and Erdélyi-Kober fractional integrals are presented

Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given

Fractional Calculus of P-transforms

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99The fractional calculus of the P-transform or pathway transform which is a generalization of many well known integral transforms is studied. The Mellin and Laplace transforms of a P-transform are obtained. The composition formulae for the various fractional operators such as Saigo operator, Kober operator and Riemann-Liouville fractional integral and differential operators with P-transform are proved. Application of the P-transform in reaction rate theory in astrophysics in connection with extended nonresonant thermonuclear reaction rate probability integral in the Maxwell-Boltzmann case and cut-off case is established. The behaviour of the kernel functions of type-1 and type-2 P-transform are also studied