Abstract-To the authors’ knowledge, previous derivations of the fractional diffusion equation are based on stochastic principles [l], with the result that physical interpretation of the resulting fractional derivatives has been elusive [2]. Herein, we develop a fmctional J%Z Iaw relating solute flux at a given point to what might be called the complete (twosided) fractional derivative of the co...

This paper presents a general solution for a space-and time-fractional diffusionwave equation defined in a bounded space domain. The space-and time-fractional derivatives are described in the Caputo sense. The application of Adomian decomposition method, developed for differential equations of integer order, is extended to derive a general solution of the space-and time-fractional diffusionwave...

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives bounded operators at them. Nonlinearity equation assumed be Lipschitz continuous dependent on lower order derivatives, which orders have same part as highest derivative. obtained abstract...

We investigate the profile of the blowing up solutions to the nonlinear nonlocal system (FDS) u′(t) + Dα0+(u − u0)(t) = |v(t)|q, t > 0, v′(t) + Dβ0+(v − v0)(t) = |u(t)|p, t > 0, where u(0) = u0 > 0, v(0) = v0 > 0, p > 1, q > 1 are given constants and Dα0+ and Dβ0+ , 0 < α < 1, 0 < β < 1 stand for the Riemann-Liouville fractional derivatives. Our method of proof relies on comparisons of the solu...

This paper is concerned with the numerical solutions of a variable-order space-time fractional reaction-diffusion model. The space-time fractional derivative is considered in the sense of Riesz-Feller, the system is defined by replacing the second order space derivatives with the variable Riesz-Feller derivatives. The problem is solved by an explicit finite difference method. Finally, simulatio...

The aim of this paper is to present a new numerical method for solving the Bagley-Torvik equation. This equation has an important role in fractional calculus. The fractional derivatives are described based on the Caputo sense. Some properties of the sinc functions required for our subsequent development are given and are utilized to reduce the computation of solution of the Bagley-Torvik equati...

In this paper, we consider some boundary value problems (BVP) for fractional order partial differential equations ‎(FPDE)‎ with non-local boundary conditions. The solutions of these problems are presented as series solutions analytically via modified Mittag-Leffler functions. These functions have been modified by authors such that their derivatives are invariant with respect to fractional deriv...

BACKGROUND Topical vitamin C derivatives have been used to treat melasma and used as a skin whitener. The aim of this study was to compare skin histology and permeation of l-ascorbic acid 2-phosphate sesquimagnesium salt (MAP-1) and magnesium l-ascorbic acid-2-phosphate (MAP-2) after fractional CO₂ laser pretreatment. METHODS The effect of fractional laser treatment on porcine skin was examin...

We study dynamic minimization problems of the calculus of variations with Lagrangian functionals containing Riemann–Liouville fractional integrals, classical and Caputo fractional derivatives. Under assumptions of regularity, coercivity and convexity, we prove existence of solutions. AMS Subject Classifications: 26A33, 49J05.

This paper presents necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo.