The theory of derivative of noninteger order goes back to Leibniz, Liouville, Riemann, Grunwald and Letnikov. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics, medicine.Classes of fractional differentiable systems have studied in [10], [4]. In the first section the fractional tangent bundle to a differentiable manifold is defined, u...

This article is devoted to the study of existence and multiplicity of positive solutions to a class of nonlinear fractional order multi-point boundary value problems of the type−Dq0+u(t) = f(t, u(t)), 1 < q ≤ 2, 0 < t < 1,u(0) = 0, u(1) =m−2∑ i=1δiu(ηi),where Dq0+ represents standard Riemann-Liouville fractional derivative, δi, ηi ∈ (0, 1) withm−2∑i=1δiηi q−1 < 1, and f : [0, 1] × [0, ∞) → [0, ...

In this study, two lagged fractional order singular neutral differential equations are considered. Using the advantage of association property Riemann -Liouville derivative, derivative appropriate Lyapunov function is calculated. Then, with help LMI, sufficient conditions for asymptotic stability zero solutions obtained.

An approximate analytical solution of transient diffusion equation with space-fractional Riemann–Liouville fractional derivative has been developed. The integral-balance method and an assumed parabolic profile with undefined exponent have been used. The spatial correlation the superdiffusion coefficient in potential power-law form has been discussed. The laws of the spatial and temporal propaga...

Abstract: In this paper, we investigate the existence of solutions for advanced fractional differential equations containing the right-handed Riemann-Liouville fractional derivative both with nonlinear boundary conditions and also with initial conditions given at the end point T of interval [0,T ]. We use both the method of successive approximations, the Banach fixed point theorem and the monot...

We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problemD0 u t μa t f t, u t , 0 < t < 1, u 0 u 1 u ′ 0 u′ 1 0, where μ > 0, a, and f are continuous, α ∈ 3, 4 is a real number, and D0 is Riemann-Liouville fractional derivative. Throughout our nonlinearity may be singular in its dependent variable. Two examples are also given to ill...

where 2 < a, b ≤ 3, 0 < ξ1 <... < ξm <1, 0 < h1 <... < hm <1, 0 <g1 <... < gm <1, 0 < δ1 <... < δm <1, ai, bi, cj, dj Î R, f, g : [0, 1] × R 3 ® R, f, g satisfies Carathéodory conditions, Dα0+ and I α 0+ are the standard Riemann-Liouville fractional derivative and fractional integral, respectively. Wang et al. Advances in Difference Equations 2011, 2011:44 http://www.advancesindifferenceequatio...

this article is devoted to the study of existence and multiplicity of positive solutions to aclass of nonlinear fractional order multi-point boundary value problems of the type−dq0+u(t) = f(t, u(t)), 1 < q ≤ 2, 0 < t < 1,u(0) = 0, u(1) =m−2∑ i=1δiu(ηi),where dq0+ represents standard riemann-liouville fractional derivative, δi, ηi ∈ (0, 1) withm−2∑i=1δiηi q−1 < 1, and f : [0, 1] × [0, ∞) → [0, ∞...

In this paper, we investigate the existence of three positive solutions for the nonlinear fractional boundary value problem Dα0+u(t) + a(t) f (t, u(t), u (t)) = 0, 0 < t < 1, 3 < α ≤ 4, u(0) = u(0) = u(0) = u(1) = 0, where Dα0+ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in t...

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in thei...