We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and theMittagLeffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit ex...

Recently the study of fractional differential equations has attracted the interest of many researcher due to their applications in various fields of sciences and engineering. The applications include viscoelasticity, electrochemistry, control theory and electromagnetism and so on, see for example [1–4]. There are also interesting applications of fractional differential equations to economics, f...

In this paper we construct new random walks connected with fractional order differential equations. Namely, the governing equations corresponding to the constructed random walks are multi-term or distributed fractional order differential equations. Nowadays the connection between random walk and fractional order dynamics is well known, see, for instance [1, 17, 26, 33, 38]. A number of construc...

In this paper, some attractivity results for fractional functional differential equations are obtained by using the fixed point theorem. By constructing equivalent fractional integral equations, research on the attractivity of fractional functional and neutral differential equations is skillfully converted into a discussion about the existence of fixed points for equivalent fractional integral ...

substantiation of the averaging method for differential equations with maxima is presented. two theorems on substantiates for differential equations with maxima are established.

This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions ...

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (...

In this paper‎, ‎we study the existence of solutions for a‎ ‎ fractional boundary value problem‎. ‎By using critical point theory‎ ‎ and variational methods‎, ‎we give some new criteria to guarantee‎ ‎ that‎ ‎ the problems have at least one solution and infinitely many solutions.

Regular and singular perturbations of fractional ordinary differential equations (fODEs) are considered. This is likely the first attempt to describe these problems. Similarities and differences between these cases and the analogous ones for classical (integer-order) differential equations are pointed out. Examples, including the celebrated Bagley-Torvik equations are discussed. Asymptotic-nume...

2 Young’s integrals and stochastic differential equations driven by fractional Brownian motions 4 2.1 Young’s integral and basic estimates . . . . . . . . . . . . . . . . . . 4 2.2 Stochastic differential equations driven by a Hölder path . . . . . . . 7 2.3 Multidimensional extension . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Fractional calculus . . . . . . . . . . . . . . . . . . . ...