Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier ...

In this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0 , 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for α ∈ (1 , +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable sub...

In this paper the exact and the approximate solutions of fuzzy fractional differential equation, in the sense of Caputo Hukuhara differentiability, with a fuzzy condition are constructed by using the fuzzy Laplace transform. The obtained solutions are expressed in the form of the fuzzy Mittag-Leffler function. The presented procedure is visualized and the graphs of the obtained approximate solu...

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigat...

In this paper, by using the coincidence degree theory, we consider the following two-point boundary value problem for fractional differential equation { D 0+x(t) = f(t, x(t), x ′(t)), t ∈ [0, 1], x(0) = 0, x′(0) = x′(1), where D 0+ denotes the Caputo fractional differential operator of order α, 1 < α ≤ 2. A new result on the existence of solutions for above fractional boundary value problem is ...

In this paper, we give some properties of the left and right finite Caputo derivatives. Such derivatives lead to finite Riesz type fractional derivative, which could be considered as the fractional power of the Laplacian operator modelling the dynamics of many anomalous phenomena in super-diffusive processes. Finally, the exact solutions of certain fractional diffusion partial differential equa...

The practical stability of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov like function along the given fractional differential equation. Comparison results using this definition for scalar fractional differential equations are presented. Several ...

Correspondence: [email protected]. tr Department of Statistics, Gaziosmanpasa University, Tasliciftlik Campus, 60250 Tokat, Turkey Abstract This paper deals with an application of the method of quasilinearization by not demanding the Hölder continuity assumption of functions involved and by choosing upper and lower solutions with initial time difference for nonlinear Caputo fractional different...

With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous...

In this article, the fractional derivatives in the sense of Caputo, and the homotopy analysis method (HAM) are used to construct the approximate solutions for the nonlinear fractional variant Bussinesq equations with respect to time fractional derivative. The HAM contains a certain auxiliary h parameter which provides us a simple way to adjust and control the convergence region and rate of conv...