Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

On nonlinear distributional and impulsive Cauchy problems

In this paper, existence results are derived for the unique, smallest, greatest, minimal and maximal solutions of nonlinear distributional Cauchy problems. Dependence of solutions on the data is also studied. The obtained results are applied to impulsive differential equations. Main tools are fixed point results in function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions.     

On accurate product integration rules for linear fractional differential equations

This paper addresses the numerical solution of linear fractional differential equations with a forcing term. Competitive and highly accurate Product Integration rules are derived by starting from an equivalent formulation in terms of a Volterra integral equation with a generalized Mittag-Leffler function in the kernel. The error analysis is reported and aspects related to the computational complexity are treated. Numerical tests confirming the theoretical findings are presented.     

Fractional integrodifferentiation in Hölder classes of arbitrary order

Hölder classes of variable order (x) are introduced and it is shown that the fractional integralI ₀₊ has Hölder order (x)+ (0 < , ₊, +₊ < 1, ₊ = sup (x)).     

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann-Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray-Schauder type.     

The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces

In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.     

The monotonic property and stability of solutions of fractional differential equations

In this paper we improve on the monotone property of Lemma 1.7.3 in Lakshmikantham et al. (2009) [5] for the case g(t,u)=λu with a nonnegative real number λ. We also investigate the Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.     

Cauchy problem for fractional diffusion equations

We consider an evolution equation with the regularized fractional derivative of an order α∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.     

Trotter products and reaction-diffusion equations

In this paper, we study a class of generalized diffusion-reaction equations of the form , where A is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference equation for numerical solutions.     

Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients

A class of weighted finite difference methods (WFDMs) for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients is presented. Their stability and convergence properties are considered. It is proven that the WFDMs are unconditionally-stable for , and conditionally-stable for , here r is the weighting parameter subjected to 0≤r≤1. Some convergence results are given. These methods, problems and results generalize the corresponding methods, problems and results given in [7], [8] and [10]. Some numerical examples are provided to show the effectiveness of the methods with different weighting parameters.     

Singular conformable sequential differential equations with distributional potentials

In this paper we consider the singular conformable sequential equation with distributional potentials. We present Weyl’s theory in the frame of conformable derivatives. Moreover we give two theorems on limit-point case.     

Explicit methods for fractional differential equations and their stability properties

The use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated. In this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed. Some numerical experiments are presented in order to validate theoretical results.     

On some explicit Adams multistep methods for fractional differential equations

In this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams–Moulton methods and they represent a way for extending classical Adams–Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k$k$-step methods, k=1,…,5$k=1,\dots ,5$, are computed and plots of stability regions in the complex plane are presented.     

A method for solving differential equations of fractional order

In this paper, we consider Caputo type fractional differential equations of order 0<α<1$0<\alpha <1$ with initial condition x(0)=_x0$x\left(0\right)=x_0$. We introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations. Generalization of the technique to finite systems is also given. Finally, we give some examples to illustrate the applications of our results.     

Nonnegativity of solutions of nonlinear fractional differential-algebraic equations

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.     

Controllability of nonlinear fractional dynamical systems

In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results.     

Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions

In this paper, the fractional differential transform method is developed to solve fractional integro-differential equations with nonlocal boundary conditions. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply.