The minimax approach for a class of variable order fractional differential equation

This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1 . The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.     

Tempered fractional differential equation: variational approach

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: (FD) where α∈(1/2,1), λ>0, are the left and right tempered fractional derivatives, is the fractional Sobolev spaces, and . Assuming that f satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.     

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

Instrumental variable estimation for a linear stochastic differential equation driven by a mixed fractional Brownian motion

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by mixed fractional Brownian motion.     

Oscillatory behavior of a fractional partial differential equation

In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

The decay rate for a fractional differential equation

We consider the fractional differential equation

     

On a universal differential equation for the analytic terms of C-superpositions on the real line

It is proved that every continuous function on the real line can be approximated uniformly (in the sense of a specific norm) by superpositions of analytic functions, which are solutions of a single universal differential equation. Every superposition is some function belonging to . This improves a former result of the author, from which the superpositions are known to be continuous.     

Numerical simulations for a variable order fractional cable equation

In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the RiemannLiouville and the Gr¨unwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.     

On a fractional sublinear elliptic equation with a variable coefficient

We study the existence and uniqueness of bounded weak solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the proof of uniqueness relies on uniqueness of weak solutions to an associated fractional porous medium equation with variable density.     

On variational methods to non-instantaneous impulsive fractional differential equation

In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax–Milgram Theorem.     

On the continuation of solutions of a fractional partial differential equation

We study the continuation of solutions of a fractional partial differential equation. We show that a solution can be uniquely continued into a domain that is uniquely determined by the boundary part supporting the initial conditions and outside which the continuation is no longer unique.     

Solving a nonlinear fractional differential equation using Chebyshev wavelets

Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.