Partial Differential Equations Of Fractional Order Research Articles

Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.

New image denoising model based on fractional-order partial differential equation

Combining fractional order differential theory with total variation method,a new image denoising model was proposed,which was based on fractional Partial Differential Equation(PDE).Current Total Variation(TV) model was good at denoising and keeping the characteristics of image edge.This new model effectively inherited these advantages;at the same time,it used the fractional differential amplitude-frequency and retained texture details of smooth region effectively.The simulation results show that on one hand,compared with the existing denoising methods,the new model can not only suppress noise better,but also keep the characteristics of image edge better;especially,it is better than the existing integer order partial differential methods since it can retain more texture details.On the other hand,seen from the comparative experiments of PSNR,the model is more effective and practical on image denoising.

Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay

A class of nonlinear fractional order partial differential equations with delay c ∂ α u ( x , t ) ∂ t α = a ( t ) △ u ( x , t ) + f ( t , u ( x , τ 1 ( t ) ) , … , u ( x , τ l ( t ) ) ) , t ∈ [ 0 , T 0 ] be investigated in this paper, where c D α is the standard Caputo’s fractional derivative of order 0 ≤ α ≤ 1 , and l is a positive integer number, the function f is defined as f ( t , u 1 , … , u l ) : R × R × ⋯ , × R → R , and x ∈ Ω is a M dimension space. Using Lebesgue dominated convergence theorem, Leray–Schauder fixed point theorem and Banach contraction mapping theorem, we obtain some sufficient conditions for the existence of the solutions of the above fractional order partial differential equations.

Fractional order partial hyperbolic functional differential equations with state-dependent delay

This paper deals with the existence and uniqueness of solutions of some classes of partial functional and neutral functional hyperbolic differential equations with state-dependent delay involving the Caputo fractional derivative. Our results will be obtained by using suitable fixed point theorems.

Global Uniqueness Results for Fractional Order Partial Hyperbolic Functional Differential Equations

We investigate the global existence and uniqueness of solutions for some classes of partial hyperbolic differential equations involving the Caputo fractional derivative with finite and infinite delays. The existence results are obtained by applying some suitable fixed point theorems.

Regularity of a fractional partial differential equation driven by space-time white noise

We will deal with one dimensional stochastic fractional order partial differential equation driven by space-time white noise. The existence and uniqueness of the solution and especially some regularities of the solution are investigated. The regularities of the solution in its time and space variables depend on the relation of the fractional order of its operator and coefficients.

Fractional Green's function for fractional partial differential equations

Fractional order partial differential equations, as generalization of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In this paper we derive an explicit representation of the fractional Green's function for a class of fractional partial differential equations, then we present their solutions by integrals in terms of fractional Green's function. The solution of the fractional advection-dispersion equation is given as special case. The new approach introduces a promising tool for solving fractional partial differential equations.

Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods

Fractional order partial differential equations, as generalization of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of applications. In this paper we present a collection of numerical algorithms for the solution of nonlinear partial differential equations with space- and time-fractional derivatives. The fractional derivatives are considered in the Caputo sense. Two numerical examples are given to demonstrate the effectiveness of the present methods. Results show that the numerical schemes are very effective and convenient for solving nonlinear partial differential equations of fractional order.

Applications of variational iteration and homotopy perturbation methods to fractional evolution equations

In this paper, variational iteration and homotopy perturbation methods that developed for integer-order differential equations are directly extended to derive explicit and numerical solutions of various evolution equations with time-fractional derivatives. The results reveal that the two methods are very effective and convenient for solving nonlinear partial differential equations of fractional order.

Vector Additive Decomposition for 2D Fractional Diffusion Equation

Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients. In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.

Finite difference approximations for two-sided space-fractional partial differential equations

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and ...

Finite difference methods for two-dimensional fractional dispersion equation

Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In this paper we discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. First-order consistency, unconditional stability, and (therefore) first-order convergence of the method are proven using a novel shifted version of the classical Grünwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.

Finite difference approximations for two-sided space-fractional partial differential equations

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. We examine the case when a left-handed or a right-handed fractional spatial derivative may be present in the partial differential equation. Stability, consistency, and (therefore) convergence of the methods are discussed. The stability (and convergence) results in the fractional PDE unify the corresponding results for the classical parabolic and hyperbolic cases into a single condition. A numerical example using a finite difference method for a two-sided fractional PDE is also presented and compared with the exact analytical solution.

Anomalous diffusion and fractional stable distributions

A new set of distributions, called fractional stable distributions, is described. A subset of this set is represented by stable laws, while its particular case is the Gaussian distribution. These distributions arise as solutions to fractional-order partial differential equations that represent a generalization of the ordinary diffusion equation to the case of anomalous diffusion. The properties of multidimensional fractional stable densities are described, their expressions in terms of special functions are presented, and physical problems that lead to these densities are discussed.

On the Cauchy problem for some fractional order partial differential equations

In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D t α ∂u(x,t) ∂t −∑ |q|=2ma q(x,t)D x qu(x,t) =∑ |q|<2mb q(x,t)D x qu(x,t), where 0< α⩽1 and ∑ | q|=2 m a q ( x, t) D x q is a uniformly elliptic partial differential operator. The existence and uniqueness of the solution of the considered Cauchy problem is studied, also a continuity property is considered.