A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative

A one dimensional fractional diffusion model with the Riemann–Liouville fractional derivative is studied. First, a second order discretization for this derivative is presented and then an unconditionally stable weighted average finite difference method is derived. The stability of this scheme is established by von Neumann analysis. Some numerical results are shown, which demonstrate the efficiency and convergence of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method.     

A new fractional analytical approach via a modified Riemann–Liouville derivative

This work suggests a new analytical technique called the fractional homotopy perturbation method (FHPM) for solving fractional differential equations of any fractional order. This method is based on He’s homotopy perturbation method and the modified Riemann–Liouville derivative. The fractional differential equations are described in Jumarie’s sense. The results from introducing a modified Riemann–Liouville derivative in the cases studied show the high accuracy, simplicity and efficiency of the approach.     

Cauchy’s integral formula via the modified Riemann–Liouville derivative for analytic functions of fractional order

The modified Riemann–Liouville fractional derivative applies to functions which are fractional differentiable but not differentiable, in such a manner that they cannot be analyzed by means of the Djrbashian fractional derivative. It provides a fractional Taylor’s series for functions which are infinitely fractional differentiable, and this result suggests introducing a definition of analytic functions of fractional order. Cauchy’s conditions for fractional differentiability in the complex plane and Cauchy’s integral formula are derived for these kinds of functions.     

Application of homotopy analysis method for fractional Swift Hohenberg equation – Revisited

In this article, the homotopy analysis method is used to obtain the approximate analytical solutions of the non-linear Swift Hohenberg equation with fractional time derivative. The fractional derivative is described in Caputo sense. Numerical results reveal that the method is easy to implement, reliable and accurate when applied to time fractional nonlinear partial differential equations. Effects of parameters of physical importance on the probability density function and the convergence of the approximate series solution using residual error formula with the proper choices of auxiliary parameter for various fractional Brownian motions and standard motion are depicted through graphs and tables for different particular cases.     

Fractional master equation: non-standard analysis and Liouville–Riemann derivative

Fractional master equations may be defined either by means of Liouville–Riemann (L–R) fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present papers put in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.     

A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative

In this work, by means of the fixed point theorem in a cone, we establish the existence result for a positive solution to a kind of boundary value problem for a nonlinear differential equation with a Riemann–Liouville fractional order derivative. An example illustrating our main result is given. Our results extend previous work in the area of boundary value problems of nonlinear fractional differential equations [C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055].     

Positive solutions for a boundary-value problem with Riemann–Liouville fractional derivative*

In this work, we are mainly concerned with the existence of positive solutions for the fractional boundary-value problem $$ \left\{ {\begin{array}{*{20}{c}} {D_{0+}^{\alpha }D_{0+}^{\alpha }u=f\left( {t,u,{u}^{\prime},-D_{0+}^{\alpha }u} \right),\quad t\in \left[ {0,1} \right],} \hfill \\ {u(0)={u}^{\prime}(0)={u}^{\prime}(1)=D_{0+}^{\alpha }u(0)=D_{0+}^{{\alpha +1}}u(0)=D_{0+}^{{\alpha +1}}u(1)=0.} \hfill \\ \end{array}} \right. $$ Here ?? ?? (2, 3] is a real number, $ D_{0+}^{\alpha } $ is the standard Riemann?CLiouville fractional derivative of order ??. By virtue of some inequalities associated with the fractional Green function for the above problem, without the assumption of the nonnegativity of f, we utilize the Krasnoselskii?CZabreiko fixed-point theorem to establish our main results. The interesting point lies in the fact that the nonlinear term is allowed to depend on u, u??, and $ -D_{0+}^{\alpha } $ .     

First boundary-value problem in the half-strip for a parabolic-type equation with bessel operator and Riemann–Liouville derivative

The first boundary-value problem in the half-strip for a parabolic-type equation with Bessel operator and Riemann–Liouville derivative is studied. In the case of the zero initial condition, the representation of the solution in terms of the Fox H-function is obtained. The uniqueness of the solution for a class of functions vanishing at infinity is proved. It is shown that when the equation under consideration coincides with the Fourier equation, the obtained representation of the solution becomes the known representation of the solution of the corresponding problem.     

On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.     

A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional

The Swift–Hohenberg equation is a central nonlinear model in modern physics. Originally derived to describe the onset and evolution of roll patterns in Rayleigh–Bénard convection, it has also been applied to study a variety of complex fluids and biological materials, including neural tissues. The Swift–Hohenberg equation may be derived from a Lyapunov functional using a variational argument. Here, we introduce a new fully-discrete algorithm for the Swift–Hohenberg equation which inherits the nonlinear stability property of the continuum equation irrespectively of the time step. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and stability of our new algorithm. We also compare our method to other existing schemes, showing that is feasible alternative to the available methods.     

Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions

Theoretical and Mathematical Physics - We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In...     

Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann–Liouvile definition of fractional derivatives, one (Jumarie) has proposed recently an alternative referred to as a modified Riemann–Liouville definition, which directly, provides a Taylor’s series of fractional order for non differentiable functions. This fractional derivative provides a fractional calculus parallel with the classical one, which applies to non-differentiable functions; and the present short article summarizes the main basic formulae so obtained.     

Adaptation of homotopy analysis method for the numeric–analytic solution of Chen system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy analysis method (HAM) is presented, which is the multistage homotopy analysis method (MSHAM). The freedom of choosing the auxiliary linear operator and the auxiliary parameter are still present in the MSHAM. The solutions of the non-chaotic and the chaotic Chen system which is a three-dimensional system of ordinary differential equations with quadratic nonlinearities were obtained by MSHAM. Numerical comparisons between the MSHAM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for solving the non-linear chaotic and non-chaotic Chen system.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

Complex Grünwald–Letnikov,Liouville, Riemann–Liouville,and Caputo derivatives for analytic functions

The well-known Liouville, Riemann–Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald–Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann–Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann–Liouville and Caputo derivatives with the Grünwald–Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.     

The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian’s decomposition method (ADM) for solving time-fractional Fornberg–Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented.     

The first integral method for modified Benjamin–Bona–Mahony equation

In this paper, we use the first integral method for analytic treatment of the modified Benjamin–Bona–Mahony equation. Some exact new solutions are formally derived.