Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β ₁∈(0,1) and β ₂∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.     

A novel attempt for finding comparatively accurate solution for sine‐Gordon equation comprising Riesz space fractional derivative

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine‐Gordon equation (SGE). Two‐dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β(0,1) and of order α(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

A new numerical approach for single rational soliton solution of Chen‐Lee‐Liu equation with Riesz fractional derivative in optical fibers

In this paper, time‐splitting spectral approximation technique has been proposed for Chen‐Lee‐Liu (CLL) equation involving Riesz fractional derivative. The proposed numerical technique is efficient, unconditionally stable, and of second‐order accuracy in time and of spectral accuracy in space. Moreover, it conserves the total density in the discretized level. In order to examine the results, with the aid of weighted shifted Grünwald‐Letnikov formula for approximating Riesz fractional derivative, Crank‐Nicolson weighted and shifted Grünwald difference (CN‐WSGD) method has been applied for Riesz fractional CLL equation. The comparison of results reveals that the proposed time‐splitting spectral method is very effective and simple for obtaining single soliton numerical solution of Riesz fractional CLL equation.     

Analytical approximations for a population growth model with fractional order

In this paper, we apply the homotopy analysis method (HAM) to solve the fractional Volterra’s model for population growth of a species in a closed system. This technique is extended to give solutions for nonlinear fractional integro–differential equations. The whole HAM solution procedure for nonlinear fractional differential equations is established. Further, the accurate analytical approximations are obtained for the first time, which are valid and convergent for all time t. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional integro–differential equations.     

Approximate analytical solutions of Schnakenberg systems by homotopy analysis method

In this paper, the homotopy analysis method (HAM) has been employed to obtain analytical solution of a two reaction–diffusion systems of fractional order (fractional Schnakenberg systems) which has been modeling morphogen systems in developmental biology. Different from all other analytic methods, HAM provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. The fractional derivative is described in the Caputo sense. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional system are of a more general nature. Respectively, solutions of FOD spread at a faster rate than the classical differential equations, and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.     

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β ∈ (0, 1) and 2α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax-Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Basis property of eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equation. We show that these eigenfunctions form a Riesz basis in the elliptic part of the domain. In addition, we prove the Riesz basis property on [0, π/2] of the system of cosines occurring in the expressions for the eigenfunctions. Earlier, the Riesz basis property was proved for the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with continuous solution gradient.     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

Riesz potentials for Korteweg-de Vries solitons

Riesz potentials (also called Riesz fractional derivatives) are defined as fractional powers of Laplacian. They are traditionally used for studying existence and uniqueness for equations of the Korteweg-de Vries type (KdV-type henceforth). Zero mean properties are established for Riesz potentials of solutions of KdV-type equations, ${D_{x}^{\alpha}u(x,t),\, {\rm for}\, \alpha\in(0,3/2)}$ . As an important example Riesz fractional derivatives and their Hilbert transforms are computed for the well-known soliton solution of KdV. Obtained representations involve the Hurwitz Zeta function. Zero mean properties are established and asymptotic expansions are derived. A particular case of the obtained formula provides an algebraic soliton solution for extended KdV.     

Riesz potentials for Korteweg-de Vries solitons

Riesz potentials (also called Riesz fractional derivatives) are defined as fractional powers of Laplacian. They are traditionally used for studying existence and uniqueness for equations of the Korteweg-de Vries type (KdV-type henceforth). Zero mean properties are established for Riesz potentials of solutions of KdV-type equations, D_x^au(x,t), for a ? (0,3/2){D_{x}^{\alpha}u(x,t),\, {\rm for}\, \alpha\in(0,3/2)}. As an important example Riesz fractional derivatives and their Hilbert transforms are computed for the well-known soliton solution of KdV. Obtained representations involve the Hurwitz Zeta function. Zero mean properties are established and asymptotic expansions are derived. A particular case of the obtained formula provides an algebraic soliton solution for extended KdV.     

Global existence and asymptotic dynamics in a 3D fractional chemotaxis system with singular sensitivity

In this paper, we investigate the global existence and asymptotic dynamics of solutions to a fractional singular chemotaxis system in three dimensional whole space. We deal with the new difficulties arising from fractional diffusion by using Riesz transform and Kato-Ponce’s commutator estimates appropriately, and establish the local existence of solution. Then with the help of combining the local existence and the a priori estimates, the global existence and uniqueness of solution with small initial data is derived. Moreover, we obtain the asymptotic decay rates of solution by the method of energy estimates.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model

In this paper, the q -homotopy analysis transform method (q -HATM) is applied to find the solution for the fractional Lakshmanan-Porsezian-Daniel (LPD) model. The LPD model is the generalization of the non-linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q -homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non-linear fractional differential equations exist in the connected areas of science and engineering.