Space-time Fractional Partial Differential Equations Research Articles

Exact Solution of Space-Time Fractional Coupled EW and Coupled MEW Equations Using Modified Kudryashov Method

In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.

B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

A preconditioned Fast Finite Difference Method for Space-Time Fractional Partial Differential Equations

AbstractWe develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.

A divide-and-conquer fast finite difference method for space–time fractional partial differential equation

Fractional partial differential equations (FPDEs) provide better modeling capabilities for challenging phenomena with long-range time memory and spatial interaction than integer-order PDEs do. A conventional numerical discretization of space–time FPDEs requires O(N2+MN) memory and O(MN3+M2N) computational work, where N is the number of spatial freedoms per time step and M is the number of time steps.We develop a fast finite difference method (FDM) for space–time FPDE: (i) We utilize the Toeplitz-like structure of the coefficient matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver to invert the coefficient matrix at each time step. (ii) We utilize a divide-and-conquer strategy, a recursive direct solver, to handle the temporal coupling of the numerical scheme. The fast method has an optimal memory requirement of O(MN) and an approximately linear computational complexity of O(NM(logN+log2M)), without resorting to any lossy compression. Numerical experiments show the utility of the method.

Construction of exact solutions to the space–time fractional differential equations via new approach

In the present article, the local fractional derivatives and the exp(−Φ(ξ)) method are used to construct the exact solutions of nonlinear space–time fractional partial differential equations. For illustrating the validity of the method, it is applied to the space–time fractional (3+1)-dimensional nonlinear Jimbo–Miwa equation and nonlinear Hirota–Satsuma coupled KdV system. This approach is an efficient mathematical tool for solving fractional differential equations and it can be applied to other nonlinear fractional differential equations.

A new analytical method for seeking traveling wave solutions of space–time fractional partial differential equations arising in mathematical physics

In this paper, based a new fractional sub-equation and the properties of the modified Riemann–Liouville fractional derivative, we propose a new analytical method named improved fractional (DαG/G) method for seeking traveling wave solutions of space–time fractional partial differential equations, which can be seen as the fractional version of the improved (G′/G) method, and is the improvement of the common fractional (DαG/G) method. Due to given traveling wave transformation, one certain space–time fractional partial differential equation can be converted into another fractional ordinary differential equation with respect to one new variable. For demonstrating the validity of this method, we apply it to seek exact traveling wave solutions for the (2+1)-dimensional space–time fractional Nizhnik–Novikov–Veselov System and the space–time fractional KP-BBM equation. With the aid of the mathematical software, some new exact traveling wave solutions including solitary wave solutions, periodic wave solutions and solutions with other forms for them are successfully found.

Intermittency fronts for space-time fractional stochastic partial differential equations in [formula omitted] dimensions

We consider time fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0, β∈(0,1), α∈(0,2], d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, W⋅(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Mijena and Nane proved in (Mijena and Nane, 2016) that: (i) absolute moments of the solutions of this equation grow exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions α=2 and d=1. In this paper we extend this result to the case α=2 and d∈{1,2,3}.

Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method

In this paper, a new generalized fractional Jacobi elliptic equation-based sub-equation method is proposed to solve space-time fractional partial differential equations in mathematical physics. This method is applied to seek exact solutions for two space-time fractional partial differential equations: the space-time fractional KP-BBM equation and the space-time fractional KdV equation. With the aid of mathematical software, a variety of exact solutions for them in the forms of the Jacobi elliptic functions are obtained.

Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients

In this paper, Haar wavelet operational matrix(HWOM) is proposed to solve initial-boundary value problems for a class of time-space fractional partial differential equations of Caputo sense with variable coefficients in both time and space(1)∑i=1nθi∂γiu(x,t)∂tγi=v(x,t)∂αu(x,t)∂xα+d(x,t)∂βu(x,t)∂xβ+q(x,t),0<x<1,0<t≤1,as an extension of Rehman and Khan's (2013) work. We obtain a matrix L instead of Qα in Rehman and Khan (2013). when dealing with boundary conditions. By utilizing the operational matrix of fractional integration and Hadamard product,we made an improvement of algorithm to deal with time-space coefficients and gave the error analysis of the HWOM for space-time dimensions. Some numerical results are paralleled with exact solutions to show the efficiency and precision of the presented technique.

Fractional Integral Transform and Application

In this special issue we publish many articles of the highest quality. The aim of these papers is to highlight the importance of fractional operators of integral transform and their applications and let the readers of this journal know about the possibilities of this new tool. In particular, various potential topics are discussed: (i) mathematical analysis of fractional integral transform theoretical methods, (ii) new methods for solving Cauchy problem for the space-time fractional partial differential equation, (iii) applications of integral transform fractional, including fractional methods, (iv) applying fractional integral transform to solution of systems of differential equations and applications in physics, mechanics, and fractional Schrodinger equation in quantum theory, (v) applications of integral transform for special functions, for example, prefunction, and (vi) applications of integral transform to fractional convolution products. As editors of this special issue, the most difficult problem that we came across was to choose the best papers from the submitted high-quality works. All the papers published in this special issue are original and contain some attractive, resourceful, and recognizable ideas. Our motivation in choosing articles for publication in this special issue was whether the submitted papers activate and inspire further scientific activities in the scope of the special issue.

Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. An algebraic method is improved to construct uniformly a series of exact solutions for some nonlinear time-space fractional partial differential equations. We construct successfully a series of some exact solutions including the elliptic doubly periodic solutions with the aid of computerized symbolic computation software package such as Maple or Mathematica. This method is efficient and powerful in solving a wide classes of nonlinear partial fractional differential equations. The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus m→0 and m→1, respectively.

A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time–space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann–Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.

Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration

This paper presents mass-time fractional partial differential equations (fPDEs) formulated in a material coordinate for swelling–shrinking soils, and space-time fPDEs formulated in Cartesian coordinates for non-swelling soils. The fPDEs are capable of incorporating mobile and immobile zones or without immobile zones. As an example of the applications, the solutions of the fPDEs are derived and used to construct equations of infiltration. The new equation of cumulative infiltration into soils with mobile and immobile zones is I(t)=At+Stβ2+β1S1tβ2+S2tβ11/(2λ-1), where A is the final infiltration rate, S is the sorptivity which differs between swelling and non-swelling soils, β2 and β1 are orders of temporal fractional derivatives for mobile and immobile zones, respectively, λ is the order of spatial fractional derivatives, and S2 and S1 are parameters incorporating relative porosities and β2 and β1, respectively. The equation of cumulative infiltration without an immobile zone is I(t)=At+Stβ/(2λ-1), where β is the order of temporal fractional derivatives. Published data are used to demonstrate the use of the new equations and derive the parameters. The transport exponent for soils with mobile and immobile zones is μ=2(β2+β1)/λ, and μ=2β/λ for soils without an immobile zone. The transport exponent is the criteria for defining flow patterns: for μ < 1, the flow process is sub-diffusion as compared to μ = 1 for classic diffusion and μ > 1 for super-diffusion.

ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR SPACE-TIME FRACTIONAL CAHN-HILLIARD EQUATION

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Cahn-Hilliard equation. The numerical results show that the approaches are easy to implement and accurate when applied to the non- linear space-time fractional derivatives Cahn-Hilliard equation. This method intro- duces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equat. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.

Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations

The maximum principle for the space and time-space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time-space Riesz-Caputo fractional dif...

Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

A New Fractional Subequation Method and Its Applications for Space-Time Fractional Partial Differential Equations

A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this method.

Analytical approximate solution for nonlinear space—time fractional Klein—Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space—time fractional derivatives Klein—Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space—time fractional derivatives Klein—Gordon equation. This method introduces a promising tool for solving many space—time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.