Space-time Fractional Partial Differential Equations Research Articles

Maximum Principle for Space and Time-Space Fractional Partial Differential Equations

In this paper, new estimates of the sequential Caputo fractional derivatives of a function at its extremum points are obtained.We derive comparison principles for the linear fractional differential equations, then apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion admits at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Y. Luchko [Fract. Calc. Appl. Anal. 14 (2011)]. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.

An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative, and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains, and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.

Initial boundary value problems for space-time fractional conformable differential equation

<abstract> In this paper, the authors study a initial boundary value problems (IBVP) for space-time fractional conformable partial differential equation (PDE). Several inequalities of fractional conformable derivatives at extremum points are presented and proved. Based on these inequalities at extremum points, a new maximum principle for the space-time fractional conformable PDE is demonstrated. Moreover, the maximum principle is employed to prove a new comparison principle and estimation of solutions. Beside that, the uniqueness and continuous dependence of the solution of the space-time fractional conformable PDE are demonstrated. </abstract>

New types of soliton solutions for space-time fractional cubic nonlinear Schrodinger equation

New definitions for traveling wave transformation and using of new conformable fractional derivative for converting fractional nonlinear evolution equations into the ordinary differential equations are presented in this study. For this aim we consider the time and space fractional derivatives cubic nonlinear Schrodinger equation. Then by using of the efficient and powerful method the exact traveling wave solutions of this equation are obtained. The new definition introduces a promising tool for solving many space-time fractional partial differential equations.

Numerical algorithm to Caputo type time–space fractional partial differential equations with variable coefficients

This article assigns to a new numerical algorithm as an appropriate tool that deals with the time–space fractional partial differential equations in Caputo sense with variable coefficients. In the current algorithm, firstly, we presume that the approximate solution of the main problem is expandable along space variable via shifted Chebyshev polynomials with time-dependent coefficients. In the second step, we employ operational matrices of space-fractional derivatives to transform (reduce) the expanded problem to a system of time-fractional ordinary differential equations (FODEs) with initial value conditions. Indeed, the solutions of this system are required to obtain the time-dependent coefficients of the mentioned expansion. To solve this system, we define some independent secondary initial value problems and solve them analytically. At the final step, we find an optimal linear combination of this particular solutions to obtain an approximate solution of the main problem such that the residual error function forced to vanish in an average sense over the desired region, and the approximate solution satisfies in initial/boundary conditions of the main problem. The convergence property of the presented algorithm is demonstrated by the residual error analysis. The reliability and accuracy of the new algorithm are confirmed by solving some illustrative test problems. In order to perform a premier analysis with more details for the convergence property of the new algorithm, we compute the observed convergence order indicators in each test problem. Moreover, we evaluate our computed results with other numerical schemes in the literature to emphasize the promising performance of the proposed algorithm.

Solving Some Fractional Partial Differential Equations by Invariant Subspace and Double Sumudu Transform Methods

In this paper, several types of space-time fractional partial differential equations has been solved by using most of special double linear integral transform ”double Sumudu ”. Also, we are going to argue the truth of these solutions by another analytically method “invariant subspace method”. All results are illustrative numerically and graphically.

Invariant subspaces and exact solutions for some types of scalar and coupled time-space fractional diffusion equations

We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated through time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of the fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota-Satsuma KdV equation. Also, we explicitly presented how to derive more than one exact solution of the equations as mentioned above using the invariant subspace method.

Stable numerical results to a class of time-space fractional partial differential equations via spectral method

In this paper, we are concerned with finding numerical solutions to the class of time–space fractional partial differential equations:Dtpu(t,x)+κDxpu(t,x)+τu(t,x)=g(t,x),1<p<2,(t,x)∈[0,1]×[0,1],under the initial conditions.u(0,x)=θ(x),ut(0,x)=ϕ(x),and the mixed boundary conditions.u(t,0)=ux(t,0)=0,where Dtp is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t. Further, Dxp is the arbitrary derivative in Caputo sense with order p corresponding to the variable space x. Using shifted Jacobin polynomial basis and via some operational matrices of fractional order integration and differentiation, the considered problem is reduced to solve a system of linear equations. The used method doesn’t need discretization. A test problem is presented in order to validate the method. Moreover, it is shown by some numerical tests that the suggested method is stable with respect to a small perturbation of the source data g(t,x). Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well.

Construction of bright-dark solitary waves and elliptic function solutions of space-time fractional partial differential equations and their applications

In this article, several kinds of wave solutions such as solitons, solitary waves and Jacobi elliptic function solutions in which many are novel of nonlinear fractional partial differential equations (NLFPDEs) in mathematical physics, has obtained with the aid of improved extended auxiliary equation method. The efficiency of the current technique is demonstrated by applications to three NLFPDEs, namely, (2+1)-dimensional space-time fractional Zoomeron equation, space-time fractional Benjamin–Bona–Mahony and space-time fractional modified third-order KdV equations. Several kinds of exact solutions are constructed which have key applications in applied sciences. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The resulting numerous solutions will also be helpful in applied mechanics and various fields of sciences. This powerful technique can be applied to other wave models that can arise in mathematical physics.

Space-time fractional stochastic partial differential equations with Lévy noise

AbstractWe consider non-linear time-fractional stochastic heat type equation$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg] \end{array} $$and$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)\stackrel{\cdot}{N }(t,x,h)\bigg] \end{array} $$in (d + 1) dimensions, where α ∈ (0, 2] and d &lt; min{2, β−1}α, ν &gt; 0, $\begin{array}{} \partial^\beta_t \end{array} $ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, $\begin{array}{} I^{1-\beta}_t \end{array} $ is the fractional integral operator, N(t, x) are Poisson random measure with Ñ(t, x) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.

Legendre Collocation Method for the Nonlinear Space–Time Fractional Partial Differential Equations

In this study, nonlinear space–time fractional partial differential equations with variable coefficients are considered. The fractional derivatives are described in the conformable sense and Caputo sense. The numerical approach is based on the shifted Legendre polynomials and the finite difference method. Several illustrative examples are given to observe the validity, effectiveness and accuracy of the present numerical method.

The Traveling Wave Solutions of Space-Time Fractional Partial Differential Equations by Modified Kudryashov Method

In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.

Applications of Laplace-Adomian decomposition method for solving time-fractional advection dispersion equation

In this paper, a space-time fractional partial differential equation, obtained from the standard partial differential equation by replacing the second order space-derivative by a fractional derivative of order β > 0 and the first order time-derivative by a fractional derivative of order α > 0 has been recently treated by a number of authors. A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the first order derivative in time by a fractional derivative in time of order α (0 < α ≤ 1). In the present paper, the solution of the analytical dispersion equation is derived using Laplace-Adomian Decomposition Method (LADM). This method has higher convergences as the solutions both of fractional order and integral are obtained in the form of series. In this method the Caputo derivatives are used to define fractional order derivatives. To confirm validity of this method illustrative examples are given.

Polynomials for numerical solutions of space-time fractional differential equations (of the Fokker–Planck type)

Purpose Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type. Design/methodology/approach The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations. Findings Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique. Originality/value Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative

The computational methods based on operational matrices are promising tools to tackle the fractional order differential equations and they have gained increasing interest among the mathematical community. Herein, an efficient and precise computational algorithm based on a new kind of polynomials together with the collocation technique is presented for time-space fractional partial differential equations with the Riesz derivative. The method is proposed with the aid of a new operational matrix of the derivative using Chelyshkov polynomials (CPs) in the Caputo sense. The operational matrices of the derivative, exact and approximate, are derived via two different ways for integer and non-integer orders. The fractional problems under study have been converted into the corresponding nonlinear algebraic system of equations and solved by means of the collocation technique. The convergence and error bound are analyzed for the suggested computational method while a comparative study is included in our work to show the accuracy and efficiency of said method. The attained results confirm that the suggested technique is very accurate, efficient and reliable. As a suitable tool, it could be adopted to obtain the solutions for a class of the fractional order partial differential (linear and nonlinear) equations arising in engineering and applied sciences.

A New Approach for the Solution of Space-Time Fractional Order Heat-Like Partial Differential Equations by Residual Power Series Method

The main concern of this article has been to apply the Residual Power Series Method (RPSM) effectively to find the exact solutions of fractional-order space-time dependent nonhomogeneous partial differential equations in the Caputo sense. Our first step is to reduce fractional-order space-time dependent non-homogeneous partial differential equations to fractional-order space-time dependent homogeneous partial differential equations before applying the proposed method. Obtaining fractional power series solutions of the problem and reproducing the exact solution is the main step. The illustrative examples reveal that RPSM is a very significant and powerful method for obtaining the solution of any-order time-space fractional non-homogeneous partial differential equations in the form of fractional power series.

The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations

In this paper, we develop a generalized bifurcation method to study exact solutions of nonlinear space-time fractional partial differential equations (PDEs), which is based on the bifurcation theory of dynamical systems. We present the procedure of the method and illustrate it with application to the space-time fractional Drinfel’d–Sokolov–Wilson equation. We identify all bifurcation conditions and derive the phase portraits of the system, from which we obtain different new exact solutions, and more interestingly, we find the so-called M/W-shaped solitary wave solutions. The results demonstrate the efficiency of the method in deriving exact solutions of space-time fractional PDEs.

An advanced method with convergence analysis for solving space-time fractional partial differential equations with multi delays

This study considers the space-time fractional partial differential equations with multi delays under a unique formulation, proposing a numerical method involving advanced matrix system. This matrix system is made up of the matching polynomial of complete graph together with fractional Caputo and Jumarie derivative types. Also, the derivative types are scrutinized to determine which of them is more proper for the method. Convergence analysis of the method is established via an average value of residual function using double integrals. The obtained solutions are improved with the aid of a residual error estimation. A general computer program module, which contains few steps, is developed. Tables and figures prove the efficiency and simplicity of the method. Eventually, an algorithm is given to illustrate the basis of the method.

Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients

In this paper, the Lie group analysis method is applied to carry out the Lie point symmetries of some space–time fractional systems including coupled Burgers equations, Ito’s system, coupled Korteweg–de-Vries (KdV) equations, Hirota–Satsuma coupled KdV equations and coupled nonlinear Hirota equations with time-dependent variable coefficients with the Riemann–Liouville derivative. Symmetry reductions are constructed using Lie symmetries of the systems. To the best of our knowledge, nobody has so far derived the invariants of space–time nonlinear fractional partial differential equations with time-dependent coefficients.

Fast finite difference methods for space-time fractional partial differential equations in three space dimensions with nonlocal boundary conditions

Fractional partial differential equations (FPDEs) provide very competitive tools to model challenging phenomena involving anomalous diffusion or long-range memory and spatial interactions. However, numerical methods for space-time FPDEs generate dense stiffness matrices and involve numerical solutions at all the previous time steps, and so have O(N2+MN) memory requirement and O(MN3+M2N) computational complexity where N and M are the numbers of spatial unknowns and time steps, respectively.We develop and analyze a finite difference method (FDM) for space-time FPDEs in three space dimensions with a combination of Dirichlet and fractional Neumann boundary conditions, in which a shifted Grünwald discretization is used in space and an L1 discretization is used in time so the derived FDM has virtually first-order accuracy. We then derive different fast FDMs by carefully analyzing the structure of the stiffness matrix of numerical discretization as well as the coupling in the time direction. The resulting fast FDMs have an almost linear computational complexity and linear storage with respect to the number of spatial unknowns as well as time steps. Numerical experiments are presented to demonstrate the utility of the methods.