Space Fractional Differential Equations Research Articles

On the numerical solution to space fractional differential equations using meshless finite differences

We derive a discretization of the Caputo and Riemann–Liouville spatial derivatives by means of the meshless Generalized Finite Difference Method, which is based on moving least squares. The conditional convergence of the method is proved and several examples over one dimensional irregular meshes are given.

Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains

Recently, several numerical methods have been developed for solving time-fractional differential equations not only on rectangular computational domains but also on convex and non-convex non-rectangular computational geometries. On the other hand, due to the existence of integrals in the definition of space-fractional operators, there are few numerical schemes for solving space-fractional differential equations on irregular regions. In this paper, we develop a novel numerical solution based on the machine learning technique and a generalized moving least squares approximation for two-dimensional fractional PDEs on irregular domains. The scheme is constructed on the monomials, and this is the strength of this technique. Moreover, it will be used to approximate the space derivatives on convex and non-convex non-rectangular computational domains. The numerical results are extended to solve the fractional Bloch–Torrey equation, fractional Gray–Scott equation, and fractional Fitzhugh–Nagumo equation.

Two Linearized Schemes for One-Dimensional Time and Space Fractional Differential Equations

This paper investigates the solution to one-dimensional fractional differential equations with two types of fractional derivative operators of orders in the range of (1,2). Two linearized schemes of the numerical method are constructed. The considered FDEs are equivalently transformed by the Riemann–Liouville integral into their integro-partial differential problems to reduce the requirement for smoothness in time. The analysis of stability and convergence is rigorously discussed. Finally, numerical experiments are described, which confirm the obtained theoretical analysis.

On nonexistence of solutions to some time space fractional evolution equations with transformed space argument

Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a [Formula: see text] system of equations with transformed space arguments.

SOLUTIONS FOR THE FRACTIONAL MATHEMATICAL MODELS OF DIFFUSION PROCESS

In this research we present two new approaches with Laplace transformation to form the truncated solution of space-time fractional differential equations (STFDE) with mixed boundary conditions. Since order of the fractional derivative of time derivative is taken between zero and one we have a sub-diffusive differential equation. First, we reduce STFDE into either a time or a space fractional differential equation which are easier to deal with. At the second step the Laplace transformation is applied to the reduced problem to obtain truncated solution. At the final step using the inverse transformations, we get the truncated solution of the problem we consider it. Presented examples illustrate the applicability and power of the approaches, used in this study.

An Artificial Neural Network Approach for Solving Space Fractional Differential Equations

The linear algebraic system generated by the discretization of fractional differential equations has asymmetry, and the numerical solution of this kind of problems is more complex than that of symmetric problems due to the nonlocality of fractional order operators. In this paper, we propose the artificial neural network (ANN) algorithm to approximate the solutions of the fractional differential equations (FDEs). First, we apply truncated series expansion terms to replace unknown function in equations, then we use the neural network to get series coefficients, and the obtained series solution can make the norm value of loss function reach a satisfactory error. In the part of numerical experiments, the results verify that the proposed ANN algorithm can make the numerical results achieve high accuracy and good stability.

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton–Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one- and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

Series Solutions of High-Dimensional Fractional Differential Equations

In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

Radial point interpolation collocation method based approximation for 2D fractional equation models

In this work, we studied the radial point interpolation collocation method (RIPCM) for the solution of the partial differential models with fractional derivatives. In the present meshless method, polynomial basis function and two novel types local support fields were considered. The (time-)space fractional differential equations with single-term or multi-term time derivatives in irregular domain were well simulated numerically by the proposed RIPCM method. Numerical tests were carried out to verify the accuracy and demonstrate the strength of the RIPCM in solving fractional derivative models in irregular domain.

Formulation of space-angular fractional radiative transfer equation in participating finite slab clumpy media

Space-angular fractional radiative transfer equation has been derived depending on the invariance of the functional potential and the Euler–Lagrange functional of the radiative-transfer intensity. Pomraning–Eddington technique is employed to unzip the space-fractional radiative transfer through a clumpy medium. The medium is considered as an anisotropic scattering participating finite slab with specular reflecting boundaries. The fractional differential operator is taken in terms of the Jumarie Riemann–Liouville representation. The resultant space-fractional equation has diffusion parameter depends on both the single scattering albedo and the anisotropic parameter of the medium. Laplace transformation method is used to solve the obtained space-fractional differential equation, whose solution is obtained in terms of the Mittag–Leffler function. Numerical calculations are carried out to investigate the effects of the fractional exponent and the other radiative-transfer parameters on the radiation density and net flux through the medium. In addition, the reflectivity and transmissivity from the medium boundaries are calculated and represented graphically.

A novel approach for the solution of fractional diffusion problems with conformable derivative

AbstractThe truncated solution of space–time fractional differential equations, including conformable derivative is constructed by the help of residual power series method (RPSM). At the first step the space–time fractional differential equations are transformed into space fractional differential equations or time fractional differential equations by means of a specific transformation. Then the solutions are obtained in the form of fractional power series by semi‐analytical method RSPM. Finally, the illustrative examples show that this method is applicable, for constructing the truncated analytic solutions for space–time fractional differential equations.

Sequential space fractional diffusion equation’s solutions via new inner product

In this study, we determine the analytic solutions of sequential space fractional differential equations with Dirichlet boundary conditions and initial conditions in one dimension. We constructed a Fourier series solution for the eigenfunctions of a corresponding Sturm–Liouville eigenvalue problem, including fractional derivative in Caputo sense using the separation of variables. We defined a new inner product with a weighted function to get coefficients in the Fourier series.

A Finite Difference Method for Space Fractional Differential Equations with Variable Diffusivity Coefficient

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

A New Algorithm of Residual Power Series (RPS) Technique

Approximate analytic solutions of time–space fractional heat and wave equations are described. A new algorithm of Residual power series technique is introduced to obtain approximate solutions of such problems. The solution was obtained without reducing fractional differential equations to time fractional or space fractional differential equations. Some interesting results are presented to verify the efficiency and reliability of the developed algorithm.

A linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation: Numerical simulations of Gordon-type solitons

In this paper, we construct a novel linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation. By using Gauss–Legendre quadrature rule to discretize the distributed integral terms in both the spatial and temporal directions, we first approximate the original distributed-order fractional problem by the multi-term time–space fractional differential equation. Then, we employ the finite difference method for the discretization of the multi-term Caputo fractional derivatives and apply the Legendre–Galerkin spectral method for the spatial approximation. The main advantage of the proposed scheme is that the implementation of the iterative method is avoided for the nonlinear term in the fractional problem. Additionally, numerical experiments are conducted to validate the accuracy and stability of the scheme. Our approach is show-cased by solving several three-dimensional Gordon-type models of practical interest, including the fractional versions of sine-, sinh-, and Klein–Gordon equations, together with the numerical simulations of the collisions of the Gordon-type solitons. The simulation results can provide a deeper understanding of the complicated nonlinear behaviors of the 3D Gordon-type solitons.

RETRACTED: Stochastic simulation for time and space fractional differential equations

In this paper, we aim to study the stochastic simulation for time and space fractional differential equations. First, we prove that the stochastic solution of the time and space fractional differential equation is a stable subordinated process driven by the stable Lévy motion. Then, the absorbent term is employed for this equation; we find the corresponding parent process yields to be driven by a tempered stable process. At last, we design an algorithm to simulate the trajectory of the proposed process. The Monte Carlo methods are also employed to get the approximated solution of the fractional differential equations. The contribution of this paper is to establish the relation between the time and space fractional differential equations and stochastic process, and provide the stochastic simulation algorithm for this fractional differential equations.

A New Approach for the Approximate Analytical Solution of Space-Time Fractional Differential Equations by the Homotopy Analysis Method

The motivation of this study is to construct the truncated solution of space-time fractional differential equations by the homotopy analysis method (HAM). The first space-time fractional differential equation is transformed into a space fractional differential equation or a time fractional differential equation before the HAM. Then the power series solution is constructed by the HAM. Finally, taking the illustrative examples into consideration we reach the conclusion that the HAM is applicable and powerful technique to construct the solution of space-time fractional differential equations.

Machine Learning of Space-Fractional Differential Equations

Data-driven discovery of “hidden physics''---i.e., machine learning of differential equation models underlying observed data---has recently been approached by embedding the discovery problem into a...

Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACTWe propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.