Space Fractional Partial Differential Equations Research Articles

An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray–Scott system

This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order α∈(1,2) in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form u=0 in R∖Ω. The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order α: from self-replication to standing waves and from travelling waves to self-replication.

Exact Solutions of Systems of Nonlinear Time-Space Fractional Partial Differential Equations Using an Iterative Method

Abstract Fractional partial differential equations are useful tools to describe transportation, anomalous, and non-Brownian diffusion. In the present paper, we propose the Daftardar-Gejji and Jafari method along with its error analysis for solving systems of nonlinear time–space fractional partial differential equations (PDEs). Moreover, we solve a variety of nontrivial time–space fractional systems of PDEs. The obtained solutions either occur in exact form or in the form of a series, which converges to a closed form. The proposed method is free from linearization and discretization and does not include any tedious calculations. Moreover, it is easily employable using the computer algebra system such as Mathematica, Maple, etc.

Physics-informed neural network algorithm for solving forward and inverse problems of variable-order space-fractional advection–diffusion equations

A new physics-informed neural network (PINN) algorithm is proposed to solve variable-order space-fractional partial differential equations (PDEs). For the forward problem, PINN algorithm based on a power series expansion is established to solve the space-fractional advection–diffusion equations with variable coefficients in one and two dimensions. The loss function is constructed by taking the coefficients in the power series expansion as the output of the network. The learning rate range is also analyzed to ensure that the error is reduced with respect to the training time. For the inverse problem, a novel dual-network architecture based on neural network parallelism is designed to estimate the order of variable fractional operator. One of the networks outputs the coefficients of the power series expansion, which is a function that depends only on time variables, and the other is fitted to the order of variable fractional operator, which is a function that is related to both space and time variables. Compared with the forward problem, we make more comparisons between the approximated solution and the exact solution at some interior points when constructing the loss function to improve the fitting ability of the inverse problem. The exact solutions are easy to find and can be readily obtained with a fine mesh. Several numerical examples are illustrated with graphs. These results confirm the effectiveness of our method in solving variable-order space-fractional partial differential equations.

A generalized midpoint-based boundary value method for unstable partial differential equations

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

Double-Preconditioning Techniques for Fractional Partial Differential Equation Solvers

This paper is devoted to the numerical computation of the solution to stationary space fractional Partial Differential Equations (PDEs). To this end, we derive some algorithms for solving fractional algebraic linear systems resulting from the discretization of these space fractional PDEs. The proposed approach is based on an efficient computation of Cauchy’s integrals allowing to estimate positive real powers of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of MA, hence allowing in some cases for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy’s integrals. Therefore, the strategy is independent of the way the Cauchy integrals are computed, although in this paper, for the sake of simplicity, we use standard quadrature rules. Based on this approach, we derive some solvers for stationary deterministic or stochastic space fractional Poisson equations which are tested in a series of numerical experiments.

Generalization of the Modified Bernstein Polynomials Method for Solving Fractional Coupled Equal Width Wave Equations

The objective of this research paper is to find the approximate solution to the fractional partial differential equations, which are difficult to find the exact solution, and therefore, by relying on the modified Bernstein polynomial, the study suggested generalizing the method to be available for both space and time fractional partial differential equations. In order to investigate the ability of the method to reach the approximate solution, the method was applied to the coupled of space-time-fractional of the equal width wave equations(FCEWE) using Caputo’s definition of the fractional derivative, the results of the application showed the ability of the method to find the approximate solution.

Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology

In this work, the solution of Riesz space fractional partial differential equations of parabolic type is considered. Since fractional-in-space operators have been applied to model anomalous diffusion or dispersion problems in the area of mathematical physics with success, we are motivated in this paper to model the standard Brownian motion with the fractional order operator in the sense of the Riesz derivative. We formulate two viable, efficient and reliable high-order approximation schemes for the Riesz derivative which incorporated both the left- and right-hand sides of the Riemann-Liouville derivatives. The proposed methods are analyzed for both stability and convergence. Finally, the methods are used to explore the dynamic richness of pattern formation in two important fractional reaction-diffusion equations that are still of recurring interest. Experimental results for different values of the fractional parameters are reported.

Reference Tracking and Observer Design for Space Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media

This paper considers a class of space fractional partial differential equations (FPDEs) that describe gas pressures in fractured media. First, the well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the considered FPDEs are investigated. Then, the reference tracking problem is studied to track the pressure gradient at a downstream location of a channel. This requires manipulation of gas pressure at the downstream location and the use of pressure measurements at an upstream location. To achiever this, the backstepping approach is adapted to the space FPDEs. The key challenge in this adaptation is the non-applicability of the Lyapunov theory which is typically used to prove the stability of the target system as, the obtained target system is fractional in space. In addition, a backstepping adaptive observer is designed to jointly estimate both the system's state and the disturbance. The stability of the closed loop (reference tracking controller/observer) is also investigated. Finally, numerical simulations are given to evaluate the efficiency of the proposed method.

Investigation to analytic solutions of modified conformable time–space fractional mixed partial differential equations

In this paper, the invariant subspace method (ISM) is developed to obtain the exact solution of linear and nonlinear time and space fractional mixed partial differential equations involving modified conformable fractional derivative (MCFD). Moreover, the method is successfully extended to the coupled system of modified conformable fractional differential equations. Variety of time-space fractional mixed PDEs are considered and solved to illustrate the established result of ISM. Finally, a comparison between exact solution of fractional PDE in the sense of MCFD and Caputo fractional derivative (Ca-FD) is presented through some examples of time–space fractional mixed partial differential equations.

Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations

<abstract><p>In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). The proposed scheme uses Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. Besides, the proposed GFPDEs represent a great generalization of significant types of fractional partial differential equations (FPDEs) and their applications, which contain many previous reports as a special case. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of differential equations which can be solved numerically. In addition, the classical fourth-order Runge-Kutta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple, easy to implement and effective numerical method.</p></abstract>

Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations

<abstract><p>In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.</p></abstract>

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

Numerical solution of a class of space fractional nonlinear vibration equations with periodic boundary conditions by the Fourier spectral method

Nonlinear vibration arises everywhere in engineering. So far there is no method to track the exact trajectory of a space fractional nonlinear oscillator; therefore, a sophisticated numerical method is much needed to elucidate its basic properties. For this purpose, a numerical method that combines the Fourier spectral method with the Runge–Kutta method is proposed. Its accuracy and efficiency have been demonstrated numerically. This approach has full physical understanding and numerical access; thus, it can be used to solve many types of nonlinear space fractional partial differential equations with periodic boundary conditions.

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.

A space-time finite element method for solving linear Riesz space fractional partial differential equations

In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C0-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

A GEOMETRICALLY CONVERGENT PSEUDO–SPECTRAL METHOD FOR MULTI–DIMENSIONAL TWO–SIDED SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

<abstract> In this study, we present a geometrically convergent numerical method for solving multi-dimensional two-sided space-time fractional differential equations. The approach represents the solutions of the differential equations in terms of the shifted Chebyshev polynomials. The expansions are evaluated at the shifted Chebyshev-Gauss-Lobatto nodes. We present the approximations for left-sided integration, and the left- and right- sided differentiation. The performance of the method is demonstrated using some two-sided space fractional partial differential equations in one and two dimensions. The numerical results obtained show that the method is accurate and computationally efficient. A theoretical analysis of the convergence of the method is presented, where it is shown that, given a continuously differentiable solution, the numerical solution converges for a sufficiently large number of grid points. </abstract>

Numerical solution of the multi-term variable-order space fractional nonlinear partial differential equations

Numerical solution of the multi-term variable-order space fractional nonlinear 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.

A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation

Purpose This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc. Design/methodology/approach Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients. Findings A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed. Originality/value Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.

Pricing of Margin Call Stock Loan Based on the FMLS

In common stock loan, lenders face the risk that their loans will not be repaid if the stock price falls below loan, which limits the issuance and circulation of stock loans. The empirical test suggests that the log-return series of stock price in the US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the model of the margin call stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of the margin call stock loan can be described by a space-fractional partial differential equation with a time-varying free boundary condition. We transform the free boundary problem to a linear complementarity problem, and the fully-implicit finite difference method that we used is unconditionally stable in both the integer and fractional order. The numerical experiments are carried out to demonstrate differences of the margin call stock loan model under the FMLS and the standard normal distribution. Last, we analyze the impact of key parameters in our model on the margin call stock loan evaluation and give some reasonable explanation.