Space-fractional Reaction Diffusion Equations Research Articles

Highly Efficient Numerical Algorithm for Nonlinear Space Variable-Order Fractional Reaction–Diffusion Models

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.

Modeling NO Biotransport in Brain Using a Space-Fractional Reaction-Diffusion Equation.

Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical processes in the body such as the regulation of cerebral blood flow and pressure. Infection and inflammatory processes such as those caused by COVID-19 produce a disequilibrium in the NO bioavailability and/or a delay in the interactions of NO with other molecules contributing to the onset and evolution of cardiocerebrovascular diseases. A link between the SARS-CoV-2 virus and NO is introduced. Recent experimental observations of intracellular transport of metabolites in the brain and the NO trapping inside endothelial microparticles (EMPs) suggest the possibility of anomalous diffusion of NO, which may be enhanced by disease processes. A novel space-fractional reaction-diffusion equation to model NO biotransport in the brain is further proposed. The model incorporates the production of NO by synthesis in neurons and by mechanotransduction in the endothelial cells, and the loss of NO due to its reaction with superoxide and interaction with hemoglobin. The anomalous diffusion is modeled using a generalized Fick’s law that involves spatial fractional order derivatives. The predictive ability of the proposed model is investigated through numerical simulations. The implications of the methodology for COVID-19 outlined in the section “Discussion” are purely exploratory.

Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions

To achieve the efficient and accurate long-time integration, we propose a fast and stable high-order numerical method for solving fractional-in-space reaction–diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the modified fourth-order exponential time differencing Runge–Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh–Nagumo, Allen–Cahn, Gierer–Meinhardt, Gray–Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.

Fast compact implicit integration factor method with non-uniform meshes for the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation

In this paper, we propose a fast compact implicit integration factor (FcIIF) method with non-uniform time meshes for solving the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation. The weighted and shifted Grüwald-Letnikov (WSGD) approximation is employed to the spatial discretization of the equation, and a system of nonlinear ordinary differential equations (ODEs) in matrix form is obtained. Since the cIIF method can provide excellent stability properties with good efficiency by decoupling the treatment of the diffusion and reaction terms, a fast cIIF (FcIIF) method with non-uniform time meshes is developed to solve the resultant nonlinear system of ODEs. Compared with the cIIF method, the proposed FcIIF method avoids the direct calculation of dense exponential matrices and requires less computational cost. The stability, accuracy and effectiveness of the proposed method are verified by the linear stability analysis and various numerical experiments.

A bounded numerical solver for a fractional FitzHugh–Nagumo equation and its high-performance implementation

In this paper, we propose a high-performance implementation of a space-fractional FitzHugh–Nagumo model. Our implementation is based on a positivity- and boundedness-preserving finite-difference model to approximate the solutions of a Riesz space-fractional reaction-diffusion equation. The model generalizes the FitzHugh–Nagumo model. The stability and convergence of the difference scheme are thoroughly discussed. Moreover, we prove the existence and uniqueness of numerical solutions, positivity, boundedness and consistency of the model. The scheme is based on weighted and shifted Grunwald differences. The conjugate gradient method is used then to solve the sparse matrix system. The MPI and PETSc libraries are used for the computational implementation. We investigate the influence of some computer factors on the performance of our implementation and scalability. More precisely, we consider the number of cores, the size of the computation mesh and the orders of the fractional derivatives. Tests are evaluated on a ccNUMA architecture with two CPUs.

Formula omitted]-Bounds of solutions of a system of strongly coupled space–time fractional evolution equations

We obtain L∞-Bounds for the solution of a system of time–space fractional reaction–diffusion equations modeling the geographical spread of an infectious disease or a rumor or an information within a large but finite population. The analysis is based on the application of the maximum principle to a judicious Lyapunov function. The large time behavior of the solution of the system is also discussed.

An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions

A split-step predictor-corrector method, based on Adams-Moulton formula, is presented for the solution of nonlinear distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions. The distributed-order space-fractional equation is first discretized to a multi-term space-fractional equation by applying a quadrature rule. Multi-term space-fractional equation is spatially discretized by matrix transfer technique and a linearly implicit split-step predictor-corrector method is applied for time-stepping to avoid solving nonlinear systems at each time step. The method is shown to be unconditionally stable and second order convergent. Numerical experiments are performed to confirm the stability and second order convergence of the method. The split-step predictor-corrector method is also compared with a second order IMEX predictor-corrector method. The IMEX predictor-corrector method is found to incur oscillatory behavior when implemented on problems with nonsmooth initial conditions or with mismatch between the initial and boundary conditions. Our method, which is an L-stable method, produces reliable and oscillation free results for such problems.

Numerical simulation of fractional-order reaction–diffusion equations with the Riesz and Caputo derivatives

The present paper deals with the numerical solution of space–time-fractional reaction–diffusion problems used to model some complex phenomena that are governed by dynamic of anomalous diffusion. The time- and space-fractional reaction–diffusion equation is modeled by replacing the first-order derivative in time and the second-order derivative in space, respectively, with the Caputo and Riesz operators. We propose an adaptable numerical scheme for the approximation of the derivatives. Accuracy of the method is justified by reporting both the maximum error and relative error in 2D with analytical solutions given. The effectiveness and applicability of the proposed methods are tested on two of practical problems that are of current and recurring interests in one and two dimensions to cover pitfalls that may arise.

Fourth-order time stepping methods with matrix transfer technique for space-fractional reaction-diffusion equations

In this paper, we develop two high-order methods for space-fractional reaction-diffusion equations. The schemes are based on fourth-order Matrix Transfer Technique (MTT) for spatial discretization and fourth-order Exponential Time Differencing Runge–Kutta (ETDRK) for temporal discretization. The ETDRK schemes are based on diagonal and sub-diagonal Padé approximations to the matrix exponential functions. It is observed that the A-stable scheme incurs unwanted oscillations due to high-frequency components present in the solution. These oscillations diminish as the order of the space-fractional derivative decreases. We propose a reliability constraint which is dependent on the order of the space-fractional derivative to avoid these oscillations. However, the L-stable scheme is oscillation-free for any time step. Partial fraction splitting technique is used to compose computationally efficient versions of the schemes. The amplification factor of the schemes is investigated by plotting their stability regions. Convergence analysis is performed on numerical experiments to demonstrate the fourth-order accuracy of the developed schemes in space and time. Numerical experiments made on multidimensional space-fractional reaction-diffusion equations show the reliability, stability, and efficiency of the schemes.

A Split-Step Predictor–Corrector Method for Space-Fractional Reaction–Diffusion Equations with Nonhomogeneous Boundary Conditions

A split-step second-order predictor–corrector method for space-fractional reaction–diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor–corrector method is also compared with an IMEX predictor–corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor–corrector method to avoid unwanted oscillations and show its validity numerically.

Fourier spectral exponential time differencing methods for multi-dimensional space-fractional reaction–diffusion equations

In this paper, we propose two numerical methods for solving the space-fractional reaction–diffusion equations, which are based on a Fourier spectral approach in space and exponential time differencing schemes in time. The advantages of the approaches are that they attain spectral convergence, produce a full diagonal representation of the fractional operator, and the extension to multiple spatial dimensions is the same as the one-dimensional space. That can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non-locality of the fractional operator, which results in full, dense matrices. Moreover, the proposed schemes are second-order convergent, unconditionally stable, and highly efficient due to the predictor–corrector feature when comparing them with the existing method. It is observed that the scheme based on using Padé(1,1) approximations to the matrix exponential function introduces oscillations with non-smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases. However, the scheme based on Padé(0,2) approximations to the matrix exponential function is oscillation-free for any time step. Numerical experiments for well-known models from the literature are performed to show the reliability and effectiveness of the proposed methods.

High-order time stepping Fourier spectral method for multi-dimensional space-fractional reaction–diffusion equations

This paper introduces a high-order time stepping scheme, that is based on using Fourier spectral in space and a fourth-order diagonal Padé approximation to the matrix exponential function for solving multi-dimensional space-fractional reaction–diffusion equations. The resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non-linear system at each time step while maintaining the stability of the scheme. The non-locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices. This scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator. It also attains spectral convergence for multiple spatial dimensions. The stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions, which provides an indication of the stability of the method. The convergence analysis is performed empirically to show that the scheme is fourth-order accurate in time, as expected. Numerical experiments on reaction–diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space-fractional reaction–diffusion equations and to validate the effectiveness of the scheme.

High-order solvers for space-fractional differential equations with Riesz derivative

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document} . The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time–space fractional reaction–diffusion equation

In this letter, we consider the numerical approximation of a two-dimensional distributed-order time–space fractional reaction–diffusion equation. The time- and space-fractional derivatives are considered in the senses of Caputo and Riesz, respectively. By using the composite mid-point quadrature, the original fractional problem is approximated by a multi-term time–space fractional differential equation. Then the multi-term Caputo fractional derivatives are discretized by the L2-1σ formula. We apply the Legendre–Galerkin spectral method for the spatial approximation. Two numerical experiments with smooth and non-smooth initial conditions, respectively, are performed to illustrate the robustness of the proposed method. The results show that: our scheme can arrive at the spectral accuracy (resp. algebraic accuracy) in space for the problem with smooth (resp. non-smooth) initial condition. For both of these two cases, our scheme can lead to the second-order accuracies in time. Additionally, the convergence rates in both spatial and temporal distributed-order variables are two.

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain $(-\infty, \infty)$ , which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.

A second-order efficient L-stable numerical method for space fractional reaction–diffusion equations

ABSTRACTIn this article we are proposing a strongly stable computationally efficient method in time to numerically solve reaction–diffusion equations with space fractional derivative. The Riesz space fractional derivative is discretized using the second-order fractional centred difference method. Time stepping scheme is based on second-order exponential time differencing Runge–Kutta method. Second-order positivity preserving Padé approximations is used to develop the proposed L-stable method. The computation efficiency of the method is significantly enhanced by using partial fractions splitting technique. The method is shown to be stable and reliable. Solution profiles as well as convergence tables in time are presented for various values of diffusion rates α.

Linearly implicit predictor–corrector methods for space-fractional reaction–diffusion equations with non-smooth initial data

In this paper, linearly implicit predictor–corrector methods are proposed for solving space-fractional reaction–diffusion equations with non-smooth initial data. The methods are based on Matrix Transfer Technique for spatial discretization and are shown to be unconditionally stable. It is observed that the linearly implicit predictor–corrector method derived by using (1,1)-Padé approximation to matrix exponential function incurs oscillatory behavior for some time steps. These oscillations are due to high frequency components present in the solution and are diminished as the order of the space-fractional derivative decreases (slow diffusion). We present a priori reliability constraint to avoid these unwanted oscillations and generalize the constraints for all (m,m)-Padé approximants, m∈Z+, to the matrix exponential functions. These time stepping constraints are seen to be dependent on the order of the space-fractional derivative. The linearly implicit predictor–corrector method based on the (0,2)-Padé approximations to the matrix exponential function is shown to be oscillation-free for any time step. Error estimates are obtained for the methods and are theoretically shown to be second-order convergent. Computational complexity of the algorithms is discussed for solving multidimensional space-fractional reaction–diffusion systems. Several numerical experiments are performed to support our theoretical observations and to show the effectiveness, reliability, and efficiency of the methods.

Lattice Boltzmann model for the Riesz space fractional reaction-diffusion

In this paper, a Riesz space fractional reaction-diffusion equation with non-linear source term is considered on a finite domain. This equation is commonly used to describe anomalous diffusion in thermal science. To solve the diffusion equation, a new fractional lattice Boltzmann method is proposed. Firstly, a difference approximation for the global spatial correlation of Riesz fractional derivative is derived by applying the numerical discretization technique, and a brief convergence analysis is presented. Then the global spatial correlation process is inserted into the evolution process of the standard lattice Boltzmann method. With combining Taylor expansion, Chapman-Enskog expansion and the multi-scales expansion, the governing evolution equation is recovered from the continuous Boltzmann equation. Three numerical examples are provided to confirm our theoretical analysis and illustrate the effectiveness of our method at last.