Table of Content

30 December 2019, Volume 1 Issue 4

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Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models

Qiang Du, Jan S. Hesthaven, Changpin Li, Chi, Wang Shu, Tao Tang

2019, 1(4):  503-504.  doi:10.1007/s42967-019-00045-6

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Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

Yanyong Wang, Yubin Yan, Ye Hu

2019, 1(4):  505-523.  doi:10.1007/s42967-019-00036-7

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We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-α), where h is the space step size and α∈(1, 2) is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders O(τ+h^3-α+h^β), where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald-Letnikov formula or Lubich's higher order approximation schemes.

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

Kamran Kazmi, Abdul Khaliq

2019, 1(4):  525-544.  doi:10.1007/s42967-019-00030-z

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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.

A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations

Xuping Wang, Zhizhong Sun

2019, 1(4):  545-563.  doi:10.1007/s42967-019-00025-w

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In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is O(τ² + h⁴), where τ and h are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.

Modeling and Computing of Fractional Convection Equation

Changpin Li, Qian Yi

2019, 1(4):  565-595.  doi:10.1007/s42967-019-00019-8

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In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendrof method in time, and the secondorder numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efciency of the derived methods are verifed by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

Zhiping Mao, Zhen Li, George Em Karniadakis

2019, 1(4):  597-619.  doi:10.1007/s42967-019-00031-y

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Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a fock, and use a fractional partial diferential equation (fPDE) to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal focking dynamics, we learn the efective nonlocal infuence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one- and two-dimensional nonlocal focking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the feld data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the infuence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one- and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the fnite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method. The proposed method ofers new insights into how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting efective governing equations for nonlocal focking dynamics directly from particle trajectories.

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

Yubo Yang, Fanhai Zeng

2019, 1(4):  621-637.  doi:10.1007/s42967-019-00033-w

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In this paper, a new type of the discrete fractional Grönwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Grönwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.