arXiv subjects

Changpin Li

Publications and source records attributed to Changpin Li.

17 recordsLinked to original sources

arXiv2024

$H^1$-analysis of H3N3-2\textbf{$_σ$}-based difference method for fractional hyperbolic equations

A novel H3N3-2$_σ$ interpolation approximation for the Caputo fractional derivative of order $α\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$α$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficients' properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_σ$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme are derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_σ$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.

math.NA↗

arXiv2023

Spectral approximation of $ψ$-fractional differential equation based on mapped Jacobi functions

Fractional calculus with respect to function $ψ$, also named as $ψ$-fractional calculus, generalizes the Hadamard and the Riemann-Liouville fractional calculi, which causes challenge in numerical treatment. In this paper we study spectral-type methods using mapped Jacobi functions (MJFs) as basis functions and obtain efficient algorithms to solve $ψ$-fractional differential equations. In particular, we setup the Petrov-Galerkin spectral method and spectral collocation method for initial and boundary value problems involving $ψ$-fractional derivatives. We develop basic approximation theory for the MJFs and conduct the error estimates of the derived methods. We also establish a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. Numerical examples confirm the theoretical results and demonstrate the effectiveness of the spectral and collocation methods.

math.NA↗

arXiv2023

Two kinds of numerical algorithms for ultra-slow diffusion equations

In this article, two kinds of numerical algorithms are derived for the ultra-slow (or superslow) diffusion equation in one and two space dimensions, where the ultra-slow diffusion is characterized by the Caputo-Hadamard fractional derivative of order $α\in (0,1)$. To describe the spatial interaction, the Riesz fractional derivative and the fractional Laplacian are used in one and two space dimensions, respectively. The Caputo-Hadamard derivative is discretized by two typical approximate formulae, i.e., L2-1$_σ$ and L1-2 methods. The spatial fractional derivatives are discretized by the 2-nd order finite difference methods. When L2-1$_σ$ discretization is used, the derived numerical scheme is unconditionally stable with error estimate $\mathcal{O}(τ^{2}+h^{2})$ for all $α\in (0, 1)$, in which $τ$ and $h$ are temporal and spatial stepsizes, respectively. When L1-2 discretization is used, the derived numerical scheme is stable with error estimate $\mathcal{O}(τ^{3-α}+h^{2})$ for $α\in (0, 0.3738)$. The illustrative examples displayed are in line with the theoretical analysis.

math.NA↗

arXiv2022

Monte Carlo method for parabolic equations involving fractional Laplacian

We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploit- s the idea of weak approximation of related stochastic differential equations driven by the symmetric stable Lévy process with jumps. We utilize the jump- adapted scheme to approximate Lévy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numeri- cal scheme by removing the small jumps of the Lévy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high dimensional parabolic equations.

math.NA↗

arXiv2022

A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation

We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tation of the fractional Poisson equation. We propose effcient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2-10 dimensions verify our theory and the effciency of the modified walk-on-sphere method.

math.NA↗

arXiv2022

Fractional SEIR Model and Data-Driven Predictions of COVID-19 Dynamics of Omicron Variant

We study the dynamic evolution of COVID-19 cased by the Omicron variant via a fractional susceptible-exposedinfected-removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is therefore more concealed, which causes a relatively slow increase in the detected cases of the new infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo-Hadamard fractional derivative is adopted to refined the classical SEIR model. Based on the reported data, we infer the fractional order, timedependent parameters, as well as unobserved dynamics of the fractional SEIR model via fractional physics-informed neural networks (fPINNs). Then, we make short-time predictions using the learned fractional SEIR model.

math.NA↗

arXiv2021

Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation

We develop fractional buffer layers (FBLs) to absorb propagating waves without reflection in bounded domains. Our formulation is based on variable-order spatial fractional derivatives. We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain. In particular, we first design proper FBsL for the one-dimensional one-way and two-way wave propagation. Then, we extend our formulation to two-dimensional problems, where we introduce a consistent variable-order fractional wave equation. In each case, we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time. We compare our results with the perfectly matched layer (PML) method and show the effectiveness of FBL in accurately suppressing any erroneously reflected waves, including corner reflections in two-dimensional rectangular domains. FBLs can be used in conjunction with any discretization method appropriate for fractional operators describing wave propagation in bounded or truncated domains.

math.NA↗

arXiv2016

Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations

It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives play a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

math.NA↗

arXiv2016

A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application

Compared to the the classical first-order Gr\"unwald-Letnikov formula at time $t_{k+1} (\textmd{or}\, t_{k})$, we firstly propose a second-order numerical approximate scheme for discretizing the Riemann-Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank-Niclson technique applied to the time-fractional differential equations. The established formula has the following form $$ \begin{array}{lll} \displaystyle \,_{\mathrm{RL}}{{{\mathrm{D}}}}_{0,t}^{\alpha}u\left(t\right)\left|\right._{t=t_{k+\frac{1}{2}}}= \tau^{-\alpha}\sum\limits_{\ell=0}^{k} \varpi_{\ell}^{(\alpha)}u\left(t_k-\ell\tau\right) +\mathcal{O}(\tau^2),\,\,k=0,1,\ldots, \alpha\in(0,1), \end{array} $$ where the coefficients $\varpi_{\ell}^{(\alpha)}$ $(\ell=0,1,\ldots,k)$ can be determined via the following generating function $$ \begin{array}{lll} \displaystyle G(z)=\left(\frac{3\alpha+1}{2\alpha}-\frac{2\alpha+1}{\alpha}z+\frac{\alpha+1}{2\alpha}z^2\right)^{\alpha},\;|z|<1. \end{array} $$ Applying this formula to the time fractional Cable equations with Riemann-liouville derivative in one or two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

math.NA↗

arXiv2015

High-order numerical algorithms for Riesz derivatives via constructing new generating functions

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order $O(\tau^2+h^2)$, where $\tau$ and $h$ are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

math.NA↗

arXiv2015

High-order fractional-compact finite difference method for Riesz spatial telegraph equation

In this paper, we establish even order compact numerical schemes (4th-order, 6th-order, 8th-order, 10th-order) for Riesz derivatives by using the symmetrical fractional centred difference operator. Then we apply the derived 4th-order algorithm to the Riesz spatial telegraph equation. We carefully study the stability and convergence by matrix method, and show that convergence orders in temporal and spatial directions are both 4th order. Numerical experiments are displayed which support the compact difference schemes for Riesz derivatives and the Riesz spatial telegraph equation.

math.NA↗

arXiv2014

Multi-UAV-based Optimal Crop-dusting of Anomalously Diffusing Infestation of Crops

This paper presents a UAV-based optimal crop-dusting method to control anomalously diffusing infestation of crops. Two anomalous diffusion models are considered, which are, respectively, time-fractional order diffusion equation and space-fractional order diffusion equation. Our problem formulation is motivated by real-time pest management by using networked unmanned cropdusters where the pest spreading is modeled as a fractional diffusion equation. We attempt to solve the optimal dynamic location of actuators by using Centroidal Voronoi Tessellations. A new simulation platform (FO-DiffMAS-2D) for measurement scheduling and controls in fractional order distributed parameter systems is also introduced in this paper. Simulation results are presented to show the effectiveness of the proposed method as well as the role of fractional order in the overall control performance.

eess.SY↗

arXiv2014

High-order Numerical Methods for Riesz Space Fractional Turbulent Diffusion Equation

Numerical methods for fractional calculus attract increasing interests due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz space fractional turbulent diffusion equation. Numerical experiments are displayed which support the theoretical analysis.

math.NA↗

arXiv2014

High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation

In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference operator. We apply these methods to fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by Fourier method in details. The convergence orders of these numerical schemes are $\mathcal {O}(\tau^2+h^6)$ and $\mathcal {O}(\tau^2+h^8)$, respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.

math.NA↗

arXiv2014

Fractional differentiation matrices with applications

In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices are investigated numerically. The spectral collocation schemes are illustrated to solve the fractional ordinary differential equations and fractional partial differential equations. Numerical examples are also presented to illustrate the effectiveness of the derived methods, which show better performances over some existing methods.

math.NA↗

arXiv2005

Synchronization of Limit Sets

In this Letter, we derive a sufficient condition of synchronizing limit sets (attractors and repellers) by using the linear feedback control technique proposed here. There examples are included. The numerical simulations and computer graphics show that our method work well.

nlin.CD↗

arXiv2005

Synchronization in delayed discrete-time complex networks

In this paper, we study synchronization in the delayed discrete-time complex networks. Several criterions of synchronization stability for such networks are established. And illustrative examples are presented. The numerical simulations coincide with the theoretical analysis.

nlin.CD↗