Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions

Finite difference methods for fractional differential equation are ever proposed. However, precise error orders have not been analyzed for the methods higher than first order accuracy. This paper proposes a few finite difference methods for fractional diffusion equations and shows our methods have second order accuracy under the conditions that the solution functions have higher order than second order at boundaries. In addition, we show that the accuracy may decrease in the case that the solution functions have lower order than second order at boundaries when we use second order accuracy scheme. In this paper, we treat schemes based on Grunwald-Letnikov definition and apply them to three kinds of fractional diffusion equations using Riemann-Liouville derivative operator including time-fractional diffusion equation, space-fractional diffusion equation and time-space-fractional diffusion equation. Finally, we show the simulation results which indicate that our methods are stable and have successfully second order accuracy under the assumed conditions.

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Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations

Journal of Computational Physics ◽

10.1016/j.jcp.2013.10.040 ◽

2014 ◽

Vol 258 ◽

pp. 305-318 ◽

Cited By ~ 78

Author(s):

Hong Wang ◽

Ning Du

Keyword(s):

Finite Difference ◽

Dimensional Space ◽

Finite Difference Methods ◽

Three Dimensional ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Fractional Diffusion Equations ◽

Alternating Direction ◽

Difference Methods ◽

Three Dimensional Space

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On the numerical solution and convergence study for the system of nonlinear fractional diffusion equations

Canadian Journal of Physics ◽

10.1139/cjp-2013-0464 ◽

2014 ◽

Vol 92 (12) ◽

pp. 1658-1666 ◽

Cited By ~ 13

Author(s):

M.M. Khader

Keyword(s):

Porous Media ◽

Approximate Solution ◽

Finite Difference ◽

Finite Difference Method ◽

Fractional Derivative ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Plasma Transport ◽

Fractional Diffusion Equations ◽

Convergence Study

In this article, an implementation of an efficient numerical method for solving the system of coupled nonlinear fractional diffusion equations (NFDEs) is introduced. The proposed system has many applications, such as porous media and plasma transport. The fractional derivative is described in the Caputo sense. The method is based upon a combination between the properties of the Legendre approximations and finite difference method (FDM). The proposed method reduces NFDEs to a system of ordinary differential equations that are solved using FDM. Special attention is given to the study of the convergence analysis and deducing the upper bound of the error of the resulting approximate solution. A numerical example is given to show the validity and the accuracy of the proposed method.

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Biharmonic navigation using radial basis functions

Robotica ◽

10.1017/s0263574721000709 ◽

2021 ◽

pp. 1-12

Author(s):

Xu-Qian Fan ◽

Wenyong Gong

Keyword(s):

Finite Difference ◽

Boundary Conditions ◽

Radial Basis Functions ◽

Real World ◽

Biharmonic Equation ◽

Finite Difference Methods ◽

Basis Functions ◽

Large Size ◽

Radial Basis ◽

Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

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