﻿ 变空间分数阶扩散方程微分阶数的数值反演 Numerical Inversion for the Fractional Order in the Variable-Order Space-Fractional Diffusion Equation

Vol.06 No.04(2017), Article ID:21470,10 pages
10.12677/AAM.2017.64069

Numerical Inversion for the Fractional Order in the Variable-Order Space-Fractional Diffusion Equation

Di Liu, Shuxiang Wang

Guangzhou Maritime University, Guangzhou Guangdong

Received: Jul. 2nd, 2017; accepted: Jul. 21st, 2017; published: Jul. 25th, 2017

ABSTRACT

We consider a variable-order fractional advection-diffusion equation. Explicit approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, the homotopy regularization algorithm is applied to solve the inverse problem, and numerical examples are presented.

Keywords:Variable-Order Fractional Diffusion Equation, Inverse Problem, Homotopy Regularization Algorithm, Numerical Inversion

1. 引言

2. 正问题及其数值求解

(1)

(2)

(3)

2.1. 差分格式

(4)

(5)

(6)

(7)

(8)

2.2. 稳定性和收敛性分析

(9)

2.3. 数值算例

(1) 考察时间空间步长对正问题的影响，结果列于表1表2表3

(2) 当时方程解析解与数值解的比较，如图1

(3) 当时真实解与数值解得比较，如图2

Table 1. The impact of space tme step of the direct problem

Table 2. The effect of time step on error ()

Table 3. The influence of space step size on error ()

Figure 1. Comparison of real solution and numerical solution for t = 0.5

Figure 2. Comparison of real solution and numerical solution (t = 0.3, 0.5, 0.7, 0.9)

3. 反问题及其数值反演

(10)

(11)

(1) 先取，即，考察初始迭代对反演算法的影响.计算结果列于表4，其中表示初始迭代值，表示反演解，表示反演解与真解的误差，为迭代次数。

(2)时，取初始值，选取不同点的观测数据作为附加数据，考虑对反演算法的影响，如表5

(3) 当在不同维数的逼近空间展开时，分别在维数为的逼近空间

Table 4. The effect of initial iteration selection on inversion results

4. 结束语

(a) (b) (c) (d)

Figure 3. Inverse solution and real solution of different approximation spaces. (a) Q = 2; (b) Q = 3; (c) Q = 3; (d) Q = 4

Table 5. The influence of the observation data of different points on the inversion results is selected

Numerical Inversion for the Fractional Order in the Variable-Order Space-Fractional Diffusion Equation[J]. 应用数学进展, 2017, 06(04): 589-598. http://dx.doi.org/10.12677/AAM.2017.64069

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