﻿ 变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 Time-Fractional Diffusion Equation

Advances in Applied Mathematics
Vol.04 No.04(2015), Article ID:16356,10 pages
10.12677/AAM.2015.44041

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

Di Liu, Chunlong Sun, Gongsheng Li*, Xianzheng Jia

*通讯作者。

School of Sciences, Shandong University of Technology, Zibo Shandong

Received: Oct. 22nd, 2015; accepted: Nov. 7th, 2015; published: Nov. 12th, 2015

Copyright © 2015 by authors and Hans Publishers Inc.

ABSTRACT

An implicit finite difference scheme is introduced to solve the variable-order time-fractional diffusion equation, and an inverse problem of determining the variable fractional order is set forth using the additional measurements at one interior point. The homotopy regularization algorithm is applied to solve the inverse problem, and numerical examples are presented. The computational and inversion results demonstrate that the variable order has important influence on the problem, and that the computations become effective when the variable order goes to 1.

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)

， (4)

2.1. 差分格式

，其中分别是空间和时间步长。记，在处，变分数阶导数离散为：

(5)

. (6)

，将(5)，(6)带入方程(1)，并略去高阶项得到

. (7)

. (8)

. (9)

，可得以矩阵形式表示的差分格式

(10)

, (11)

，这里表示取对角阵。

,.

. (12)

. (13)

, (14)

,. (15)

, (16)

2.2. 数值算例

3. 反问题及反演算法

, (17)

Table 1. The solutions errors with space/time steps at

Figure 1. The exact and numerical solutions at for

, (18)

. (19)

. (20)

, (21)

(22)

,

.(23)

, , (24)

, (25)

. (26)

4. 数值反演

, (27)

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2，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) 当在不同维数的逼近空间展开时，其有不同的表示形式。由于截断误差的存在，理论上逼近空间维数越高，则反演结果应该越精确。分别在维数为的逼近空间中反演，初始迭代相应取为，反演解与真解的图像分别绘于图2的(a)，(b)，(c)及(d)。

3) 实际问题中，附加数据往往带有某种误差，对于扰动数据实施反演算法是反问题数值方法研究的重要内容。设带扰动的附加数据表示为

, (28)

5. 结束语

Table 2. The inversion results with initial iterations

Table 3. The inversion results with noisy data

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

Figure 2. The exact and inversion solutions for the fractional order in different approximate spaces. (a) Q = 2; (b) Q = 3; (c) Q = 4; (d) Q = 5

Numerical Inversion for the Fractional Order in the Variable-Order Time-Fractional Diffusion Equation[J]. 应用数学进展, 2015, 04(04): 326-335. http://dx.doi.org/10.12677/AAM.2015.44041

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