This paper deals with an inverse problem for identifying multiparameters in 1D space fractional advection dispersion equation (FADE) on a finite domain with final observations. The parameters to be identified are the fractional order, the diffusion coefficient, and the average velocity in the FADE. The forward problem is solved by a finite difference scheme, and then an optimal perturbation regularization algorithm is introduced to determine the three parameters simultaneously. Numerical inversions are performed both with the accurate data and noisy data, and several factors having influences on realization of the algorithm are discussed. The inversion solutions are in good approximations to the exact solutions demonstrating the efficiency of the proposed algorithm.
Many diffusion processes in nature and engineering, such as contaminants transport in the soil, oil flow in porous media, long distance transport of pollutants in the groundwater, and so forth, are referred to as anomalous diffusion, where the particle plume spreads faster or slower than predicted by the classical diffusion model. In recent two decades, the space FADE has been found to be an efficient mathematical model to describe such anomalous diffusion phenomena (see [
There are quite a few research works on the fractional differential equations, refer to [
In this paper, we will deal with an inverse problem of simultaneously determining the fractional order, the diffusion coefficient, and the average velocity in (
In Section
For
Let the space step be
Denote
For the above difference scheme, similarly as done in [
The implicit difference scheme defined by (
In this subsection, an example is presented to show effectiveness of the finite difference scheme (
In the numerical simulation, we will take
Errors in the solutions at

20  40  80  100  200  400 
E_{rr}  0.0088  0.0055  0.0031  0.0026  0.0014 

Errors in the solutions at

20  40  80  100  200  400 
E_{rr}  0.0165  0.0105  0.0060  0.0049  0.0026  0.0014 
The exact and numerical solutions with
From Tables
In many cases for the anomalous diffusion model given by (
As we know, most of inversion algorithms are based on regularization strategies so as to overcome illposedness of the inverse problem, and different kinds of inverse problems could need different approximate methods on the basis of conditional wellposedness analysis (see [
Denote
Suppose that
Taking Taylor’s expansion for
It is not difficult to testify that minimizing functional (
Taking the space domain as
It is important to choose an optimal regularization parameter by theoretical analysis in solving inverse problems, where regularization methods are needed. However, it is still a feasible approach to the choice of regularization parameter by trial and error especially for moderate illposed inverse problems. The regularization parameter here is selected by numerical testification, and its influence on the inversion algorithm for given differential step is discussed in this subsection.
Suppose that the fractional order is
Influence of regularization parameter on the algorithm with


E_{rr} 



(1.5998012, 1.0004052, 1.0003096) 

10.190/59 

(1.5998300, 1.0003477, 1.0002653) 

9.516/52 

(1.5998732, 1.0002609, 1.0001987) 

8.281/45 

(1.5998862, 1.0002350, 1.0001787) 

7.156/41 

(1.5999263, 1.0001542, 1.0001167) 

6.000/33 

(1.5999617, 1.0000825, 1.0000618) 

4.250/24 

(1.5999665, 1.0000747, 1.0000552) 

3.391/19 

(1.5999779, 1.0000495, 1.0000365) 

3.375/19 

(1.5999780, 1.0000493, 1.0000364) 

3.406/19 

(1.5999780, 1.0000492, 1.0000364) 

3.344/19 

(1.5999780, 1.0000492, 1.0000364) 

3.157/19 
0  Divergent 
It is noticeable that the inversion results have little changes if utilizing any positive regularization parameters smaller than
Influence of regularization parameter on the algorithm with


E_{rr} 


5  (1.6015151, 0.99697413, 0.99767406) 

32.657/176 
4.5  (1.6013808, 0.99724171, 0.99787964) 

29.687/163 
4  (1.6012283, 0.99754585, 0.99811333) 

27.704/150 
3.5  (1.6010527, 0.99789612, 0.99838251) 

24.703/137 
3  (1.6009217, 0.99815749, 0.99858340) 

22.500/122 
2.5  (1.6007547, 0.99849085, 0.99883965) 

20.094/108 
2.4  (1.6007389, 0.99852247, 0.99886397) 

19.313/105 
2.3  (1.6006934, 0.99861329, 0.99893378) 

18.985/103 
2.2  (1.6006824, 0.99863531, 0.99895072) 

18.469/100 
2.1  (1.6006235, 0.99875285, 0.99904107) 

17.640/96 
≤2  Divergent 
The solutions errors with regularization parameter and differential step.
From Tables
By the above computations together with (
Also set the exact solution to be
Influence of numerical differential step on the algorithm with


E_{rr} 


≥0.9  Divergent  
0.85  (1.6001245, 0.99973957, 0.99978970) 

6.563/36 
0.8  (1.6001340, 0.99969424, 0.99977646) 

6.453/35 
0.7  (1.6001260, 0.99971463, 0.99979024) 

6.172/34 
0.6  (1.6001232, 0.99972209, 0.99979564) 

5.937/32 
0.5  (1.6001146, 0.99974254, 0.99981050) 

5.359/29 
0.4  (1.6000995, 0.99977749, 0.99983595) 

3.672/20 
0.3  (1.5999046, 1.0002119, 1.0001567) 

4.984/27 
0.2  (1.5999303, 1.0001532, 1.0001138) 

5.203/28 
0.1  (1.5999617, 1.0000825, 1.0000618) 

4.375/24 
0.05  (1.5999773, 1.0000477, 1.0000361) 

4.484/24 
0.04  (1.5999772, 1.0000475, 1.0000360) 

5.344/29 
≤0.03  Divergent 
From Table
In this subsection, we will investigate influence of the fractional order on the inversion algorithm with the convergent criterion given as
Influence of fractional order on the inversion algorithm.



E_{rr} 


1.9  (1.9, 1, 1)  (1.90000, 1.00000, 1.00000) 

3.547/21 
1.8  (1.8, 1, 1)  (1.80000, 1.00000, 1.00000) 

3.859/23 
1.7  (1.7, 1, 1)  (1.70000, 1.00000, 1.00000) 

4.344/26 
1.6  (1.6, 1, 1)  (1.60000, 1.00000, 1.00000) 

5.500/33 
1.5  (1.5, 1, 1)  (1.50000, 1.00000, 1.00000) 

7.797/47 
1.4  (1.4, 1, 1)  (1.40000, 1.00000, 1.00000) 

13.031/79 
1.3  (1.3, 1, 1)  (1.30000, 1.00001, 1.00001) 

24.437/148 
1.2  (1.2, 1, 1)  (1.24150, 0.838272, 0.845473) 

87.734/531 
From Table
It is difficult to perform an inversion algorithm in the case of using random noisy data, especially for inverse problems arising from the fractional diffusion. Noting computational errors and data noises, the additional information utilized for real inverse problems is often given as
Generally speaking, in case of using regularization strategy to damp the noises, an optimal regularization parameter should be chosen according to the noise level. However, the situation becomes simple for the inverse problem considered here. Without loss of generality, we set the noise levels be
Inversion results with regularization parameters for






0.01  (1.4992660, 1.0017420, 1.0015202) 

7.2  39.9 
0.1  (1.4992650, 1.0017420, 1.0015203) 

8.7  47 
0.5  (1.4992640, 1.0017427, 1.0015217) 

13.8  75.1 
Inversion results with regularization parameters for






0.01  (1.4947065, 1.0158355, 1.0141558) 

8.4  42.0 
0.1  (1.4947063, 1.0158360, 1.0141562) 

8.3  49.1 
0.5  (1.4947055, 1.0158378, 1.0141577) 

13.1  79.0 
From Tables
Inversion results with random noises using







(1.4841, 1.0568, 1.0515) 

11.9  58.3 

(1.4947, 1.0158, 1.0142) 

8.3  49.1 

(1.4973, 1.0073, 1.0065) 

8.8  47.4 

(1.4993, 1.0017, 1.0015) 

8.7  47.0 

(1.4997, 1.0008, 1.0007) 

8.4  46.8 

(1.4999, 1.0001, 1.0001) 

8.1  46.9 

(1.5000, 1.0000, 1.0000) 

8.1  47.0 
From Table
(i) An inverse problem for identifying multiparameters of the fractional order, the diffusion coefficient, and the average velocity simultaneously in the FADE with final observations is investigated. An implicit finite difference scheme is employed to solve the forward problem, and an optimal perturbation regularization algorithm is applied to reconstruct the three parameters. By the numerical simulations, we conclude that the optimal perturbation regularization algorithm is of numerical stability, and it is efficient for multiparameters identification problem arising from the FADE.
(ii) There are few factors that have influences on the inversion algorithm, but the regularization parameter and the numerical differential step here seem to play a much more important role in the realization of the algorithm. The inversion algorithm can be performed successfully for the regularization parameter and the numerical differential step belonging to suitable intervals, respectively. However, the inversion algorithm may be a failure if the numerical differential step is taking too small values. In addition, it seems to be insensitive to the choice of the regularization parameter in concrete realization of the algorithm showing that the inverse problem discussed here is of moderate illposedness.
(iii) By the inversion computations, we find that the fractional order in the FADE has some influence on the forward problem and the inverse problem. If the fractional order
The authors thank for the referees and the editor for their fruitful suggestions and comments. This work is supported by the National Natural Science Foundation of China (no. 11071148) and the Natural Science Foundation of Shandong Province, China (no. ZR2011AQ014).