Identification Problem of Source Term of a Reaction Diffusion Equation

—This paper will give the numerical difference scheme with Dirichlet boundary condition， and prove stability and convergence of the difference scheme, final numerical experiment results also confirm effectiveness of the algorithm.


INTRODUCTION
Source term inversion in groundwater pollution is a class of inverse problems [7,8], and is also important field of inverse problem research.Scholars have done a large amount of work and obtained many results.Reference [2] presented a new gradient regularization algorithm to solve an inverse problem of determining source terms in one-dimension solute transport with final observations, and reference [3] proposed a implicit method to solve a class of space-time fractional order diffusion equations with variable coefficient.However, when fractional derivative replaces second derivative in diffusion equations, there is anomalous diffusion phenomenon.In this paper, we give the numerical difference scheme in source term identification with Dirichlet boundary condition, and we prove the stability and convergence of the difference scheme, also verify the practicality and effectiveness of the algorithm through numerical experiment.
In this paper, we use difference scheme to solve the forward problem, and when solving the inverse problem, we use the gradient regularization method based on Tikhonov regularization strategy.Here, the additional information for source term identification is set as the final observations, and suppose that the source term function are only concerned with the space variable and has nothing to do with the time variable.
In fact, the solute transport model can be described by the following equation [ by introducing fractional derivative and adding initial boundary conditions [5], Eqs.(1) will be modified as the following problem where 0 1,1 Where ( ) 0, ( ) 0, ( ) 0

i k i k i k y x t y x t y x t Oh xh
, by using Grünwald's improved formula [4], we have that ,  2) y represents numerical solution of Eqs. ( 6)-( 8), then we have Since local truncation error is ( ), Oh   thus the difference scheme is consistent [10].Eqs.(14) will be repalced by When 0, ]   ( 1) , ( ) 0.
( , , , ) , and from Lemma 2.1, we further obtain that % The desired result follows. Since Consequently, we can obtain the following result when kT   .y is numerical solution of implicit difference scheme, then there exists a constant The inverse problem, which is composed of Eqs. ( 2)-( 5), is to solve nonlinear operator equation where 12 ( ) ( ( ), ( ), , ( )) , .
It is easy to prove that solving the local minimum values of Eqs.V. NUMERICAL EXPERIMENTS In order to verify the effectiveness of the algorithm in the source term identification, we do the following numerical experiment [7].For simplicity, we set part variables as follows ( )      Through the above numerical experiment, we find that the inversion results and exact solution are almost the same, and this shows that the above algorithm is feasible and very effective.

Theorem 2 . 3
Suppose that( , ) functions on K , then a tiny disturbing ; , ) y b x x t denotes the solution of initial boundary value problem for 1 ( ), bx where 10 ( above, until satisfies the precision requirement.
To better simulate the errors generated by actual data, and verify the effectiveness of the algorithm, we choose the error level.According to the above algorithm, we do 8 times numerical experiments, and obtain the inversion results under different error level  (see TABLEⅢ Figure 1 when  =0.01.

Figure 1 .
Figure 1.The comparison of inversion results and exact solutions TABLE Ⅲ. THE INVERSION RESULTS DIFFERENT ERROR LEVEL Times  =0.01  =0.05  =0.1

TABLE Ⅰ .
THE INVERSION RESULTS UNDER DIFFERENT REGULARIZATION COEFFICIENT

TABLE Ⅱ .
THE INVERSION RESULTS UNDER DIFFERENT INITIAL VALUE