Regularization for a Riesz-Feller space fractional backward diffusion problem with a time-dependent coefficient

In the present paper, we consider a backward problem for a space-fractional diffusion equation (SFDE) with a time-dependent coefficient. Such the problem is obtained from the classical diffusion equation by replacing the second-order spatial derivative with the Riesz-Feller derivative of order   2 , 0   . This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Therefore, we propose one new regularization solution to solve it. Then, the convergence estimate is obtained under a priori bound assumptions for exact solution.


INTRODUCTION
The fractional differential equations appear more and more frequently in physical, chemical, biology and engineering applications. Nowadays, fractional diffusion equation plays important roles in modeling anomalous diffusion and subdiffusion systems [2], description of fractional random walk, unification of diffusion [3], and wave propagation phenomenon [4]. It is well known that the SFDE is obtained from the classical diffusion equation in which the second-order space derivative is replaced with a space-fractional partial derivative.
where the fractional spatial derivative x D   is the Here, we wish to determine the temperature ( , ) u x t from temperature measurements ( ). Gx  Since the measurements usually contain an error, we now could assume that the measured data function We assume that F satisfies the Lipschitz condition for some constant In case of the source function 0 F  and ( ) 1, t

 
Problem (1) has been proposed by some authors. Zheng and Wei [7] used two methods, the spectral regularization and modified equation methods, to solve this problem. In [6], they developed an optimal modified method to solve this problem by an a priori and an a posteriori strategy. In 2014, Zhao et al [8] applied a simplified Tikhonov regularization method to deal with this problem. After then, a new regularization method of iteration type for solving this problem has been introduced by Cheng et al [1]. Although we have many works on the linear homogeneous case of the backward problem, the nonlinear case of the problem is quite scarce. For the nonlinear problem, the solution u is complicated and defined by an integral equation such that the right hand side depends on . u This leads to studying nonlinear problem is very difficult, so in this paper we develop a new appropriate technique.
The remainder of this paper is organized as follows. In Section 2, we propose the regularizing scheme for Problem (1). Then, in Section 3, we show that the regularizing scheme of Problem (1) is wellposed. Finally, the convergence estimate is given in Section 4.

Let
() G  denote the Fourier transform of the integrable function ( ), Gx which defined by In terms of the Fourier transform, we have the following properties for the Riesz-Feller spacefractional derivative [5]

THE WELL POSEDNESS OF PROBLEM (9)
First, we consider the following Lemma which is used in the proof of the main results.

 
As an immediate consequence of (a), making the change , sT  we have (b). Next, we prove (c). In fact from (b), we obtain This completes the proof. □ We are now in a position to prove the following theorem. Proof. We divide it into two steps.
Step1. The existence and the uniqueness of a solution of Problem (9).
Let us define the norm on It is easily be seen that For 0, tT  using the inequality On the other hand, letting .

  
Thus, we obtain This completes the proof of Step 2 and also the proof of the theorem. □

CONVERGENCE ESTIMATE
)F s u s ds On the other hand, we get ()  [1,6,8].

CONCLUSION
In this paper, we use the new regularization solution to slove a Riesz-Feller space-fractional backward diffusion problem with a time-dependent coefficient. The convergence result has been obtained under a priori bound assumptions for the exact solution and the suitable choices of the regularization parameter.