Original article
An optimal regularization method for space-fractional backward diffusion problem

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Abstract

In this paper, a space-fractional backward diffusion problem (SFBDP) in a strip is considered. By the Fourier transform, we proposed an optimal modified method to solve this problem in the presence of noisy data. The convergence estimates for the approximate solutions with the regularization parameter selected by an a priori and an a posteriori strategy are provided, respectively. Numerical experiments show that the proposed methods are effective and stable.

Introduction

Fractional partial differential equations in mathematical physics models have become increasing popular in recent years. Various models using fractional partial differential equations have been successfully applied to describe problems in biology, physics, chemistry and biochemistry, and finance, refer to [18], [8], [1], [12], [10], [4]. These new fractional-order models are more adequate than the integer-order models, because the fractional order derivatives and integrals enable the description of the memory and hereditary properties of different substance [10]. This is the most significant difference of the fractional order models in comparison with the integer order models.

In this paper, we consider a space-fractional diffusion equation, obtained from the standard diffusion equation by replacing the standard space partial derivatives with a space fractional partial derivatives. In physics, the space-fractional diffusion equations are used to model anomalous diffusion or dispersion, where a particle speeds at a rate inconsistent with the classical Brownian motion models [8]. If the initial concentration distribution and boundary conditions are given, a complete recovery of the unknown solution is attainable from solving a well-posed forward problem [10], [9], [17], [7], [19].

However, in some practical problems, it is difficult to specify the initial distribution and boundary conditions. So an ill-posed problem of space-fractional diffusion equation is raised: we have to determine the initial distribution from the knowledge of the final distribution and boundary conditions. This is referred to as the space-fractional backward diffusion problem (SFBDP), which is an ill-posed problem, refer to Section 2. In general, no solution which satisfies the space-fractional diffusion equation, the final data and the boundary conditions, exist. Further, even if a solution exists, it will not be continuously dependent on the boundary and the final data. Thus it is impossible to solve the SFBDP using classical numerical method and requires special techniques to be employed. To the authors’ knowledge, the results for SFBDP are very few. Zheng and Wei [20] used two methods: spectral regularization method and modified equation method to solve this problem. [16] firstly gave a conditional stability estimates for this problem. Our main aim is to present and analyze a regularization method based on modifying the exact “kernel” in the frequency domain. We give the convergence estimates under an a priori and an a posteriori parameter choices and show that this method is an optimal regularization method.

The paper is organized as follows. In Section 2, we present the model of problem and propose an optimal regularization method. The convergence estimates are given in Section 3. In Section 4, the numerical implementation of the method and two numerical examples are given. Finally, we give a conclusion in Section 5.

Section snippets

Model problem and regularization

Consider the following space-fractional backward diffusion problem (SFBDP):

ut(x,t)=xDμαu(x,t),x,t(0,T),u(x,t)|x±=0,t(0,T),u(x,T)=g(x),x,where xDμα is the Riesz–Feller fractional derivative (with respect to x) of order α(0 < α  2) and skewness μ(|μ|  min {α, 2  α}, μ  ±1) which is defined by the Fourier transform in [6], i.e.,F{xDμαg(x);ω}=ψαμ(ω)gˆ(ω),ψαμ(ω)=|ω|αei(sign(ω))μπ/2,where the Fourier transform and inverse Fourier transform of function g(x) are respectively written as,gˆ(ω)=F{g(x);ω}

Convergence estimates

It is well known that for any ill-posed problem some a priori assumption on the exact solution is needed. Otherwise, the convergence of the regularized solution will not be obtained or the convergence rate can be arbitrarily slow [2]. For simplicity, we assume the following a priori bound holds:

u(·,0)pE,p0,where E is a positive constant and ∥ ·  p denotes the norm of function space Hp() defined byHp()=f(x)|fL2(),fp:=12π(1+|ω|2)p|fˆ(ω)|2dω1/2<.

Numerical implementation

In this section, two numerical examples are given to verify the effect of the proposed methods. For simplicity, we always fix the maximum time T = 1 and x  (−10, 10). For noisy data, we usegδ=g+ϵg·randn(size(g)),whereg=(g1,g2,,gN)=(g(x1),g(x2),,g(xN)),xj=10+20(j1)N1,j=1,2,,N.We calculate the noise level δ:=gδgl2=(1/N)i=1N|giδgi|2. The function “randn” generates a random number which is normally distributed with mean 0, variance σ2 = 1 and standard deviation σ = 1, “randn(size(g))” returns

Conclusion

In this paper, we consider a modified kernel regularization method for solving the space-fractional backward diffusion problem, which can be implemented by the fast Fourier transform. The convergence rates under the regularization parameter selected by the a priori and a posteriori strategy are provided respectively. From the mathematical analysis and numerical simulations, it can be seen that both the a priori and a posteriori parameter choice rules are effective.

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This paper was supported by the NSF of China (10971089 and 11171136) and the Fundamental Research Funds for the Central Universities (lzujbky-2013-k02).

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