The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation
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The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation Dinh Nguyen Duy Hai1,2 · Dang Duc Trong1
Received: 27 April 2017 / Revised: 29 September 2017 / Accepted: 7 December 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract In this paper, we reconstruct the solution u(x, t) of the backward space-fractional diffusion problem with a locally Lipschitzian nonlinear source ⎧ γ ⎨ ut (x, t) = x Dθ u(x, t) + (x, t, u(x, t)), (x, t) ∈ R × (0, T ), u(x, t)|x→±∞ = 0, t ∈ (0, T ), ⎩ u(x, T ) = GT (x), x ∈ R. This problem is severely ill-posed in the Hadamard sense, hence, a regularization is in order. In the paper, we introduce one spectral regularization method and establish stability error estimates with optimal order under an a priori choice of regularization parameter. Finally, numerical implementations are given to show the effectiveness of the proposed regularization methods. Keywords Space-fractional backward diffusion problem · Ill-posed problem · Regularization · Error estimate Mathematics Subject Classification (2010) 26A33 · 47A52 · 47J06 · 65M32
Dang Duc Trong
[email protected] Dinh Nguyen Duy Hai [email protected] 1
Department of Mathematics and Computer Science, Ho Chi Minh City National University, 227 Nguyen Van Cu, Dist 5, Ho Chi Minh City, Vietnam
2
Faculty of Basic Science, Ho Chi Minh City University of Transport, Number 2, D3 street, Ward 25, Binh Thanh District, Ho Chi Minh City, Vietnam
D. N. D. Hai, D. D. Trong
1 Introduction Fractional derivatives calculus and fractional differential equations have been used recently to solve a range of problems in chemistry, physics, biology, mechanical engineering, signal processing and systems identification, etc (see [6–8], among others). Hence, in the last decades, the study of fractional diffusion equations has attracted intense attention. As known, the space-fractional diffusion equation (SFDE) is obtained from the classical diffusion equation in which the term of second-order spatial derivative is replaced by a space-fractional partial derivative: γ ut (x, t) = x Dθ u(x, t) + (x, t, u(x, t)), (x, t) ∈ R × (0, T ), (1) u(x, t)|x→±∞ = 0, t ∈ (0, T ), γ
where the derivative x Dθ is the Riesz-Feller fractional derivative of order γ (0 < γ < 2) and skewness θ (|θ| ≤ min{γ , 2 − γ }, θ = ±1), which is defined by (γ + θ)π ∞ f (x + s) − f (x) (1 + γ ) γ sin D f (x) = ds x θ π 2 s 1+γ 0 (γ − θ)π ∞ f (x − s) − f (x) + sin ds . 2 s 1+γ 0 In the last few years, almost all papers dealing with the SFDE have focused on direct problem and boundary value problems [5, 9, 16]. However, it is usually encountered into solving the function u(x, t) any earlier time t ∈ (0, T ) by the final data u(x, T ) in the practical problems, which will lead to some inverse problems. Concretely, in the present paper, we will consider the SFDE (1) subject to the final condition u(x, T ) = GT (x), x ∈ R,
(2)
where GT (x) is given inexactly. γ For γ = 2, the der
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