An inverse problem for an inhomogeneous time-fractional diffusion equation: a regularization method and error estimate
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An inverse problem for an inhomogeneous time-fractional diffusion equation: a regularization method and error estimate Nguyen Huy Tuan1 · Luu Vu Cam Hoan2,3 · Salih Tatar4 Received: 21 March 2018 / Revised: 10 September 2018 / Accepted: 15 November 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we study an inverse problem for an inhomogeneous time-fractional diffusion equation in the one-dimensional real-positive semiaxis domain. Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative. After we show that the inverse problem is severely ill posed, we apply a modified regularization method based on the solution in the frequency domain to solve the inverse problem. A convergence estimate is also derived. We present two numerical examples to show the efficiency of the method. Keywords Regularization method · Inverse problem · Caputo fractional derivative · Convergence estimate Mathematics Subject Classification 35K05 · 35K99 · 47J06 · 47H10
Communicated by Domingo Alberto Tarzia. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s40314-0190776-x) contains supplementary material, which is available to authorized users.
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Salih Tatar [email protected] Nguyen Huy Tuan [email protected] Luu Vu Cam Hoan [email protected]
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Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Basic Science, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam
3
Department of Mathematics and Computer Science, University of Science-VNU.HCMC, Ho Chi Minh City, Vietnam
4
Department of Mathematics and Computer Science, Alfaisal University, Riyadh, Saudi Arabia
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N. H. Tuan et al.
1 Introduction In this paper, we consider the problem of recovering the temperature from the following inverse problem: ⎧ γ ⎨ −au x (x, t) =0 Dt u(x, t) + F(x, t, u(x, t)), x > 0, t > 0, u(1, t) = g(t), t ≥ 0, (1) ⎩ u(x, 0) = lim x→+∞ u(x, t) = 0, where a is the constant diffusivity coefficient, F(x, t, u(x, t)) is the nonlinear source term γ and 0 Dt u(x, t) is the Caputo time fractional derivative of order 0 < γ < 1 defined by t ∂ 1 γ (t − τ )−γ u(x, τ ) dτ, (2) 0 Dt u(x, t) := (1 − γ ) 0 ∂τ where is the Gamma function. This was intended to properly handle initial values (Caputo 1967; Chen et al. 2012; Eidelman et al. 2004) since its Laplace transform(LT) s β f˜(s) − s β−1 f (0) incorporates the initial value in the same way as the first derivative. Here, f˜(s) is the usual Laplace transform. It is well known that the Caputo derivative has a continuous spectrum (Chen et al. 2012), with eigenfunctions given in terms of the Mittag-Leffler function E β (z) :=
∞ k=0
zk . (1 + βk)
In fact, it is easy to see that f (t) = E β (−λt β ) solves the eigenvalue equation ∂ β f (t) = −λ f (t), ∂t β
f (0) = 1
for any λ > 0. This is easily verifi
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