Regularization methods for unknown source in space fractional diffusion equation

Tian, Wenyi; Li, Can; Deng, Weihua; Wu, Yue

doi:10.1016/j.matcom.2012.08.011

Cited by 20 publications

(15 citation statements)

References 23 publications

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“…For ς ∈ H ∕2 and for any T ∈ [0, ∞), we shall prove that (1). For any ∈ [0, ′ ∕2), there exists a constant M > 0 independent of t and | − ′ | d such that…”

Section: Theoremmentioning

confidence: 93%

“…We remind the mean value theorem in several variables. If the domain D 0 is a convex domain in R d , and g is a differentiable function, then for every 1…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

“…Herein

, {normalΔ}_{a, b}^{α} w = ({}, a_{- \infty} D_{x}^{α - 1} w - b_{x} D_{\infty}^{α - 1} w)

, where

_{- \infty} D_{x}^{α}

and

_{x} D_{\infty}^{α}

are the left and right Riemann‐Liouville fractional derivatives of order α , respectively. We have (see Tian)

\hat{{normalΔ}_{a, b}^{α} w} (ω) = | ω {false|}^{α} (\cos (α π / 2) + i (a - b) \sin (\pm α π / 2)) \hat{w} (ω), \forall ω \in R .

Putting ℓ = ( α , θ ), we have an other example

A_{ℓ} w = -_{x} D_{θ}^{α} w

where

_{x} D_{θ}^{α}

is the Riesz‐Feller derivative of order α and skewness θ . One has

\hat{_{x} D_{θ}^{α} w} (ω) = - (\cos (θ π / 2) + i sign (ω) \sin (θ π / 2)) | ω {false|}^{α} \hat{w} (ω) .

The parameter vector ℓ can include many important parameters a...…”

Section: Introductionmentioning

confidence: 99%

“…Nonhomogeneous and nonlinear backward parabolic problems were considered in a lot of papers. () Recently, a lot of fractional ill‐posed problem were studied (see, eg, other studies()). However, in the present paper, we consider the ill‐posedness depended on the parameters.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

Dien

Duc

2019

Math Methods in App Sciences

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In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed.

show abstract

“…For ς ∈ H ∕2 and for any T ∈ [0, ∞), we shall prove that (1). For any ∈ [0, ′ ∕2), there exists a constant M > 0 independent of t and | − ′ | d such that…”

Section: Theoremmentioning

confidence: 93%

“…We remind the mean value theorem in several variables. If the domain D 0 is a convex domain in R d , and g is a differentiable function, then for every 1…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

“…Herein

, {normalΔ}_{a, b}^{α} w = ({}, a_{- \infty} D_{x}^{α - 1} w - b_{x} D_{\infty}^{α - 1} w)

, where

_{- \infty} D_{x}^{α}

and

_{x} D_{\infty}^{α}

are the left and right Riemann‐Liouville fractional derivatives of order α , respectively. We have (see Tian)

\hat{{normalΔ}_{a, b}^{α} w} (ω) = | ω {false|}^{α} (\cos (α π / 2) + i (a - b) \sin (\pm α π / 2)) \hat{w} (ω), \forall ω \in R .

Putting ℓ = ( α , θ ), we have an other example

A_{ℓ} w = -_{x} D_{θ}^{α} w

where

_{x} D_{θ}^{α}

is the Riesz‐Feller derivative of order α and skewness θ . One has

\hat{_{x} D_{θ}^{α} w} (ω) = - (\cos (θ π / 2) + i sign (ω) \sin (θ π / 2)) | ω {false|}^{α} \hat{w} (ω) .

The parameter vector ℓ can include many important parameters a...…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

Dien

Duc

2019

Math Methods in App Sciences

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show abstract

“…In [32], the authors used the boundary element method combined with a generalized Tikhonov regularization method to identify the Downloaded by [University of Colorado at Boulder Libraries] at 17:08 27 December 2014 unknown source which depends only on the time variable for the time-fractional diffusion equation. In [33], the authors used the Fourier method to identify the unknown source which depends only on the spatial variable for the space-fractional diffusion equation, but the regularization parameter is a priori choice rule.…”

Section: Introductionmentioning

confidence: 99%

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

Yang

2014

Inverse Problems in Science and Engineering

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In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

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Identification of a Time-Dependent Right-Hand Side of an Unsteady Equation with a Fractional Power of an Elliptic Operator

Vabishchevich

2020

Large-Scale Scientific Computing

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Regularization methods for unknown source in space fractional diffusion equation

Cited by 20 publications

References 23 publications

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

Identification of a Time-Dependent Right-Hand Side of an Unsteady Equation with a Fractional Power of an Elliptic Operator

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