A non-local boundary value problem method for parabolic equations backward in time

Hào, Ðinh Nho; Duc, Nguyen Van; Sahli, Hichem

doi:10.1016/j.jmaa.2008.04.064

Cited by 51 publications

(15 citation statements)

References 12 publications

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“…Thus, our present paper significantly contributes to the method of non-local boundary methods for ill-posed problems. We note also that this method has been successfully applied to parabolic equations backward in time [4,5,11,14,15,20].…”

Section: Introductionmentioning

confidence: 86%

A non-local boundary value problem method for the Cauchy problem for elliptic equations

2009

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Let H be a Hilbert space with norm • , A : D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equationswith a 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

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Section: Introductionmentioning

confidence: 86%

A non-local boundary value problem method for the Cauchy problem for elliptic equations

2009

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“…It changes the final time condition by a new approximate condition to deduce a well-posed initial boundary value problem for partial differential equation. The best convergence rate in L 2 norm is O(δ 1/2 ), see [7,8,26] for solving backward problems of parabolic equations and see [32,43] for solving backward problems of fractional diffusion equations. If γ = 1, it is called a modified quasiboundary value method, refer to [4] for solving the backward problem for parabolic equation and see [36] for solving the inverse source problem of fractional diffusion equation where an optimal convergence order O(δ 2/3 ) in L 2 norm is obtained.…”

Section: A Generalized Quasi-boundary Value Regularization Methods An...mentioning

confidence: 99%

A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation

Wei

Luo

2022

Inverse Problems

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This paper is devoted to identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. By the series expression of the solution of the direct problem, the inverse source problem can be formulated by a first kind of Fredholm integral equation. The existence and uniqueness, ill-posedness and a conditional stability in Hilbert scale for the considered inverse problem are provided. We propose a generalized quasi-boundary value regularization method to solve the inverse source problem and also prove that the regularized problem is well-posed. Further, two kinds of convergence rates in Hilbert scale for the regularized solution can be obtained by using an a priori and an a posteriori regularization parameter choice rule, respectively. The numerical examples in one-dimensional case and two-dimensional case are given to confirm our theoretical results for the constant coefficients problem. We also propose a finite difference method based on a variant of L1 scheme to solve the regularized problem for the variable coefficients problem and give its convergence rate. One finite difference method based on a convolution quadrature is provided to solve the regularized problem for comparison. The numerical results for three examples by two algorithms are provided to show the effectiveness and stability of the proposed algorithms.

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“…The quasi-boundary value regularization method, also called nonlocal boundary value regularization method in Hào et al [22], is a classical regularization technique by replacing the boundary condition or final condition by a new approximate condition. This regularization method has been used to solve many inverse problems such as the inverse source problem [23], the Cauchy problem [24,25], and the backward problem [26,27]. In addition, there are two improved versions of the quasi-boundary value regularization method, the modified quasi-boundary value regularization method [28], and the generalized quasi-boundary value regularization method [19].…”

Section: Introductionmentioning

confidence: 99%

The iterative generalized quasi‐boundary value method for an inverse source problem of time fractional diffusion‐wave equation in a cylinder

Shi,

Cheng,

Lyu

2023

Math Methods in App Sciences

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This paper is devoted to solve an inverse source problem of time fractional diffusion‐wave equation in a cylinder. The ill‐posedness, existence, and uniqueness of the inverse source problem are proved. We propose an iterative generalized quasi‐boundary value regularization method to deal with the inverse source problem. The convergence rates of the regularized solution are obtained under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule. Numerical examples are presented to show the validity and stability of the proposed method.

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A non-local boundary value problem method for parabolic equations backward in time

Cited by 51 publications

References 12 publications

A non-local boundary value problem method for the Cauchy problem for elliptic equations

A non-local boundary value problem method for the Cauchy problem for elliptic equations

A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation

The iterative generalized quasi‐boundary value method for an inverse source problem of time fractional diffusion‐wave equation in a cylinder

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