Đặng Đức Trọng - Academia.edu (original) (raw)

Papers by Đặng Đức Trọng

Research paper thumbnail of Uniqueness of elastic continuation in a semilinear elastic body

Vietnam Journal of Mechanics, Mar 31, 2003

The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic sol... more The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic solid with a displacement-dependent tension modulus, generalizing an earlier result by Ang, Ikehata, Trong and Yamamoto for a nonhomogeneous linear elastic solid.

Research paper thumbnail of The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders

arXiv (Cornell University), Feb 16, 2020

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of ... more We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. For t = 0, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Hölder type.

Research paper thumbnail of On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation

Computers & Mathematics with Applications, 2020

Abstract We investigate the linear but ill-posed inverse problem of determining a multi-dimension... more Abstract We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale H r ( R n ) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale H r ( R n ) . Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales H r ( R n ) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.

Research paper thumbnail of Stepwise regularization method for a nonlinear Riesz–Feller space-fractional backward diffusion problem

Journal of Inverse and Ill-posed Problems, 2019

In this paper, we consider the backward diffusion problem for a space-fractional diffusion equati... more In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

Research paper thumbnail of 2. Regularization of moment problems by truncated expansion and by the Tikhonov method

Lecture Notes in Mathematics, 2002

Contents. 2.1 Method of truncated expansion 2.1.1 A construction of regularized solutions 2.1.2 C... more Contents. 2.1 Method of truncated expansion 2.1.1 A construction of regularized solutions 2.1.2 Convergence of regularized solutions and error estimates 2.1.3 Error estimates using eigenvalues of the Laplacian 2.2 Method of Tikhonov 2.2.1 Case 1: exact solutions in L2(W</font >)L^2(\Omega ) 2.2.2 Case 2: exact solutions in La</font >*(W</font >), 1 \leqq a</font >* \leqq ¥</font >L^{\alpha^*}(\Omega ),\ 1 \leqq \alpha^*

Research paper thumbnail of 3. Backus-Gilbert regularization of a moment problem

Lecture Notes in Mathematics, 2002

Contents. 3.1 Introduction 3.2 Backus-Gilbert solutions and their stability 3.2.1 Definition of t... more Contents. 3.1 Introduction 3.2 Backus-Gilbert solutions and their stability 3.2.1 Definition of the Backus-Gilbert solutions 3.2.2 Stability of the Backus-Gilbert solutions 3.3 Regularization via Backus-Gilbert solutions 3.3.1 Definitions and notations 3.3.2 Main results

Research paper thumbnail of 7. Regularization of some inverse problems in heat conduction

Lecture Notes in Mathematics, 2002

Contents. 7.1 The backward heat equation 7.2 Surface temperature determination from borehole meas... more Contents. 7.1 The backward heat equation 7.2 Surface temperature determination from borehole measurements: a two-dimensional problem 7.3 An inverse two-dimensional Stefan problem: identification of boundary values 7.4 Notes and remarks

Research paper thumbnail of 6. Regularization of some inverse problems in potential theory

Lecture Notes in Mathematics, 2002

Contents. 6.1 Analyticity of harmonic functions 6.2 Cauchy’s problem for the Laplace equation 6.3... more Contents. 6.1 Analyticity of harmonic functions 6.2 Cauchy’s problem for the Laplace equation 6.3 Surface temperature determination from borehole measurements (steady case)

Research paper thumbnail of Backward heat equations with locally lipschitz source

Applicable Analysis, 2014

Research paper thumbnail of Two regularization methods for backward heat problems with new error estimates

Nonlinear Analysis: Real World Applications, 2011

We introduce two new methods for solving a backward heat conduction problem. For these two method... more We introduce two new methods for solving a backward heat conduction problem. For these two methods, we give a stability analysis with new error estimates. Meanwhile, we investigate the roles of the regularization parameters in these two methods. Numerical results show that our algorithm is effective.

Research paper thumbnail of Sinc approximation of the heat distribution on the boundary of a two-dimensional finite slab

Nonlinear Analysis: Real World Applications, 2008

Research paper thumbnail of Sharp estimates for approximations to a nonlinear backward heat equation

Nonlinear Analysis: Theory, Methods & Applications, 2010

A nonlinear backward heat problem for an infinite strip is considered. The problem is illposed in... more A nonlinear backward heat problem for an infinite strip is considered. The problem is illposed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.

Research paper thumbnail of A nonlinear parabolic equation backward in time: Regularization with new error estimates

Nonlinear Analysis: Theory, Methods & Applications, 2010

Consider a nonlinear backward parabolic problem in the form ut+Au(t)=f(t,u(t)),u(T)=g, where A is... more Consider a nonlinear backward parabolic problem in the form ut+Au(t)=f(t,u(t)),u(T)=g, where A is a positive self-adjoint unbounded operator. Based on the fundamental solution to the parabolic equation, we propose to solve this problem by the Fourier truncated method, which generates a well-posed integral equation. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to

Research paper thumbnail of A Hausdorff‐like moment problem and the inversion of the Laplace transform

Mathematische Nachrichten, 2006

We consider the problem of finding u ∈ L 2(I ), I = (0, 1), satisfying ∫I u (x )x dx = μ k , wher... more We consider the problem of finding u ∈ L 2(I ), I = (0, 1), satisfying ∫I u (x )x dx = μ k , where k = 0, 1, 2, …, (α k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of On a backward parabolic problem with local Lipschitz source

Journal of Mathematical Analysis and Applications, 2014

We consider the regularization of the backward in time problem for a nonlinear parabolic equation... more We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t +Au(t) = f (u(t), t), u(1) = ϕ, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β > 0) is well-posed and that its solution U β (t) converges on [0, 1] to the exact solution u(t) as β → 0 +. These results extend some earlier works on the nonlinear backward problem.

Research paper thumbnail of Approximate and analytical inversion formulas in heat conduction on multidimensional spaces

Journal of Inverse and Ill-posed Problems, 2005

In this paper we shall give practical real inversion formulas of heat conduction on multidimensio... more In this paper we shall give practical real inversion formulas of heat conduction on multidimensional spaces and show their numerical experiments by using computers.

Research paper thumbnail of A new version of quasi-boundary value method for a 1-D nonlinear ill-posed heat problem

Journal of Inverse and Ill-posed Problems, 2009

In this paper, a simple and convenient new regularization method which is called modified quasi-b... more In this paper, a simple and convenient new regularization method which is called modified quasi-boundary value method for solving nonlinear backward heat equation is given. Some new quite sharp error estimates between the approximate solution are provided and generalize the results in our paper [17, 19, 20]. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

Research paper thumbnail of Coefficient identification for a parabolic equation

Inverse Problems, 1994

ABSTRACT

Research paper thumbnail of A new regularized method for two dimensional nonhomogeneous backward heat problem

Applied Mathematics and Computation, 2009

We consider the problem of finding, from the final data uðx; y; TÞ ¼ gðx; yÞ, the initial data uð... more We consider the problem of finding, from the final data uðx; y; TÞ ¼ gðx; yÞ, the initial data uðx; y; 0Þ of the temperature function uðx; y; tÞ; ðx; yÞ 2 I ¼ ð0; pÞ Â ð0; pÞ; t 2 ½0; T satisfying the following system u t À u xx À u yy ¼ f ðx; y; tÞ; ðx; y; tÞ 2 I Â ð0; TÞ; uð0; y; tÞ ¼ uðp; y; tÞ ¼ uðx; 0; tÞ ¼ uðx; p; tÞ ¼ 0 ðx; y; tÞ 2 I Â ð0; TÞ: The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.

Research paper thumbnail of A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

Applied Mathematics and Computation, 2010

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamenta... more In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].

Research paper thumbnail of Uniqueness of elastic continuation in a semilinear elastic body

Vietnam Journal of Mechanics, Mar 31, 2003

The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic sol... more The authors prove a theorem on uniqueness of elastic continuation in a nonhomogeneous elastic solid with a displacement-dependent tension modulus, generalizing an earlier result by Ang, Ikehata, Trong and Yamamoto for a nonhomogeneous linear elastic solid.

Research paper thumbnail of The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders

arXiv (Cornell University), Feb 16, 2020

We consider a class of nonlinear fractional equations having the Caputo fractional derivative of ... more We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. For t = 0, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Hölder type.

Research paper thumbnail of On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation

Computers & Mathematics with Applications, 2020

Abstract We investigate the linear but ill-posed inverse problem of determining a multi-dimension... more Abstract We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale H r ( R n ) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale H r ( R n ) . Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales H r ( R n ) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.

Research paper thumbnail of Stepwise regularization method for a nonlinear Riesz–Feller space-fractional backward diffusion problem

Journal of Inverse and Ill-posed Problems, 2019

In this paper, we consider the backward diffusion problem for a space-fractional diffusion equati... more In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

Research paper thumbnail of 2. Regularization of moment problems by truncated expansion and by the Tikhonov method

Lecture Notes in Mathematics, 2002

Contents. 2.1 Method of truncated expansion 2.1.1 A construction of regularized solutions 2.1.2 C... more Contents. 2.1 Method of truncated expansion 2.1.1 A construction of regularized solutions 2.1.2 Convergence of regularized solutions and error estimates 2.1.3 Error estimates using eigenvalues of the Laplacian 2.2 Method of Tikhonov 2.2.1 Case 1: exact solutions in L2(W</font >)L^2(\Omega ) 2.2.2 Case 2: exact solutions in La</font >*(W</font >), 1 \leqq a</font >* \leqq ¥</font >L^{\alpha^*}(\Omega ),\ 1 \leqq \alpha^*

Research paper thumbnail of 3. Backus-Gilbert regularization of a moment problem

Lecture Notes in Mathematics, 2002

Contents. 3.1 Introduction 3.2 Backus-Gilbert solutions and their stability 3.2.1 Definition of t... more Contents. 3.1 Introduction 3.2 Backus-Gilbert solutions and their stability 3.2.1 Definition of the Backus-Gilbert solutions 3.2.2 Stability of the Backus-Gilbert solutions 3.3 Regularization via Backus-Gilbert solutions 3.3.1 Definitions and notations 3.3.2 Main results

Research paper thumbnail of 7. Regularization of some inverse problems in heat conduction

Lecture Notes in Mathematics, 2002

Contents. 7.1 The backward heat equation 7.2 Surface temperature determination from borehole meas... more Contents. 7.1 The backward heat equation 7.2 Surface temperature determination from borehole measurements: a two-dimensional problem 7.3 An inverse two-dimensional Stefan problem: identification of boundary values 7.4 Notes and remarks

Research paper thumbnail of 6. Regularization of some inverse problems in potential theory

Lecture Notes in Mathematics, 2002

Contents. 6.1 Analyticity of harmonic functions 6.2 Cauchy’s problem for the Laplace equation 6.3... more Contents. 6.1 Analyticity of harmonic functions 6.2 Cauchy’s problem for the Laplace equation 6.3 Surface temperature determination from borehole measurements (steady case)

Research paper thumbnail of Backward heat equations with locally lipschitz source

Applicable Analysis, 2014

Research paper thumbnail of Two regularization methods for backward heat problems with new error estimates

Nonlinear Analysis: Real World Applications, 2011

We introduce two new methods for solving a backward heat conduction problem. For these two method... more We introduce two new methods for solving a backward heat conduction problem. For these two methods, we give a stability analysis with new error estimates. Meanwhile, we investigate the roles of the regularization parameters in these two methods. Numerical results show that our algorithm is effective.

Research paper thumbnail of Sinc approximation of the heat distribution on the boundary of a two-dimensional finite slab

Nonlinear Analysis: Real World Applications, 2008

Research paper thumbnail of Sharp estimates for approximations to a nonlinear backward heat equation

Nonlinear Analysis: Theory, Methods & Applications, 2010

A nonlinear backward heat problem for an infinite strip is considered. The problem is illposed in... more A nonlinear backward heat problem for an infinite strip is considered. The problem is illposed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.

Research paper thumbnail of A nonlinear parabolic equation backward in time: Regularization with new error estimates

Nonlinear Analysis: Theory, Methods & Applications, 2010

Consider a nonlinear backward parabolic problem in the form ut+Au(t)=f(t,u(t)),u(T)=g, where A is... more Consider a nonlinear backward parabolic problem in the form ut+Au(t)=f(t,u(t)),u(T)=g, where A is a positive self-adjoint unbounded operator. Based on the fundamental solution to the parabolic equation, we propose to solve this problem by the Fourier truncated method, which generates a well-posed integral equation. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to

Research paper thumbnail of A Hausdorff‐like moment problem and the inversion of the Laplace transform

Mathematische Nachrichten, 2006

We consider the problem of finding u ∈ L 2(I ), I = (0, 1), satisfying ∫I u (x )x dx = μ k , wher... more We consider the problem of finding u ∈ L 2(I ), I = (0, 1), satisfying ∫I u (x )x dx = μ k , where k = 0, 1, 2, …, (α k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research paper thumbnail of On a backward parabolic problem with local Lipschitz source

Journal of Mathematical Analysis and Applications, 2014

We consider the regularization of the backward in time problem for a nonlinear parabolic equation... more We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t +Au(t) = f (u(t), t), u(1) = ϕ, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β > 0) is well-posed and that its solution U β (t) converges on [0, 1] to the exact solution u(t) as β → 0 +. These results extend some earlier works on the nonlinear backward problem.

Research paper thumbnail of Approximate and analytical inversion formulas in heat conduction on multidimensional spaces

Journal of Inverse and Ill-posed Problems, 2005

In this paper we shall give practical real inversion formulas of heat conduction on multidimensio... more In this paper we shall give practical real inversion formulas of heat conduction on multidimensional spaces and show their numerical experiments by using computers.

Research paper thumbnail of A new version of quasi-boundary value method for a 1-D nonlinear ill-posed heat problem

Journal of Inverse and Ill-posed Problems, 2009

In this paper, a simple and convenient new regularization method which is called modified quasi-b... more In this paper, a simple and convenient new regularization method which is called modified quasi-boundary value method for solving nonlinear backward heat equation is given. Some new quite sharp error estimates between the approximate solution are provided and generalize the results in our paper [17, 19, 20]. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

Research paper thumbnail of Coefficient identification for a parabolic equation

Inverse Problems, 1994

ABSTRACT

Research paper thumbnail of A new regularized method for two dimensional nonhomogeneous backward heat problem

Applied Mathematics and Computation, 2009

We consider the problem of finding, from the final data uðx; y; TÞ ¼ gðx; yÞ, the initial data uð... more We consider the problem of finding, from the final data uðx; y; TÞ ¼ gðx; yÞ, the initial data uðx; y; 0Þ of the temperature function uðx; y; tÞ; ðx; yÞ 2 I ¼ ð0; pÞ Â ð0; pÞ; t 2 ½0; T satisfying the following system u t À u xx À u yy ¼ f ðx; y; tÞ; ðx; y; tÞ 2 I Â ð0; TÞ; uð0; y; tÞ ¼ uðp; y; tÞ ¼ uðx; 0; tÞ ¼ uðx; p; tÞ ¼ 0 ðx; y; tÞ 2 I Â ð0; TÞ: The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.

Research paper thumbnail of A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

Applied Mathematics and Computation, 2010

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamenta... more In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].