Inverse coefficient problems for nonlinear parabolic differential equations

Ou, Yun; Hasanov, Alemdar; Liu, Zhen Hai

doi:10.1007/s10114-008-6384-0

Cited by 31 publications

(17 citation statements)

References 14 publications

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“…The parameters of the nonlinear heat conduction problem considered in [22] are spatially and temperature dependent. The problem researched in [23] is the same as ours, and if the nonlinear function N ¼ jruj 2 , then the problem is similar to the one discussed in [24].…”

Section: Introductionmentioning

confidence: 92%

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An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

Zhao

Liu

2012

Inverse Problems in Science and Engineering

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We investigate the problem of estimating the permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the parameter estimation is formulated as a least-square minimization problem with a regularization term. To overcome the weakness of the local convergence of traditional methods, a homotopy method is applied to solve this inverse problem. By introducing the Tikhonov regularization and adaptively choosing the homotopy parameters, a new and globally convergent algorithm is constructed. Finally, many numerical simulations are presented to show the global convergence, computational efficiency and anti-noise ability of the proposed algorithm.

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Section: Introductionmentioning

confidence: 92%

“…Parameter inversions of various nonlinear parabolic equations have been studied extensively [17][18][19][20][21][22][23][24]. Ye and Sun [17], considered the inverse problem of the parabolic equation with the control parameter.…”

Section: Introductionmentioning

confidence: 99%

An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

Zhao

Liu

2012

Inverse Problems in Science and Engineering

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“…This method is based on the invertibility of the above input-output mapping. The other method is called quasi-solution method and is given in [24]. Here, we develop the quasi-solution method for the evolutional inverse problem.…”

Section: Existence Of a Quasi-solution Of The Evolutional Inverse Promentioning

confidence: 99%

“…For example in [23], the authors have developed an inverse method, which includes minimization of a least-squares cost functional, to identify the unknown coefficient and obtained quasi-solution of the inverse problem in the appropriate class of admissible coefficients. In [24], the direct and inverse problems are analysed and the convergence of solutions for the corresponding direct problems continuously depending on the coefficient convergence are proved.…”

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confidence: 99%

Numerical solution of the nonlinear parabolic problem related to inverse elastoplastic torsional problem

Tatar

2012

Inverse Problems in Science and Engineering

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In this article, we study a class of direct and inverse coefficient problems defined for a nonlinear parabolic equation. An existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. Direct problem is solved numerically by using semi-implicit finite difference scheme and then some examples are presented related to steady-state solution of the nonlinear direct problem and ill-posedness of the inverse problem. IntroductionThe determination of unknown coefficients in partial differential equations from measured output data are well-known in the literature as inverse coefficient problems (ICPs). Especially, the nonlinear ICPs have been investigated by many authors in the past years. Among these, the torsion problems in linear elasticity have been well-studied in the classical literature [1][2][3][4]. In the case of elastoplastic torsion, the problems related to qualitative properties and an existence of a weak solution of the nonlinear direct problem are also well-studied in mathematical literature [5,6]. Within the range of monotone potential operator theory, the solvability of the nonlinear direct problem has been proved in [7], in the class of admissible coefficients. Many analytical and computational methods are given in the scientific literature, especially when small amounts of twisting are reversible and the beam returns to its original shape after releasing the twisting force. This is called the elastic behaviour. However, inverse problems in torsional deformation are less well-known and have received relatively little attention in the mathematical as well as in engineering literature [8][9][10]. The quasi-static mathematical model of torsional creep within J 2 -deformation theory of plasticity is given in [11]. In this model, one seeks the solution u(x, y), (x, y) 2 & R 2 , of the following nonlinear boundary-value problem:

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“…The ease of measurement requirement suggests that the data be measured on the boundary. ICPs for partial differential equations have been studied by many authors (see [4,5,7,8,12,17]). ICPs for elliptic hemivariational inequalities were considered by Migorski and Ochal [14,15].…”

Section: Introductionmentioning

confidence: 99%

Inverse coefficient problems for elliptic hemivariational inequalities

Xiao

Liu

2010

Chin. Ann. Math. Ser. B

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This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.

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Inverse coefficient problems for nonlinear parabolic differential equations

Cited by 31 publications

References 14 publications

An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

Numerical solution of the nonlinear parabolic problem related to inverse elastoplastic torsional problem

Inverse coefficient problems for elliptic hemivariational inequalities

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