Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral  overdetermination condition

Azizbayov, Elvin I.; Mehraliyev, Yashar T.

doi:10.15330/cmp.12.1.23-33

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…Numerous authors have studied the two-dimensional heat equation inverse problems in great detail; for instance. In references [1][2][3][4][5][6], existence and uniqueness of solution have been discussed by many researchers using different methods. While, numerical solutions were also discussed by some researchers as, in reference [7] considered method of fundamental solutions, researchers [8] applied a meshless method.…”

Section: Introductionmentioning

confidence: 99%

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Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

Qahtan,

Hussein,

Ahsan

et al. 2023

MMEP

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This study delves into the nonlocal inverse boundary-value problem for a second-order, two-dimensional parabolic equation within a rectangular domain. The primary focus is to identify the unknown coefficient and propose a resolution to the problem. The second-order, two-dimensional convection equation is addressed through the direct application of the alternating direction explicit (ADE) finite difference scheme. An adaptation of the ADE scheme is formulated to accommodate mixed boundary conditions, utilizing suitable expressions at the boundaries. Furthermore, unconditional stability is scrutinized through a series of examples. Each ADE scheme typically comprises two substeps, known as upward and downward sweeps, during which values computed at the new time level are incorporated into the discretization template. The inverse problem is restructured into a nonlinear regularized least-square optimization problem, with a defined boundary for the unknown factor, and is effectively resolved using the MATLAB subroutine lsqnonlin from the optimization toolbox. Given the typically ill-posed nature of the problem under investigation, where minor errors in the input data can significantly affect the output, Tikhonov's regularization technique is employed to produce stable and regularized results.

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Section: Introductionmentioning

confidence: 99%

“…By using two different methods of investigation into stability were used, the matrix and the von-Neumann [25,26]. The inverse problem in this paper has been shown to be uniquely solvable in reference [1]. As no numerical solution has been proposed yet, the main aim of this work is to find such a solution.…”

Section: Introductionmentioning

confidence: 99%

Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

Qahtan,

Hussein,

Ahsan

et al. 2023

MMEP

View full text Add to dashboard Cite

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“…Akhundov studied the well-posedness of the inverse problem for determining the unknown coefficient of higher derivatives of a quasilinear parabolic equation of divergent type in a multidimensional domain in his work [1]. In [4], [5], the identification of the unknown lowest coefficient and the righthand side in a second-order parabolic equation with integral overdetermination conditions is studied, and sufficient conditions for the existence and uniqueness of the classical solution to the considered inverse problem are established. J.R. Cannon and Y.P.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Nonlinear Inverse Boundary Value Problem for a Second Order Parabolic Equation With Nonlocal Boundary Conditions

Azizbayov,

Mehraliyev

2023

BMJ

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The paper studies the classical solvability of an inverse boundary value problem for a second order parabolic equation with nonlocal boundary conditions. For this purpose, first, the considered problem is reduced to an auxiliary equivalent problem in a certain sense. Then, using the Fourier method the auxiliary problem is presented as a system of integral equations. Further, by means of the contraction mappings principle the unique existence of the solution of the obtained system of integral equations is shown. At the end of investigation the existence and uniqueness theorem for the classical solution of the original inverse boundary value problem is proved based on the equivalence of these problems

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“…Authors studied the determination of the solely time-dependent diffusion coefficient in a two-dimensional parabolic equation with different boundary and additional measurements in [2,37,40]. In the references [1,4,35], and [36], authors considered the inverse time-dependent lowest term identification problem with various classical (Dirichlet, Neumann, and Robin) and non-classical boundary conditions. On contrary of these references authors investigated the solvability of the inverse problem for both space and time-dependent coefficient in [46].…”

Section: Introductionmentioning

confidence: 99%

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Huntul

Tekin²

2023

Hacettepe Journal of Mathematics and Statistics

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In this article, simultaneous identification of the timewise lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the problem is yet ill- posed (very small errors in the time-average temperature input may cause large errors in the output source and potential functions), we need to regularize the solution. Hence, regularization is needed for the retrieval of unknown terms. The 2D equation is computationally solved using the ADE scheme and reshaped as non-linear optimization of the Tikhonov regularization function. This is numerically studied by means of the MATLAB $lsqnonlin$ subroutine. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.

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Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition

Cited by 7 publications

References 12 publications

Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

Nonlinear Inverse Boundary Value Problem for a Second Order Parabolic Equation With Nonlocal Boundary Conditions

An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

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