Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

Kasemets, Kairi; Janno, Jaan

doi:10.1155/2013/297104

Cited by 4 publications

(3 citation statements)

References 26 publications

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“…In many cases equations that describe propagation of electrodynamic and elastic waves with integral convolution are reduced to one second-order hyperbolic integro-differential equation. Various problems of recovering the kernel of convolution integral in these equations were investigated [1][2][3][4][5][6][7][8][9][10][11][12]. Determination of time-and space-dependent kernels in parabolic integro-differential equations with several additional conditions was considered by many authors (see e.g., [13][14][15][16][17][18][19][20][21][22][23]).…”

Section: Introduction Formulation Of Problemmentioning

confidence: 99%

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Durdiev

Nuriddinov

2021

Journal of Siberian Federal University. Mathematics &Amp; Physics

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The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

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Section: Introduction Formulation Of Problemmentioning

confidence: 99%

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Durdiev

Nuriddinov

2021

Journal of Siberian Federal University. Mathematics &Amp; Physics

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“…In contrast, inverse problems involve the identification of the cause from the knowledge of the effect. Difficulties encountered in the solution of inverse heat conduction problems should be recognized, as, in general, they are ill-posed, [1,2], and many efficient methods have been developed in the past to solve a wide range of parabolic inverse problems, see [3][4][5][6][7][8][9] to mention only a few.…”

Section: Introductionmentioning

confidence: 99%

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

Cao

Lesnic

2018

Numerical Methods Partial

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In this article, we consider coefficient identification problems in heat transfer concerned with the determination of the space‐dependent perfusion coefficient and/or thermal conductivity from interior temperature measurements using the conjugate gradient method (CGM). We establish the direct, sensitivity and adjoint problems and the iterative CGM algorithm which has to be stopped according to the discrepancy principle in order to reconstruct a stable solution for the inverse problem. The Sobolev gradient concept is introduced in the CGM iterative algorithm in order to improve the reconstructions. The numerical results illustrated for both exact and noisy data, in one‐ and two‐dimensions for single or double coefficient identifications show that the CGM is an efficient and stable method of inversion.

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“…Often, in cases of equations describing the propagation of electrodynamic and elastic waves with integral convolution terms are reduced to one second-order hyperbolic integro-differential equation. One-and multidimensional problems of recovering the kernel of convolution integral in these equations were investigated in [5]- [24] (see, also references therein). The numerical solutions for kernel determination problems from integro-differential equations were considered in the works [25]- [27].…”

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confidence: 99%

Uniqueness for multidimensional kernel determination problems from a parabolic integro-differential equation

Durdiev

Nuriddinov

2020

Preprint

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We study two problems of determining the kernel of the integral terms in a parabolic integro-differential equation. In the first problem the kernel depends on time t and x = (x 1 , . . . , xn) spatial variables in the multidimensional integro-differential equation of heat conduction. In the second problem the kernel it is determined from one dimensional integro-differential heat equation with a time-variable coefficient of thermal conductivity. In both cases it is supposed that the initial condition for this equation depends on a parameter y = (y 1 , . . . , yn) and the additional condition is given with respect to a solution of direct problem on the hyperplanes x = y. It is shown that if the unknown kernel has the form k(x, t) =[?] i=o N a i (x)b i (t), then it can be uniquely determined.

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Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

Cited by 4 publications

References 26 publications

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

Uniqueness for multidimensional kernel determination problems from a parabolic integro-differential equation

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