The Lyusternik‒Sobolev lemma and the specific asymptotic stability of solutions of linear homogeneous Volterra-type integro-differential equations of order 3

Imanaliev, M. I.; Asanova, Kanykei; Iskandarov, S.

doi:10.1134/s1064562416040190

Cited by 2 publications

(2 citation statements)

References 2 publications

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“…The application of the method of characteristics to the solution of partial differential equations of the first order makes it possible to reduce the study of wave evolution [21]. In [22,23], methods for integrating nonlinear partial differential equations of the first order were developed. Further, many papers appeared devoted to the study of questions of the unique solvability of the Cauchy problem for different types of partial differential equations of the first order (see, for example, [24][25][26][27][28][29][30][31][32][33]).…”

Section: Problem Statementmentioning

confidence: 99%

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Yuldashev¹,

Fayziyev²

2023

Nanosystems: Phys. Chem. Math.

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In the article, the problems of unique solvability and determination of the redefinition coefficient function in the initial inverse problem for a nonlinear Whitham type partial differential equation with impulse effects are studied. The modified method of characteristics allows partial differential equations of the first order to be represented as ordinary differential equations that describe the change of unknown function along the line of characteristics. The unique solvability of the initial inverse problem is proved by the method of successive approximations and contraction mappings. The determination of the unknown coefficient is reduced to solving the nonlinear integral equation. KEYWORDS inverse problem, Whitham type equations, determination of the coefficient function, method of successive approximations, unique solvability.

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Section: Problem Statementmentioning

confidence: 99%

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Yuldashev¹,

Fayziyev²

2023

Nanosystems: Phys. Chem. Math.

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“…Note that the equation (0.1) is an integro-differential equation with a heat operator on the left and a Volterra-convolutional integral on the right sides. To get acquainted with the issues of solvability of various problems for integro-differential equations with Volterra-non-convolutional integrals, we refer the reader to the papers [22,23] (also see the literature in them).…”

Section: Introductionmentioning

confidence: 99%

On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

Durdiev

Nuriddinov

2020

Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki

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The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

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The Lyusternik‒Sobolev lemma and the specific asymptotic stability of solutions of linear homogeneous Volterra-type integro-differential equations of order 3

Cited by 2 publications

References 2 publications

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

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