On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution

Turmetov, Batirkhan; Karachik, V. V.

doi:10.3390/sym13101781

Cited by 10 publications

(14 citation statements)

References 34 publications

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“…Let us show that this function satisfies all the conditions of Problem P. Indeed, applying the operator −∆ to function (16), we obtain…”

Section: Problem P For the Casementioning

confidence: 88%

“…If condition (21) is fulfilled, the solution to problem ( 19) -(20) exists and is represented in the form (22). Substituting the value of the function v(x) into the right side of equality (16), as in the case k = 1, we can show that the resulting function u(x) satisfies all the conditions of Problem P. Finally, if we use equality (22), we obtain representation (18) for the function u(x). The theorem is proved.…”

Section: Problem P In the Case K =mentioning

confidence: 93%

“…Further, in [3], [4] for a nonlocal Laplace operator with involutively transformed arguments in rectangular domains, problems of the Cauchy and Dirichlet types were studied. In a more general case, the main boundary value problems for nonlocal analogues of the Poisson equation and spectral questions for the nonlocal Laplace operator were studied in [1], [5], [8], [16], [18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Periodic Problems for the Nonlocal Poisson Equation in the Circle

Turmetov,

Koshanova,

Muratbekova

2023

Int. J. of Appl. Math.

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This work is devoted to the study of the methods of solving the boundary value problem with transformed arguments. In the studied problems, the arguments are transformed by mapping the type of involution. Moreover, these mappings are used both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analogue of the Poisson equation. The boundary conditions are specified as a relation between the value of the desired function in the upper semicircle and its value in the lower semicircle. Two types of boundary conditions are considered. They generalize the known periodic and antiperiodic conditions for circular regions. When solving the main problems for the classical Poisson equation, auxiliary problems are obtained. Using well-known assertions for these auxiliary problems, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the studied problems are found.

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“…Let us show that this function satisfies all the conditions of Problem P. Indeed, applying the operator −∆ to function (16), we obtain…”

Section: Problem P For the Casementioning

confidence: 88%

Section: Problem P In the Case K =mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Periodic Problems for the Nonlocal Poisson Equation in the Circle

Turmetov,

Koshanova,

Muratbekova

2023

Int. J. of Appl. Math.

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“…Note that the properties and applications of nonlocal elliptic operators L 4 are studied in [28][29][30].…”

Section: Introductionmentioning

confidence: 99%

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

et al. 2023

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The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved.

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“…The correctness of boundary and initial-boundary value problems for differential equations with various types of involution, qualitative properties of their solution, as well as their spectral issues were quite well studied in [25][26][27][28]. Spectral problems for the second-order differential operator were studied in [26,27].…”

Section: Introductionmentioning

confidence: 99%

On Unique Solvability of a Multipoint Boundary Value Problem for Systems of Integro-Differential Equations with Involution

2022

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In this paper, a multipoint boundary value problem for systems of integro-differential equations with involution has been studied. To solve the studied problem, the parameterization method is used. Based on the parametrization method, the studied problem is decomposed into two parts, i.e., into the Cauchy problem and a system of linear equations. Necessary and sufficient conditions for the unique solvability of the studied problem are determined.

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On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution

Cited by 10 publications

References 34 publications

On Periodic Problems for the Nonlocal Poisson Equation in the Circle

On Periodic Problems for the Nonlocal Poisson Equation in the Circle

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

On Unique Solvability of a Multipoint Boundary Value Problem for Systems of Integro-Differential Equations with Involution

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