Basis properties of a spectral problem with spectral parameter in the boundary condition

Керимов, Н. Б.; Aliev, Z. S.

doi:10.1070/sm2006v197n10abeh003808

Cited by 26 publications

(39 citation statements)

References 8 publications

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“…After that, important developments were made by Binding et al [2], Fulton [11], Kapustin and Moiseev [18], Kerimov and Allakhverdiev [19,20]. It is useful to note the reference [22] whose results on expansion in term of eigenfunctions will be used in the present paper.…”

Section: )mentioning

confidence: 99%

“…It was shown in [22] that the system of eigenfunctions {y n (x)} (n = 0, 1, 2, ...; n ̸ = n 0 ) is a Riesz basis for L 2 [0, 1]. The system {u n (x)} (n = 0, 1, 2, ...; n ̸ = n 0 ) which is biorthogonal to the system {y n (x)} (n = 0, 1, 2, ...; n ̸ = n 0 ) has the form…”

Section: Mathematical Analysismentioning

confidence: 99%

“…It was shown in [22] that the eigenvalues and eigenfunctions have the following asymptotic behaviour:…”

Section: Mathematical Analysismentioning

confidence: 99%

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An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Hazanee

Lesnic

Ismailov

et al. 2015

Applied Mathematical Modelling

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This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with a non-classical boundary and an integral over-determination conditions. The existence, uniqueness and continuous dependence upon the data of the classical solution of the inverse problem is shown by using the generalised Fourier method. Furthermore in the numerical process, the boundary element method (BEM) together with the second-order Tikhonov regularization is employed with the choice of regularization parameter based on the generalised cross-validation (GCV) criterion. Numerical results are presented and discussed.

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Section: )mentioning

confidence: 99%

Section: Mathematical Analysismentioning

confidence: 99%

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An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Hazanee

Lesnic

Ismailov

et al. 2015

Applied Mathematical Modelling

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“…Oscillation and comparison results have been obtained in [6,7,14]. Basis properties and eigenfunction expansions have been considered in [15,16,17,24]. Problems with various singularities have been analyzed in [1,12].…”

Section: Introductionmentioning

confidence: 99%

Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions

Guliyev¹

2005

Inverse Problems

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“…Basis properties of the boundary value problem (0.1)-(0.3) in cases where h is affine or bilinear have been analyzed in [5], [6], [8].…”

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confidence: 99%

The basis property in L_pof the boundary value problem rationally dependent on the eigenparameter

Керимов

Aliyev

2006

Studia Math.

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Abstract. We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in L p of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L 2 we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness in L p we use F. Riesz's theorem.Consider the spectral problemwhere λ is the spectral parameter, q is a real-valued and continuous function on the interval [0, 1],where all the coefficients are real and a ≥ 0, In a recent paper [1] existence and asymptotics of eigenvalues and oscillation of eigenfunctions of this problem were studied. It was proved that the eigenvalues of (0.1)-(0.3) are real, simple and form a sequence λ 0 < λ 1 < · · · accumulating only at ∞ and with λ 0 < c 1 . Moreover, it was proved that if 2000 Mathematics Subject Classification: 34L10, 34B24, 34L20.

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Basis properties of a spectral problem with spectral parameter in the boundary condition

Cited by 26 publications

References 8 publications

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions

The basis property in L_pof the boundary value problem rationally dependent on the eigenparameter

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Basis properties of a spectral problem with spectral parameter in the boundary condition

Cited by 26 publications

References 8 publications

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions

The basis property in Lpof the boundary value problem rationally dependent on the eigenparameter

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Product

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The basis property in L_pof the boundary value problem rationally dependent on the eigenparameter