The two-dimensional heat equation with nonlocal boundary conditions

Ionkin, N. I.; Morozova, V. A.

doi:10.1007/bf02754498

Cited by 22 publications

(28 citation statements)

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“…Problem – has nonlocal boundary conditions and its spectral problem, that is, the problem –, is nonself‐adjoint. In , a biorthogonal system of functions is constructed

\begin{matrix} leftalign-star \\ rightalign-odd MathClass-open{\underset{↓}{\underset{︸}{Z_{0 k} : = X_{0} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}}, \underset{↓}{\underset{︸}{Z_{MathClass-open(2 m - 1 MathClass-close) k} : = X_{m} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}}, \underset{↓}{\underset{︸}{Z_{2 mk} : = X_{2 mk} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}} MathClass-close}, & align-even & rightalign-label \\ rightalign-odd MathClass-open{W_{0 k} : = {X ̃}_{0} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close), W_{MathClass-open(2 m - 1 MathClass-close) k} : = {X ̃}_{m} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close), W_{2 mk} : = {X ̃}_{2 mk} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close) MathClass-close}... \end{matrix}

…”

Section: The Direct Problemmentioning

confidence: 99%

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

Kirane

Malik

Al-Gwaiz

2012

Math Methods in App Sciences

118

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We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L2[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Problem – has nonlocal boundary conditions and its spectral problem, that is, the problem –, is nonself‐adjoint. In , a biorthogonal system of functions is constructed

\begin{matrix} leftalign-star \\ rightalign-odd MathClass-open{\underset{↓}{\underset{︸}{Z_{0 k} : = X_{0} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}}, \underset{↓}{\underset{︸}{Z_{MathClass-open(2 m - 1 MathClass-close) k} : = X_{m} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}}, \underset{↓}{\underset{︸}{Z_{2 mk} : = X_{2 mk} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close)}} MathClass-close}, & align-even & rightalign-label \\ rightalign-odd MathClass-open{W_{0 k} : = {X ̃}_{0} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close), W_{MathClass-open(2 m - 1 MathClass-close) k} : = {X ̃}_{m} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close), W_{2 mk} : = {X ̃}_{2 mk} MathClass-open(x MathClass-close) V_{k} MathClass-open(y MathClass-close) MathClass-close}... \end{matrix}

…”

Section: The Direct Problemmentioning

confidence: 99%

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

Kirane

Malik

Al-Gwaiz

2012

Math Methods in App Sciences

118

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“…If √ λ = q = α + iβ, β = 0, then the general solution of Eq. (4) satisfying the boundary condition (5) can be represented in the form (7) by Lemma 1. By substituting the expression (7) into the nonlocal condition (6), we obtain C(sin q − γ sin qξ) = 0.…”

Section: Investigation Of the One-dimensional Problemmentioning

confidence: 99%

“…Eigenvalue problems for differential operators with a nonlocal condition are a relatively new trend in this field. The papers [4][5][6][7][8][9] study the eigenvalue problem for a one-or two-dimensional differential operator with special nonlocal conditions that involve only boundary points. The eigenvalue problem with a nonlocal Bitsadze-Samarskii condition was considered in [10,11], and a problem with an integral condition was studied in [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

On the Structure of the Spectrum of a Differential Operator with a Nonlocal Condition

Sapagovas¹,

Štikonas²

2005

Diff Equat

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“…In [91], he investigate stability FDS for a parabolic equation with NBCs u(0, t) = 0, u x (1, t) = u x (0, t)…”

Section: Introductionmentioning

confidence: 99%

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

Štikonas

2014

NAMC

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The two-dimensional heat equation with nonlocal boundary conditions

Cited by 22 publications

References 0 publications

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

On the Structure of the Spectrum of a Differential Operator with a Nonlocal Condition

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

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