scholarly journals On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 468
Author(s):  
Marián Slodička
Keyword(s):  
Boundary Conditions ◽  
Semigroup Theory ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Semilinear Parabolic Equation ◽  
Point Boundary ◽  
Constructive Algorithm ◽  
Transient Problem ◽  
Steady State Problem ◽  
Semilinear Parabolic

This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms.

scholarly journals Quasilinearization for some nonlocal problems

1993 ◽  
Vol 6 (2) ◽  
pp. 117-122
Author(s):  
Yunfeng Yin
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Quadratic Convergence ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Semilinear Parabolic Equation ◽  
Monotone Sequence ◽  
Nonlocal Problems ◽  
Semilinear Parabolic

The method of generalized quasilinearization [4] is applied to study semilinear parabolic equation ut−Lu=f(t,x,u) with nonlocal boundary conditions u(t,x)=∫Ωϕ(x,y)u(t,y)dy in this paper. The convexity of f in u is relaxed by requiring f(t,x,u)+Mu2 to be convex for some M>0. The quadratic convergence of monotone sequence is obtained.

scholarly journals INVESTIGATION OF SPECTRUM CURVES FOR A STURM-LIOUVILLE PROBLEM WITH TWO-POINT NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 37-52
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Point Boundary ◽  
Sturm Liouville

The article investigates the Sturm–Liouville problem with one classical and another nonlocal two-point boundary condition. We analyze zeroes, poles and critical points of the characteristic function and how the properties of this function depend on parameters in nonlocal boundary condition. Properties of the Spectrum Curves are formulated and illustrated in figures for various values of parameter ξ.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

Exact Nonlocal Boundary Conditions in the Theory of Dielectric Waveguides

PIERS Online ◽  
2009 ◽  
Vol 5 (5) ◽  
pp. 436-440
Author(s):  
Rafail Z. Dautov ◽  
Evgeny M. Karchevskiy
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Dielectric Waveguides

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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