On the non-local boundary-value problem for a parabolic equation

Muraveĭ, L A; Filinovskii, A. V.

doi:10.1007/bf01210424

Cited by 22 publications

(21 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Gordeziani and Avalishvili [48] discussed hyperbolic equations with nonlocal boundary conditions. Mesloub and Bouziani [30] and Muravei and Philinovskii [50] also discussed the theoretical aspects of the solutions to the one-dimensional equation. The studied equation is hyperbolic and the boundary condition is of the integral type.…”

Section: Introductionmentioning

confidence: 97%

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Dehghan

2004

Numerical Methods Partial

200

123

View full text Add to dashboard Cite

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article.

show abstract

Section: Introductionmentioning

confidence: 97%

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Dehghan

2004

Numerical Methods Partial

200

123

View full text Add to dashboard Cite

show abstract

“…representing the unique solution of problem (2). One can verify directly that this solution belongs to the space W and that the operator (f, ϕ, ψ) ∈ V → u ∈ W is continuous.…”

Section: Nonlocal Problems In the Theory Of Differential Equations 139mentioning

confidence: 95%

“…We only note that the problems in question were studied in the theory of elliptic and parabolic equations (see, e.g., [1,2]). This is not the case for hyperbolic equations, even those on the plane, although some disparate attempts were made in [3].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal problems in the theory of hyperbolic differential equations

Paneah

2009

Trans. Moscow Math. Soc.

View full text Add to dashboard Cite

Abstract. There are relatively few local problems for general hyperbolic differential equations in a bounded domain on the plane, and all these problems are well studied, and, in simple cases, are included in almost any textbook on partial differential equations. On the contrary, nonlocal problems (even more general than boundary problems) remain practically not studied, although a number of problems of this type were successfully studied in connection with elliptic or parabolic equations. In the present paper, we consider two nonlocal quasiboundary problems of sufficiently general type in the characteristic rectangle for equations of the above type. In both cases we find conditions for unique solvability and (for the first time in the theory of hyperbolic equations) the conditions for problems to be Fredholm. Examples show that these conditions are sharp: if they are violated, the resulting problems may fail to have the required solvability properties. The proofs (in their nonanalytic part) are given in the framework of perturbation theory of operators in Banach spaces.

show abstract

“…Gordeziani and Avalishvili [8] discussed hyperbolic equations with nonlocal boundary conditions. Mesloub and Bouziani [5] and Muravei and Philinovskii [9] also discussed on the theoretical aspects of the solutions to the one-dimensional equation. The studied equation is hyperbolic and the boundary condition is of integral type.…”

Section: Introductionmentioning

confidence: 97%

Numerical solution of the one‐dimensional wave equation with an integral condition

Saadatmandi

Dehghan

2006

Numerical Methods Partial

View full text Add to dashboard Cite

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this research a numerical technique is developed for the one-dimensional hyperbolic equation that combine classical and integral boundary conditions. The proposed method is based on shifted Legendre tau technique. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

show abstract

On the non-local boundary-value problem for a parabolic equation

Cited by 22 publications

References 1 publication

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Nonlocal problems in the theory of hyperbolic differential equations

Numerical solution of the one‐dimensional wave equation with an integral condition

Contact Info

Product

Resources

About