scholarly journals Boundary value problems for elliptic partial differential operators on bounded domains

2007 ◽  
Vol 243 (2) ◽  
pp. 536-565 ◽  
Author(s):  
Jussi Behrndt ◽  
Matthias Langer
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Bounded Domains ◽  
Partial Differential ◽  
Partial Differential Operators

Solving 2D boundary-value problems using discrete partial differential operators

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Marcin Jaraczewski ◽  
Tadeusz Sobczyk
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Partial Differential ◽  
Content Type ◽  
Partial Derivatives ◽  
Partial Differential Operators

Purpose Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions. Design/methodology/approach This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann. Findings An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window. Research limitations/implications The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases. Practical implications This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings. Originality/value The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

Boundary-Value Problems of a Degenerate Sobolev-Type Differential Equation

1977 ◽  
Vol 20 (2) ◽  
pp. 221-228 ◽  
Author(s):  
C. V. Pao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Sobolev Type ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Type Differential Equation

AbstractThe purpose of this paper is to study a degenerate Sobolev type partial differential equation in the form of Mut + Lu = f, where M and L are second order partial differential operators defined in a domain (0, T]×Ω in Rn+1. The degenerate property of the equation is in the sense that both M and L are not necessarily strongly elliptic and their coefficients may vanish or be negative in some part of the domain (0, T]×Ω. Two types of boundary conditions are investigated.

scholarly journals Higher order elliptic boundary value problems in spaces with homogeneous norms

1981 ◽  
Vol 31 (1) ◽  
pp. 92-113 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
A Priori ◽  
Elliptic Boundary Value ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

scholarly journals INTERVAL EXTENSION STRUCTURE BASIC TYPES OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FORTH ORDER

2018 ◽  
pp. 23-31
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Interval Function ◽  
Partial Differential ◽  
Clamped Plate ◽  
Interval Extension ◽  
Interval Functions

In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the fourth order has been developed. Used Method of R-functions for constructed coordinate sequences. Constructing interval extensions of structural formulas, we consider problems (1) on the transverse bending of thin plates and 5 problems on a plate - rigidly clamped plate, loosely supported plate, elastically fixed plates, partially rigidly clamped and partially elastically fixed plates, plates, partially rigidly clamped and partially free . For the problem, the rigidly clamped plate Formula (7) is an interval structure for solving the boundary value problem (4). Here L={▁ω ▁ψ,ω ̅▁ψ,▁ω ψ ̅,ω ̅ψ ̅ },L_1={▁ω D_1 ▁φ,ω ̅D_1 ▁φ,▁ω D_1 φ ̅,ω ̅D_1 φ ̅ } L_2={▁ω^2 ▁Φ,ω ̅^2 ▁Φ,▁ω^2 Φ ̅,ω ̅^2 Φ ̅ }, [ ▁Φ,Φ ̅ ] is an indefinite interval function. For the free-supported plate problem, a solution is obtained for the interval expansion of the structure in the form (15), (17), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval function, D_2, T_2 - differential operators of the form (11) and (12). For the problem of elastically fixed plates, a solution was obtained in the interval expansion of the structure in the form (21) - (24), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval functions, D_2,T_2,D_1- differential operators of the form (11) and (12), (3). For the problem of partially rigidly clamped and partially elastically fixed plates, a solution was obtained in the interval expansion of the structure in the form of (28), (30), (32), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ]- indefinite interval functions, D_2,T_2,D_1- differential operators of the form (11) and (12), (6). For the plate problem, partially rigidly pinched and partially free, a solution is obtained in the interval extension of the structure (40), (41), (42), [ ▁(Φ_1 ),(Φ_1 ) ̅ ], [ ▁(Φ_2 ),(Φ_2 ) ̅ ] - indefinite interval functions, D_2,T_2,D_1,D_3- differential operators of the form (11), (12), (6) and (38).

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

2020 ◽  
Vol 28 (2) ◽  
pp. 237-241
Author(s):  
Biljana M. Vojvodic ◽  
Vladimir M. Vladicic
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

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