scholarly journals On the domain of selfadjoint extension of the product of Sturm-Liouville differential operators

2003 ◽  
Vol 2003 (11) ◽  
pp. 695-709
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Selfadjoint Extension ◽  
Sesquilinear Form ◽  
Sturm Liouville

The second-order symmetric Sturm-Liouville differential expressionsτ1,τ2,…,τnwith real coefficients are considered on the intervalI=(a,b),−∞≤a<b≤∞. It is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and it is an exact parallel of the regular case. This characterization is an extension of those obtained by Everitt and Zettl (1977), Hinton, Krall, and Shaw (1987), Ibrahim (1999), Krall and Zettl (1988), Lee (1975/1976), and Naimark (1968).

scholarly journals On the product of self-adjoint Sturm-Liouville differential operators in direct sum spaces

2006 ◽  
Vol 37 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Direct Sum ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1,\tau_2, \ldots, \tau_n $, with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $ L_w^2 $-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6], [7], [8], [9], [12], [14] and [15].

scholarly journals On the boundary conditions for products of Sturm-Liouville differential operators

2001 ◽  
Vol 32 (3) ◽  
pp. 187-199
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green&rsquo;s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

scholarly journals Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular C-Symmetric Differential Operators in Direct Sum Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qinglan Bao ◽  
Xiaoling Hao ◽  
Jiong Sun
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Direct Sum ◽  
Even Order ◽  
Special Case ◽  
Symmetric Differential Operators

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular C-symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

Nonlinear boundary conditions for nonlinear second order differential operators on C[0,1]

2001 ◽  
Vol 76 (5) ◽  
pp. 391-400 ◽  
Author(s):  
A. Favini ◽  
G. R. Goldstein ◽  
J. A. Goldstein ◽  
S. Romanelli
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Differential Operators ◽  
Nonlinear Boundary Conditions ◽  
Second Order

scholarly journals A CHARACTERIZATION OF SECOND-ORDER DIFFERENTIAL OPERATORS ON FINITE ELEMENT SPACES

2004 ◽  
Vol 14 (12) ◽  
pp. 1881-1892 ◽  
Author(s):  
SNORRE H. CHRISTIANSEN
Keyword(s):  
Finite Element ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Second Order ◽  
Unified Framework ◽  
Regge Calculus ◽  
Edge Elements ◽  
Computational Electromagnetism ◽  
Linear Differential

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

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