scholarly journals Functional differentiation of integral operators of special form and some questions of the inverse interpolation

2021 ◽  
Vol 57 (4) ◽  
pp. 401-416
Author(s):  
M. V. Ignatenko ◽  
L. A. Yanovich
Keyword(s):  
Special Form ◽  
Integral Operators ◽  
Functional Differentiation ◽  
Data Representation ◽  
Chebyshev System ◽  
Interpolation Theory ◽  
Explicit Formulas ◽  
Variational Derivatives ◽  
Inverse Interpolation ◽  
Derivatives Of

This article is devoted to the problem of operator interpolation and functional differentiation. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations containing the first variational derivatives of the required functional are given. For functionals defined on sets of functions and square matrices, various interpolating polynomials of the Hermitе type with nodes of the second multiplicity, which contain the first variational derivatives of the interpolated operator, are constructed. The presented solutions of the Hermitе interpolation problems are based on the algebraic Chebyshev system of functions. For analytic functions with an argument from a set of square matrices, explicit formulas for antiderivatives of functionals are obtained. The solution of some differential equations with integral operators of a special form and the first variational derivatives is found. The problem of the inverse interpolation of functions and operators is considered. Explicit schemes for constructing inverse functions and functionals, including the case of functions of a matrix variable, obtained using certain well-known results of interpolation theory, are demonstrated. Data representation is illustrated by a number of examples.

scholarly journals An Analytic and Numerical Investigation of a Differential Game

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 66
Author(s):  
Aviv Gibali ◽  
Oleg Kelis
Keyword(s):  
Differential Game ◽  
Equation Approach ◽  
Linear Quadratic ◽  
Control Cost ◽  
Dual Representation ◽  
Cost Functional ◽  
Variational Derivatives ◽  
Zero Sum ◽  
The Cost ◽  
Derivatives Of

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

Weight Functions for Notches: Constructive and Variational Definition

1997 ◽  
Vol 64 (2) ◽  
pp. 270-274 ◽  
Author(s):  
H. O. K. Kirchner
Keyword(s):  
Stress Singularity ◽  
Weight Functions ◽  
Variational Derivatives ◽  
Derivatives Of

Weight functions for notches or cracks, which express the intensity of the stress singularity at the tip as functionals of the loadings present, can be defined either as combination of eigenfunctions or as variational derivatives of energies. The two definitions are equivalent.

scholarly journals Some Properties of the Hermite Polynomials and Their Squares and Generating Functions

2016 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Explicit Formulas ◽  
Derivatives Of

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Federico C. Buroni ◽  
Andrés Sáez
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Position Vector ◽  
Mixed Complex ◽  
Elastic Solids ◽  
Explicit Formulas ◽  
Linear Elastic ◽  
Derivatives Of ◽  
Appl Math

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues pα (α=1,2,3) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

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