scholarly journals Interpolation rational integral fraction of nth order on a continuum set of nodes

2021 ◽  
pp. 101-105
Author(s):  
Ihor Demkiv ◽  
Yaroslav Baranetskyi ◽  
Halyna Berehova
Keyword(s):  
Integral Equation ◽  
Volterra Equation ◽  
Standard Form ◽  
Integral Volterra Equation ◽  
Substitution Rule ◽  
Rational Integral

The paper constructs and investigates an integral rational interpolant of the nth order on a continuum set of nodes, which is the ratio of a functional polynomial of the first degree to a functional polynomial of the (n-1)th degree. Subintegral kernels are determined from the corresponding continuum conditions. Additionally, we obtain an integral equation to determine the kernel of the numerator integral. This integral equation, using elementary transformations, is reduced to the standard form of the integral Volterra equation of the second kind. Substituting the obtained solution into expressions for the rest of the kernels, we obtain expressions for all kernels included in the integral rational interpolant. Then, in order for a rational functional of the nth order to be interpolation on continuous nodes, it is sufficient for this functional to satisfy the substitution rule. Note that the resulting interpolant preserves any rational functional of the obtained form.

LINEAR INTEGRAL VOLTERRA EQUATION OF THE FIRST KIND IN THE BANACH SPACE

2015 ◽  
Vol 4 (2) ◽  
pp. 8-13
Author(s):  
Зенина ◽  
V. Zenina ◽  
Сапронов ◽  
Ivan Sapronov ◽  
Уточкина ◽  
...  
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Volterra Equation ◽  
Polynomial Kernel ◽  
Integral Volterra Equation ◽  
Integrable Functions ◽  
Linear Integral ◽  
Space Of Integrable Functions

We construct solutions to a singular Volterra integral equation of the first kind with а polynomial kernel in the space of integrable functions whose values be-long to a Banach space.

scholarly journals On the existence and uniqueness of solution to Volterra equation on a time scale

2019 ◽  
Vol 27 (3) ◽  
pp. 177-194
Author(s):  
Bartłomiej Kluczyński
Keyword(s):  
Integral Equation ◽  
Sobolev Space ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Volterra Equation ◽  
Uniqueness Of Solution ◽  
Inversion Theorem ◽  
T Tau

AbstractUsing a global inversion theorem we investigate properties of the following operator\matrix{\matrix{ V(x)( \cdot ): = {x^\Delta }( \cdot ) + \int_0^ \cdot {v\left( { \cdot ,\tau ,x,\left( \tau \right)} \right)} \Delta \tau , \hfill \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x(0) = 0, \hfill \cr}\cr {} \cr }in a time scale setting. Under some assumptions on the nonlinear term v we then show that there exists exactly one solution {x_y} \in W_{\Delta ,0}^{1,p}\left( {{{[0,1]}_\mathbb{T}},{\mathbb{R}^N}} \right) to the associated integral equation\left\{ {\matrix{{{x^\Delta }(t) + \int_0^t {v\left( {t,\tau ,x\left( \tau \right)} \right)} \Delta \tau = y(t)\,\,\,for\,\Delta - a.e.\,\,\,t \in {{[0.1]}_\mathbb{T}},} \cr {x(0) = 0,} \cr } } \right.which is considered on a suitable Sobolev space.

scholarly journals To the solution of the singular inhomogeneous integral Volterra equation

2017 ◽  
Vol 86 (2) ◽  
pp. 20-31
Author(s):  
T.N. Bekjan ◽  
◽  
M.T. Jenaliyev ◽  
S.A. Iskakov ◽  
M.I. Ramazanov ◽  
...  
Keyword(s):  
Volterra Equation ◽  
Integral Volterra Equation

On the Eliminant of .

1897 ◽  
Vol 21 ◽  
pp. 360-368 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Integral Equation ◽  
Rational Integral

1. In a paper “On the Existence of a Root of a Rational Integral Equation,” published in the Proc. Lond. Math. Soc., xxv. pp. 173–184, the author, Professor E. B. Elliott, says (p. 184) that it is unfortunate, for the simplicity of the argument of his paper, that a proof of a certain property of this eliminant, viz., that when two linear factors have been withdrawn from it, there is left a perfect square—“is one which direct algebraical methods have not yet supplied.”In the course of the following year the want referred to received attention, a demonstration being given by Mr W. W. Taylor, in a paper entitled “Evolution of a certain Dialytic Determinant,” which was read before the same Society (see Proc. Lond. Math. Soc., xxvii. pp. 60–66).I purpose here giving another demonstration, which I think has the merit of bringing out more clearly the character of the constitution of the eliminant, and in which is followed, at the same time, that direct and expeditious course most likely to be taken by a student familiar with the theory of determinants.

scholarly journals Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Szymon Dudek ◽  
Leszek Olszowy
Keyword(s):  
Banach Space ◽  
Hausdorff Distance ◽  
Continuous Dependence ◽  
Volterra Equation ◽  
Fréchet Space ◽  
Frechet Space ◽  
Integral Volterra Equation ◽  
Quadratic Integral ◽  
Continuous Dependence Of Solutions

We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Fréchet space.

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