scholarly journals Finitely-generated solutions of certain integral equations

1994 ◽  
Vol 37 (2) ◽  
pp. 325-345 ◽  
Author(s):  
D. Porter ◽  
D. S. G. Stirling
Keyword(s):  
Integral Equation ◽  
Finite Number ◽  
Recent Work ◽  
Integral Equations ◽  
Finite Rank ◽  
Integral Operators ◽  
Compact Operators ◽  
Particular Solutions ◽  
Singular Kernels ◽  
Special Case

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

scholarly journals A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel

1981 ◽  
Vol 22 (4) ◽  
pp. 431-438 ◽  
Author(s):  
G. Vainikko ◽  
P. Uba
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Collocation Methods ◽  
Piecewise Polynomial ◽  
Arbitrary Degree ◽  
Weakly Singular Kernel ◽  
Piecewise Polynomial Approximation ◽  
Collocation Points ◽  
Weakly Singular ◽  
Singular Kernels

AbstractWe construct collocation methods with an arbitrary degree of accuracy for integral equations with logarithmically or algebraically singular kernels. Superconvergence at collocation points is obtained. A grid is used, the degree of non-uniformity of which is in good conformity with the smoothness of the solution and the desired accuracy of the method.

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

1971 ◽  
Vol 323 (1553) ◽  
pp. 201-210 ◽  
Keyword(s):  
Integral Equation ◽  
Wave Equation ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Operators ◽  
Exterior Boundary ◽  
Integral Equation Methods ◽  
Singular Kernels ◽  
Single Integral

The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.

scholarly journals Solvability of Some Integral Equations in Banach Space and Their Applications to the Theory of Viscoelasticity

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Onur Alp İlhan
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Contact Problem ◽  
Compact Operator ◽  
Integral Equations ◽  
Initial Value ◽  
Volterra Type ◽  
Isolated Point ◽  
Theory Of Viscoelasticity ◽  
Special Case

An integral equation of Volterra type with additional compact operator in Banach space is considered. A special case is an integral equation of contact problem that arises in theory of viscoelasticity of mixed Fredholm and Volterra type with spectral parameter depending on time. In case the initial value of the parameter coincides with some isolated point of the spectrum of compact operator, the conditions of solvability are established.

scholarly journals SYSTEMS OF INTEGRAL EQUATIONS WITH WEIGHTED DIFFERENCE KERNELS

2004 ◽  
Vol 47 (1) ◽  
pp. 205-230
Author(s):  
D. Porter ◽  
N. R. T. Biggs
Keyword(s):  
Integral Equation ◽  
Finite Number ◽  
Integral Equations ◽  
Equation System ◽  
Subject Classification ◽  
Finite Generation ◽  
Primary 45A05 ◽  
Systems Of Integral Equations ◽  
Single Integral ◽  
Explicit Expressions

AbstractExplicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding.AMS 2000 Mathematics subject classification: Primary 45A05

scholarly journals A note on dual integral equations involving inverse associated Weber-Orr transforms

1996 ◽  
Vol 19 (1) ◽  
pp. 161-169
Author(s):  
Nanigopal Mandal ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Dual Integral Equations ◽  
Elementary Methods ◽  
Special Case

We consider dual integral equations involving inverse associated Weber-Orr transforms. Elementary methods have been used to reduce dual integral equations to a Fredholm integral equation of second kind. Some known results are obtained as special case.

Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Edyta Hetmaniok ◽  
Iwona Nowak ◽  
Damian Słota ◽  
Roman Wituła
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear Integral ◽  
Special Case ◽  
Linear Integral Equations

AbstractIn this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

scholarly journals Localized boundary-domain integral equation formulation for mixed type problems

2010 ◽  
Vol 17 (3) ◽  
pp. 469-494 ◽  
Author(s):  
Otar Chkadua ◽  
Sergey E. Mikhailov ◽  
David Natroshvili
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Integral Equations ◽  
Variable Coefficient ◽  
Mixed Type ◽  
Mixed Boundary ◽  
Integral Operators ◽  
Mixed Boundary Value Problem ◽  
Domain Integral ◽  
Integral Equation Formulation

Abstract Some modifed direct localized boundary-domain integral equations (LBDIEs) systems associated with the mixed boundary value problem (BVP) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the corresponding localized boundary-domain integral operators in appropriately chosen function spaces.

Weakly-Singular Integral Equations for Steady State Flow in Anisotropic Porous Media Containing Discontinuity Surfaces

2005 ◽  
Author(s):  
Jaroon Rungamornrat ◽  
Mary F. Wheeler ◽  
Xiuli Gai
Keyword(s):  
Integral Equation ◽  
Porous Media ◽  
Integral Equations ◽  
Fluid Pressure ◽  
Weak Form ◽  
Flow In Porous Media ◽  
Fluid Flux ◽  
Anisotropic Permeability ◽  
Weakly Singular ◽  
Singular Kernels

In this paper, we present the development of the weakly-singular, weak-form fluid pressure and fluid flux integral equations for steady state Darcy’s flow in porous media. The integral equation for fluid flux is required for the treatment of flow in a domain which contains surfaces of discontinuities (e.g. cracks and impermeable surfaces), since the pressure integral equation contains insufficient information about the fluid flux on the surface of discontinuity. In this work, a systematic technique has been established to regularize the conventional fluid pressure and fluid flux integral equations in which the pressure equation contains a Cauchy singular kernel and the fluid flux equation contains both Cauchy and strongly-singular kernels. The key step in the regularization procedure is to construct a special decomposition for the fluid velocity fundamental solution and the strongly-singular kernel such that it is well-suited for performing an integration by parts via Stokes’ theorem. These decompositions involve weakly-singular kernels where their explicit form can be constructed, for general anisotropic permeability tensors, by the integral transform method. The resulting integral equations possess several features: they contain only weakly-singular kernels of order 1/r; their validity requires only that the pressure boundary data is continuous; and they are applicable for modeling fluid flow in porous media with a general anisotropic permeability tensor. A suitable combination of these weakly-singular, weak-form integral equations gives rise to a symmetric weak-form integral equation governing the boundary valued problem, thereby forming a basis for the weakly-singular, symmetric Galerkin boundary element method (SGBEM). As a consequence of that the integral equations are weakly-singular, the SGBEM allows standard C° elements to be employed everywhere in the discretization.

scholarly journals Random compact operators

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 515-523
Author(s):  
Reza Saadati
Keyword(s):  
Integral Equations ◽  
Operator Theory ◽  
Integral Operators ◽  
Compact Operators ◽  
Random Operator ◽  
Bounded Operators ◽  
Random Integral ◽  
Definition Of

Random compact operators are useful to study random differentiation and random integral equations. In this paper, we define the random norm of R-bounded operators and study random norms of differentiation operators and integral operators. The definition of random norm of R-bounded operators led us to study the random operator theory.

Fredholm solutions of partial integral equations

1953 ◽  
Vol 49 (2) ◽  
pp. 213-217 ◽  
Author(s):  
Abdus Salam ◽  
F. Smithies
Keyword(s):  
Integral Equation ◽  
Quantum Theory ◽  
Integral Equations ◽  
Symmetric Solution ◽  
Image Position ◽  
Theory Of Fields ◽  
Special Case

The following ‘partial’ integral equation arises in the quantum theory of fields:where f(x1x2) = f(x2x1), K(x1x2, y) = K(x2x1, y), and a symmetric solution of the equation is desired. The solution of (1) will be obtained as a special case of the solution of the following equation:

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