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# On the solution of integral equations with strongly singular kernels

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• Integral equations.

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The Physical Object ID Numbers Statement A.C. Kaya and F. Erdogan. Series NASA contractor report -- NASA CR-176687. Contributions Erdogan, F., United States. National Aeronautics and Space Administration. Format Microform Pagination 1 v. Open Library OL15390614M

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Get this from a library. On the solution of integral equations with strongly singular kernels. [A C Kaya; F Erdogan; United States. National Aeronautics and Space Administration.]. Get this from a library. On the solution of integral equations with strongly singular kernels.

[A C Kaya; F Erdogan; Langley Research Center.]. Solution of integral equations. Let us now assume that the mixed boundary value problem is reduced to the following one-dimensional integral equation: fb [ks(t,x) + k(t,x)\f(t) dt = g(x) (a x), (41) J a where the kernel k is square integrable in [a,b] and g is a known bounded function.

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations.

A mixed boundary value problem from the theory of elasticity is considered as an by: On the solution of integral equations of the ﬁrst kind with singular kernels of Cauchy-type G.

Okecha, C. Onwukwe Department of Mathematics, Statistics and Computer Science University of Calabar PMB Calabar Cross River State, Nigeria email: [email protected], [email protected] (Received JAccepted September Abstract. This paper deals with numerical solution of a singular integral equation of the second kind with special singular kernel function.

The numerical solution in this paper is based on Nystrom method. The Nystrom method is based on approximation of the integral in equation by numerical integration rule.

Convergence of the numerical solution is shown. Brunner H. () The numerical solution of integral equations with weakly singular kernels. In: Griffiths D.F. (eds) Numerical Analysis.

Lecture Notes in Mathematics, vol Cited by: Considering the weakly singular kernel linear Volterra integral equation u(t) + t integraldisplay 0 1 √ t − s u(s)ds = f(t), 0lessorequalslantt lessorequalslant1, () where f(x)= sinπx + √ 2[−cosπxS(√ 2x)+ sinπxC(√ 2x)], S(x) = integraltext x 0 cos(πt 2 2)dt, C(x) = integraltext x 0 sin(πt 2 2)dt and u(x) = sin(πx) is the exact solution of by: A Survey on Solution Methods for Integral Equations If u1(x) and u2(x) are both solutions to the integral equation, then c1u1(x) + c2u2(x) is also a solution.

The Kernel K(x;t) is called the kernel of the integral equation. The equation is called singular if:File Size: KB. strongly singular integral can be and efficient treatment for the singular integral kernels related to the Green's function. integral equation/moment method solution approach with non-free.

This is the first book to present theory, construction, and application of Liapunov functionals for integral equations with singular kernels. The study covers equations with kernels that are either singular, continuous, differentiable, or sums of these : T.

Burton. Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. Global convergence estimates are derived and a collection of numerical results is given. Key words: Volterra integral equation, weakly singular kernel, boundary singular-ity, collocation method.

These equations arise from the formulation of the mixed boundary value problems in applied physics and engineering. In particular, they play an important role in the solution of a great variety of contact and crack problems in solid mechanics.

In the first group of integral equations the kernels have a simple Cauchy-type by: Solution Methods for Integral Equations It seems that you're in USA.

We have a The Approximate Solution of Singular Integral Equations. Boundary and Initial-Value Methods for Solving Fredholm Equations with Semidegenerate Kernels. Pages Golberg, M. : Springer US. On the solution of integral equations with strongly singular kernels.

Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from Author: A.

Kaya and F. Erdogan. Delves, L.M., and Mohamed, J.L.Computational Methods for Integral Equations (Cam-bridge, U.K.: Cambridge University Press).

[2] Integral Equations with Singular Kernels Many integral equations have singularities in either the kernel or the solution or both. A simple quadrature method will show poor convergence with N if such.

Linear Integral Equations: Theory and Technique is an chapter text that covers the theoretical and methodological aspects of linear integral equations. After a brief overview of the fundamentals of the equations, this book goes on dealing with specific integral equations with separable kernels and a method of successive approximations.

solution of integral equations with strongly singular kernels is the objective. Numerical examples of the application of the Gauss-Chebyshev rule to some plane and axisymmetric crack problems are given. Introduction. The problem of numerical quadrature for the integrals with kernels. "This book is an excellent introductory text for students, scientists, and engineers who want to learn the basic theory of linear integral equations and their numerical solution." (Math.

Reviews, ) "This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a Brand: Springer-Verlag New York. This chapter discusses singular integral equations, Cauchy principal value for integrals, and the solution of the cauchy-type singular integral equation.

A kernel of the form [K(s,t) = cot(t-s)/2. () Solution of a class of Volterra integral equations with singular and weakly singular kernels. Applied Mathematics and Computation() High-order collocation methods for singular Volterra functional equations of neutral by: Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points.

In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded by: 1.

In this lecture, we discuss a method to find the solution of a singular integral equation i.e. an integral equation in which the range of integration if infinite or in which the kernel becomes. Integral Equations and their Applications WITeLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual.

Abstract: In this paper, we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a $\theta$-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both.

ON THE SOLUTION OF INTEGRAL EQUATIONS WITH STRONGLY SINGULAR KERNELS by A.C. Kaya and F. Erdogan Lehigh University, Bethlehem, PA Abstract In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x)-m, m>1.

Interpreting the integrals. In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator () = ∫ (,) (),whose kernel function K: R n ×R n → R is singular along the diagonal x = ically, the singularity is such that |K(x, y)| is of size |x − y| −n asymptotically.

[12] K.E., Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimension, in M., Golberg (ed.), Numerical Solution of Integral Equations, Plenum Press, New York, Cited by:   [7] Bechiars, J., “ Weakly singular integral equations, smoothness properties and numerical solution ”, J.

of Integral Equations 2 (), – [8] Borer, D., “Approximate solution of Fredholm integral equations of the second kind with singular kernels”, Thesis, Oregon State University, Corvallis, A direct function theoretic method is employed to solve certain weakly singular integral equations arising in the study of scattering of surface water waves by vertical barriers with gaps.

Such integral equations possess logarithmically singular kernel, and a direct function theoretic method is shown to produce their solutions involving singular integrals of similar types instead of the Author: Sudeshna Banerjea, Barnali Dutta, A.

Chakrabarti. Project Euclid - mathematics and statistics online. E.O. Tuck, Application and solution of Cauchy singular integral equations, in The application and numerical solution of integral equations (R.S. Anderssen, F.R. de Hoog and M.A.

Lukas, eds.), Sijthoff and. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical by: J.

Banaś and B. Rzepka, “On existence and asymptotic stability of solutions of a nonlinear integral equation,” Journal of Mathematical Analysis and Applications, vol. no. 1, pp. –, Cited by: 6. Singular integral equations play important roles in physics and theoretical mechanics, particularly in the areas of elasticity, aerodynamics, and unsteady aerofoil theory.

They are highly effective in solving boundary problems occurring in the theory of functions of a complex variable, potential theory, the theory of elasticity, and the theory Cited by: Here g(x,s) is called the kernel of the integral equation, f(x) is given and λ is in general a complex parameter.

In most cases it is real. And also we can assume that g(x,s) is continuous in a ≤ x,s ≤ b and f(x) is continuous in a ≤ x ≤ b. The integral equation given in (40) may be solved by using several methods.

Let us assume that yFile Size: KB. Non-Local Solution of Mixed Integral Equation with Singular Kernel. By M. Abdou, S.

Raad & W. Wahied. Alexandria University, Egypt. Abstract- In this paper, we consider a non-local mixed integral equation in position and time in the space 2. 1,1. 0, T ; T. Then, using a quadratic numerical method, we have a system of.

MT - Integral equations Introduction Integral equations occur in a variety of applications, often being obtained from a differential equation.

The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on the existence and uniqueness of the solution. Kernels of integral equations What is a kernel.

In equations (6) to (9), the function N (x,y) is called the kernel of the integral equation. Every integral equation has a kernel. Kernels are important because they are at the heart of the solution to integral equations. Analytical solutions to integral equations Example Size: 86KB.

Abstract. Singular integral equations with Cauchy kernel and piecewise--continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form (t \Gamma ø)[(t \Gamma ø) 2 \Gamma n 2 (t)" 2 ] \Gamma1, ".

0, where n(t), t 2 \Gamma, is a continuous field of unit vectors non--tangential to \Gamma. 2 N. Ioakimidis: Methods of numerical solution of singular integral equations () Ioakimidis, N.

I., Methods of numerical solution of singular integral equations with Cauchy-type kernels (in Greek). In the Proceedings of the GeneralSeminarofMathematics, Artemia-dis, N.

K., editor, Vol. 7, Academic Year –, pp. 41–. ON THE SOLUTION OF INTEGRAL EQUATIONS WITH STRONGLY SINGULAR KERNELS by A.C. Kaya and P. Erdogan Lehigh University, Bethlehem, PA Abstract In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x)~m, m>l.

Interpreting the integrals.Overview. The most basic type of integral equation is called a Fredholm equation of the first type, = ∫ (,) ().The notation follows φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function.

Note that the limits of integration are constant: this is what characterizes a Fredholm equation. A numerical method is proposed for the approximate solution of a Cauchy-type singular integral equation (or an uncoupled system of such equations) of the first or the second kind and with a generalized kernel, in the sense that, besides the Cauchy singular part, the kernel has also a Fredholm part presenting strong singularities when both its variables tend to the same end-point of the Cited by:

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