• Title/Summary/Keyword: Singular Integral Equations

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EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SINGULAR SYSTEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • Wang, Lin;Lu, Xinyi
    • Journal of applied mathematics & informatics
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    • v.31 no.5_6
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    • pp.877-894
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    • 2013
  • In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green's function, a nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii's fixed point theorem in a cone. Some examples are included to show the applicability of our results.

ON THE CONVERGENCE OF QUADRATURE RULE FOR SINGULAR INTEGRAL EQUATIONS

  • KIM, SEKI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.4 no.2
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    • pp.85-97
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    • 2000
  • A quadrature rule for the solution of Cauchy singular integral equation is constructed and investigated. This method to calculate numerically singular integrals uses classical Jacobi quadratures adopting Hunter's method. The proposed method is convergent under a reasonable assumption on the smoothness of the solution.

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Numerical solution of singular integral equation for multiple curved branch-cracks

  • Chen, Y.Z.;Lin, X.Y.
    • Structural Engineering and Mechanics
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    • v.34 no.1
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    • pp.85-95
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    • 2010
  • In this paper, numerical solution of the singular integral equation for the multiple curved branch-cracks is investigated. If some quadrature rule is used, one difficult point in the problem is to balance the number of unknowns and equations in the solution. This difficult point was overcome by taking the following steps: (a) to place a point dislocation at the intersecting point of branches, (b) to use the curve length method to covert the integral on the curve to an integral on the real axis, (c) to use the semi-open quadrature rule in the integration. After taking these steps, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. In addition, accurate results for the stress intensity factors (SIFs) at crack tips have been found in a numerical example. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

THE DISCRETE SLOAN ITERATE FOR CAUCHY SINGULAR INTEGRAL EQUATIONS

  • KIM, SEKI
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.2 no.2
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    • pp.81-95
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    • 1998
  • The superconvergence of the Sloan iterate obtained from a Galerkin method for the approximate solution of the singular integral equation based on the use of two sets of orthogonal polynomials is investigated. The discrete Sloan iterate using Gaussian quadrature to evaluate the integrals in the equation becomes the Nystr$\ddot{o}$m approximation obtained by the same rules. Consequently, it is impossible to expect the faster convergence of the Sloan iterate than the discrete Galerkin approximation in practice.

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A NOTE ON THE SOLUTION OF A NONLINEAR SINGULAR INTEGRAL EQUATION WITH A SHIFT IN GENERALIZED HOLDER SPACE

  • Argyros, Ioannis K.
    • The Pure and Applied Mathematics
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    • v.14 no.4
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    • pp.279-282
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    • 2007
  • Using the center instead of the Lipschitz condition we show how to provide weaker sufficient convergence conditions of the modified Newton Kantorovich method for the solution of nonlinear singular integral equations with Curleman shift (NLSIES). Finer error bounds on the distances involved and a more precise information on the location of the solution are also obtained and under the same computational cost than in [1].

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A NOTE ON THE SOLUTION OF A NONLINEAR SINGULAR INTEGRAL EQUATION WITH A SHIFT IN GENERALIZED $H{\ddot{O}}LDER$ SPACE

  • Argyros, Ioannis K.
    • East Asian mathematical journal
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    • v.23 no.2
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    • pp.257-260
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    • 2007
  • Using the center instead of the Lipschitz condition we show how to provide weaker sufficient convergence conditions of the modified Newton Kantorovich method for the solution of nonlinear singular integral equations with Curleman shift (NLSIES). Finer error bounds on the distances involved and a more precise information on the location of the solution are also obtained and under the same computational cost than in [1].

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Existence and Non-Existence of Positive Solutions of BVPs for Singular ODEs on Whole Lines

  • LIU, YUJI;YANG, PINGHUA
    • Kyungpook Mathematical Journal
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    • v.55 no.4
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    • pp.997-1030
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    • 2015
  • This paper is concerned with integral type boundary value problems of second order singular differential equations with quasi-Laplacian on whole lines. Sufficient conditions to guarantee the existence and non-existence of positive solutions are established. The emphasis is put on the non-linear term $[{\Phi}({\rho}(t)x^{\prime}(t))]^{\prime}$ involved with the nonnegative singular function and the singular nonlinearity term f in differential equations. Two examples are given to illustrate the main results.

Theoretical analysis of overlay resisting crack propagation in old cement concrete pavement

  • Pan, Baofeng;Gao, Yuanyuan;Zhong, Yang
    • Structural Engineering and Mechanics
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    • v.52 no.4
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    • pp.829-841
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    • 2014
  • The main purpose of this study is to determine the effect of overlay on the crack propagation. In order to simplify the problem, a cement concrete pavement is modeled as an elastic plate on Winkler foundation. To derive the singular integral equations, the Fourier transform and dislocation density function are used. Lobatto-Chebyshev integration formula, as a numerical method, is used to solve the singular integral equations. The numerical solution of stress intensity factor at the crack tip is derived. In order to examine the effect of overlay for resisting crack propagation, numerical analyses are carried out for a cement concrete pavement with an embedded crack and a concrete pavement with an asphalt overlay. Results show the significant factors that influence the crack propagation.