Showing papers on "Singular integral published in 1979"
Book•
15 Jan 1979
TL;DR: In this article, a rigorous theory of singular perturbations is presented. But the theory is not applicable to non-linear problems, such as linear problems with singularities on subsets of lower dimensions.
Abstract: Asymptotic Definitions and Properties. Functions with Singularities on Subsets of Lower Dimension (Boundary Layers). Matching Relations and Composite Expansions. Heuristic Analysis of Singular Perturbations. Linear Problems. Heuristic Analysis Continued. Non-linear Problems. Foundations for a Rigorous Theory of Singular Perturbations. Elliptic Singular Perturbations. Bibliography.
403 citations
TL;DR: In this article, a direct approach is employed to obtain a general formulation of plate bending problems in terms of a pair of singular integral equations involving displacement, normal slope, bending moment and shear on the plate boundary.
263 citations
257 citations
TL;DR: In this article, the Placzek lemma is used to establish a system of singular integral equations and constraints that is solved uniquely for a half-space to yield the exact exit distribution.
Abstract: The Placzek lemma is used to establish a system of singular integral equations and constraints that is solved uniquely for a half-space to yield the exact exit distribution. These singular integral...
140 citations
Book•
01 Jan 1979
82 citations
TL;DR: In this article, a new mufti-parameter singular perturbation problem is formulated and sufficient conditions for uniform asymptotic stability are derived, and the behavior of solution is investigated.
Abstract: A new mufti-parameter singular perturbation problem is formulated. Sufficient conditions for uniform asymptotic stability are derived, and asymptotic behavior of solution is investigated.
63 citations
TL;DR: In this paper, an algorithm for the approximate solution of a complete singular integral equation (with Cauchy principal value integral) taken over the arc $( - 1,1) is described.
Abstract: An algorithm is described for the approximate solution of a complete singular integral equation (with Cauchy principal value integral) taken over the arc $( - 1,1)$. The coefficients in the dominant part of this equation are not necessarily restricted to be constants so that the approximate solution of a wide class of singular integral equations is possible. No restriction is placed on the index of the equation.
54 citations
Book•
01 Oct 1979
54 citations
TL;DR: In this paper, the continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived, and it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end points.
Abstract: Continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived. The equations studied in this paper arise from e.g. potential problems or problems of radiative equilibrium. Under reasonable assumptions it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end-points.
TL;DR: In this article, the authors further investigate relationships between various conditions on singular kernels which imply continuity of the corresponding operator, and further investigate the relationship between singular kernels and their corresponding operators.
Abstract: The purpose of the paper is to further investigate relationships between various conditions on singular kernels K which imply continuity of the corresponding operator
TL;DR: In this paper, the authors considered the plane elasticity problem of an arbitrary system of curvilinear cracks in an isotropic elastic half-plane bonded to another halfplane consisting of a different elastic material and reduced to a complex Cauchy type singular integral equation along the cracks.
Abstract: The plane elasticity problem of an arbitrary system of curvilinear cracks in an isotropic elastic half-plane bonded to another half-plane consisting of a different isotropic elastic material is formulated by using the complex variable technique and reduced to a complex Cauchy type singular integral equation along the cracks. The special cases of a half-plane with a stress-free boundary and of a periodic array of curvilinear cracks are also treated. The numerical techniques available for the solution of complex Cauchy type singular integral equations are presented and a discussion on them is made. Finally, three applications to some special cases of straight or circular-arc-shaped cracks are made.
TL;DR: In this paper, the numerical solution of singular integral equations with singular integral equation with with is studied, and the authors propose a method to find the solution of the singular integral system with with.
Abstract: (1979). Numerical solution of systems of singular integral equations with. Applicable Analysis: Vol. 9, No. 1, pp. 37-52.
TL;DR: In this paper, the problem of calculating the stresses near the tip of a radial crack at the edge of a circular hole in an infinite elastic solid when the crack and the hole are loaded in an arbitrary fashion is reduced to a pair of singular integral equations which are solved numerically.
TL;DR: In this article, the Gauss-Chebyshev method was applied to two crack problems in plane isotropic elasticity and the numerical results obtained illustrate the powerfulness of the method.
Abstract: As is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.
TL;DR: In this article, 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 the kernel has also a Fredholm part presenting strong singularities when both its variables tend to the same end point of the integration interval.
TL;DR: A new class of singular integral operators arises that are infinitely smoothing in the interior and preserve singular supports at the boundary and whose kernels are products of isotropic kernels with parabolic ones.
Abstract: This paper is devoted to the derivation of an approximate formula for the kernel of the Neumann operator N on strongly pseudo-convex domains. In particular, a new class of singular integral operators arises that are infinitely smoothing in the interior and preserve singular supports at the boundary and whose kernels are products of isotropic kernels with parabolic ones.
01 Jan 1979
TL;DR: In this article, the problem of finding the stress distribution near a penny-shaped crack situated at the interface of two bonded dissimilar elastic solids is solved numerically by reducing them to a set of algebraic equations.
Abstract: The paper deals with the problem of finding the stress distribution near a penny-shaped crack situated at the interface of two bonded dissimilar elastic solids. The crack is opened by the interaction of plane harmonic longitudinal elastic wave, incident normally on the crack. The problem is first reduced to a set of simultaneous dual integral equations which are further transformed to a set of simultaneous singular integral equations. These are solved numerically by reducing them to a set of algebraic equations. The solution is used to calculate the stress-intensity factors and the size of the overlapping zones at the edge of the crack.
01 Jan 1979
TL;DR: A survey of numerical methods for solving Cauchy singular integral equations on both open and closed arcs in the plane is given in this paper, where a generalization of a Galerkin method due to Karpenko is presented.
Abstract: We present a survey of numerical methods for solving Cauchy singular integral equations on both open and closed arcs in the plane. For completeness, necessary theory is reviewed, particularly the method of regularization. For closed arcs we discuss collocation methods based on piecewise polynomial or rational representations of the solution. Emphasis here, as for the open arc case, is on regularizable equations. For open arcs a detailed discussion is given of a degenerate kernel method developed recently by Dow and Elliott. In addition to this, a generalization of a Galerkin method due to Karpenko is presented. Attention is drawn to the relation of Cauchy singular equations and solving rectangular systems of linear equations. The possibility of exploiting this for the direct solution of such equations is discussed, and some direction for future research is given.
TL;DR: In this paper, the authors considered the one-to-one property of the mapping of singular equations of the 1st and 2nd kinds, when the line of integration is a segment, and showed that the algebraic system is uniquely solvable for fairly large n, and that approximate solutions converge to the exact solution in spaces with a weight.
Abstract: THE APPROXIMATE solution of singular equations of the 1st and 2nd kinds, when the line of integration is a segment, is considered. By contraction of the domain of definition or range of values of the operator, the one-to-one property of the mapping is established. Versions of the Bubnov-Galerkin method are used for the approximation solution. Chebyshev and Jacobi polynomials are used as coordinate elements. It is shown that the algebraic system is uniquely solvable for fairly large n , and that the approximate solutions converge to the exact solution in spaces with a weight. The process is stable.
TL;DR: In this paper, the stable analytic extrapolation of the scattering amplitudes with positive imaginary parts from the whole boundary or from a part of it to interior points was investigated, using the Lagrange technique in convex optimization.
Abstract: We investigate the stable analytic extrapolation of the scattering amplitudes with positive imaginary parts from the whole boundary or from a part of it to interior points. By using the Lagrange technique in convex optimization, we derive a system of singular integral equations which completely solve the problem. In the case of the extrapolation from the whole boundary this system reduces to the Fredholm equation recently obtained by Aubersonet al., using different considerations, in connection with a related extremum problem. Our results give also the correct analytic-extrapolation formulae in the limiting case of scattering data which are exactly the restriction to the boundary of some analytic function.
TL;DR: In this article, a method for solving certain systems of generalized Abel integral equations by constructing an equivalent system of singular integral equations is presented, and an application is then given to a class of simultaneous dual relations of a type arising in bimedia fracture problems in elasticity.
Abstract: A method is presented for solving certain systems of generalized Abel integral equations by constructing an equivalent system of singular integral equations. An application is then given to a class of simultaneous dual relations of a type arising in bimedia fracture problems in elasticity. The equations discussed in this paper generalize those considered in an earlier paper of Lowengrub and Walton [SIAM J. Math. Anal., this issue, pp. 794–807].