scholarly journals Numerical methods of computation of singular and hypersingular integrals

2001 ◽  
Vol 28 (3) ◽  
pp. 127-179 ◽  
Author(s):  
I. V. Boikov
Keyword(s):  
Numerical Methods ◽  
Singular Integrals ◽  
Analytical Form ◽  
Approximate Methods ◽  
Hypersingular Integrals ◽  
Fixed Singularity

In solving numerous problems in mathematics, mechanics, physics, and technology one is faced with necessity of calculating different singular integrals. In analytical form calculation of singular integrals is possible only in unusual cases. Therefore approximate methods of singular integrals calculation are an active developing direction of computing in mathematics. This review is devoted to the optimal with respect to accuracy algorithms of the calculation of singular integrals with fixed singularity, Cauchy and Hilbert kernels, polysingular and many-dimensional singular integrals. The isolated section is devoted to the optimal with respect to accuracy algorithms of the calculation of the hypersingular integrals.

Solution of heat and mass transfer problems with non-linear coefficients

2019 ◽  
Vol 5 (4) ◽  
pp. 10-20
Author(s):  
Boris G. Aksenov ◽  
Yuri E. Karyakin ◽  
Svetlana V. Karyakina
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Numerical Methods ◽  
Heat And Mass Transfer ◽  
Unknown Function ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Maximum Deviation ◽  
Approximate Methods ◽  
Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

scholarly journals Neutral coated inclusions of finite conductivity

2011 ◽  
Vol 468 (2140) ◽  
pp. 954-970 ◽  
Author(s):  
Paweł Jarczyk ◽  
Vladimir Mityushev
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Singular Integrals ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Closed Curve ◽  
Finite Conductivity ◽  
The Core ◽  
Fixed Curve ◽  
Shaped Inclusion

We discuss the conductivity of two-dimensional media with coated neutral inclusions of finite conductivity. Such an inclusion, when inserted in a matrix, does not disturb the uniform external field. We are looking for shapes of the core and coating in terms of the conformal mapping ω ( z ) of the unit disc onto coated inclusions. The considered inverse problem is reduced to an eigenvalue problem for an integral equation containing singular integrals over a closed curve L 1 on the transformed complex plane. The conformal mapping ω ( z ) is constructed via eigenfunctions of the integral equation. For each fixed curve L 1 , the boundary of the core is given by the curve ω ( L 1 ). The boundary of the coating is obtained by the mapping of the unit circle. It is justified that any shaped inclusion with a smooth boundary can be made neutral by surrounding it with an appropriate coating. Shapes of the neutral inclusions are obtained in analytical form when L 1 is an ellipse.

Buckling of Bars of Variable Rigidity with Varying Axial Load

1963 ◽  
Vol 67 (635) ◽  
pp. 734-736
Author(s):  
K. T. Sundara Raja Iyengar ◽  
S. Anantharamu
Keyword(s):  
Numerical Methods ◽  
Approximate Method ◽  
Convenient Method ◽  
Axial Load ◽  
Difficult Problem ◽  
Variable Rigidity ◽  
Rigorous Solution ◽  
Approximate Methods ◽  
Buckling Loads ◽  
End Conditions

The Evaluation of buckling loads of columns presents a difficult problem whose rigorous solution may be very difficult particularly when the variation of moment of inertia or the axial load does not follow a simple law. Hence approximate methods such as the variational and numerical methods will have to be employed. An approximate method is suggested in this note which offers a convenient method for the calculation of buckling loads of bars with any type of end conditions.

scholarly journals Integro-differential equations with singular and hypersingular integrals

2019 ◽  
Vol 54 (4) ◽  
pp. 404-407 ◽  
Author(s):  
E. I. Zverovich ◽  
A. P. Shilin
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Singular Integrals ◽  
Closed Curve ◽  
Special Structure ◽  
Finite Part ◽  
Hypersingular Integrals ◽  
Hadamard Finite Part

Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.

scholarly journals Analytic formulas for the natural frequencies of hinged structures with taking into account the dead weight

2018 ◽  
Vol 230 ◽  
pp. 02016 ◽  
Author(s):  
Yurii Krutii ◽  
Mykola Suriyaninov ◽  
Victor Vandynskyi
Keyword(s):  
Exact Solution ◽  
Solid Mechanics ◽  
Physical Phenomenon ◽  
Analytical Form ◽  
Variable Coefficients ◽  
Vibration Frequencies ◽  
Bending Vibrations ◽  
Approximate Methods ◽  
Dead Weight ◽  
The Dead

Free bending vibrations of hinged vertical uniform rod with taking into account the dead weight are investigated. The research is based on the exact solution of the partial differential vibration equation with variable coefficients. This approach allows to get more reliable picture of rod’s vibration because only the exact solution carries information of a qualitative nature and forms the most complete picture of the physical phenomenon under consideration. The frequencies equation of problem was written in dimensionless form and the way of its root finding is shown. It has been shown that the problem of determination the nature frequencies of structures is reduced to finding corresponding dimensionless vibration coefficients from equation. The formulas for the first three vibration frequencies of structures were obtained in analytical form. An analytic relationship between the frequencies with and without taking into account the dead weight of the structures was established. The nature of the dependence of frequencies on the value of the longitudinal load was revealed. The presence of conclusive analytical formulas for determining the vibration frequencies of hinged vertical structures with taking into account the dead weight is a real alternative for using the approximate methods for solving this class of problems of solid mechanics.

Transient Response of a Nonlinear System by a Bilinear Approximation Method

1956 ◽  
Vol 23 (4) ◽  
pp. 635-641
Author(s):  
E. I. Ergin
Keyword(s):  
Numerical Methods ◽  
Nonlinear System ◽  
Transient Response ◽  
Bilinear Approximation ◽  
Bilinear Method ◽  
Approximate Methods ◽  
Line Segments ◽  
Transient Load ◽  
Nonlinear Elastic ◽  
Improved Accuracy

Abstract A line-segment approximation for a nonlinear elastic force has been developed which makes it possible to obtain analytical solutions to a number of transient-load problems which formerly could be solved only by graphical or numerical methods. It is shown that for many problems involving even large nonlinearities, two line segments only are sufficient to give satisfactory accuracy. For some problems where the classical analytical approximate methods of Lindstedt or of Kryloff and Bogoliuboff can be applied, it is shown that the bilinear method leads to an improved accuracy.

Approximate solution of random ordinary differential equations

1978 ◽  
Vol 10 (1) ◽  
pp. 172-184 ◽  
Author(s):  
William E. Boyce
Keyword(s):  
Approximate Solution ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Approximate Methods ◽  
Expository Paper

This is a largely expository paper on approximate methods of solving random ordinary differential equations, with an emphasis on direct numerical methods. Two methods are discussed in some detail and several others are mentioned briefly.

Comparison of numerical methods for light propagating in fibers with high steps of refractive index

2012 ◽  
Vol 20 (1) ◽  
Author(s):  
K. Zegadło ◽  
M. Karpierz
Keyword(s):  
Refractive Index ◽  
Numerical Methods ◽  
Light Propagation ◽  
Chromatic Dispersion ◽  
Three Dimensional ◽  
Complex Structures ◽  
Two Dimensional ◽  
Approximate Methods ◽  
Fast Development ◽  
Numerical Tools

AbstractFast development of complex structures like microstructural fibers, photonic nanowires or slot waveguides requires numerical tools to predict a light propagation. There are many works concerning weakly guided case, but the microstructural fibers need algorithm for a high step of the refractive index. In this paper, three approximate methods are compared. The comparison concerns a structure consisting of circular cores surrounded by cladding for different values of the refractive index steps. Application of these methods in chromatic dispersion case is also presented. It is shown that certain conditions prefer two dimensional scalar algorithms (based on approximated methods) than three dimensional ones. This allows us to implement more efficient and less complicated methods.

Null-Field Approach for the Multi-inclusion Problem Under Antiplane Shears

2006 ◽  
Vol 74 (3) ◽  
pp. 469-487 ◽  
Author(s):  
Jeng-Tzong Chen ◽  
An-Chien Wu
Keyword(s):  
Boundary Layer ◽  
Fourier Series ◽  
Analytical Form ◽  
Fundamental Solutions ◽  
Inclusion Problem ◽  
Boundary Layer Effect ◽  
Hypersingular Integrals ◽  
Null Field ◽  
Layer Effect ◽  
Two Inclusions

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein et al.’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.

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