Two-dimensional Singular Integral Equations Research Articles

Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel

In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.

The method for calculating singular integrals in problems of axially symmetric Stokes flows

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

Кругове теплоактивне міжфазне включення в кусково-однорідному трансверсального-ізотропному просторі

An exact solution of the stationary thermoelasticity problem about interfacial circular absolutely rigid inclusion, which is under conditions of complete adhesion and under conditions of smooth contact with transversely homogeneous spaces, is constructed. The task with the help of the constructed discontinuous solution, by the method of singular integral relations, is reduced to a system of singular integral equations (SIE). An exact solution has been built for the specified systems of two-dimensional singular integral equations. As a result, dependences jumps of stresses and displacement on temperature, equivalent load, main moments and thermomechanical characteristics of transversally isotropic materials. The influence of the type of contact interaction on the behavior of the solutions is established. In particular, it has been shown that the stresses in the neighborhood of the inclusion with a smooth contact have a root singularity, and with complete coupling, the root singularity, which is amplified by oscillation. The behavior of the generalized intensity coefficient (GCIN) was studied for the combination of various transversely isotropic materials at different power and temperature loads.

An Efficient Operational Matrix Method for Solving a Class of Two-Dimensional Singular Volterra Integral Equations

In this paper, we consider a spectral method to solve a class of two-dimensional singular Volterra integral equations using some basic concepts of fractional calculus. This method uses a modification of hat functions for finding a numerical solution of the considered equation. Some properties of the modification of hat functions are presented. The main contribution of this work is to introduce the fractional order operational matrix of integration for the considered basis functions. Making use of the Riemann-Liouville fractional integral operator helps us to reduce the main problem to a system of linear algebraic equations which can be solved easily. After that, error analysis of the method is discussed. Finally, numerical examples are included to confirm the accuracy and applicability of the suggested method.

On 3D problem of vertically uniform heat flow disturbed by an anticrack in a transversely isotropic magnetoelectroelastic space

Abstract The static equilibrium of a magnetoelectroelastic transversely isotropic space with a rigid sheet-like inclusion (an anticrack) of arbitrary shape in the isotropy plane under a remote normally incident uniform steady heat flow is studied. Considered is the case where the anticrack is thermally insulated and magnetoelectrically impermeable. By constructing appropriate harmonic potentials in the derived general solution, the governing two-dimensional singular integral equation is obtained in terms of a normal stress discontinuity across the inclusion. An analogy between the thermo-magnetoelectroelastic anticrack problem and its thermoelastic counterpart is exploited. A complete solution in terms of elementary functions is obtained and discussed in the simple case of a circular rigid disc-inclusion.

On 3D problem of an anticrack under vertically uniform heat flow in a transversely isotropic electro-thermo-elastic space

Abstract A static thermoelectroelastic problem for a transversely isotropic space with an arbitrary shaped rigid sheet-like inclusion (an anticrack) in the isotropy plane obstructing a uniform steady heat flow is investigated. The anticrack is treated as isothermal and electrically impermeable. A general method of solving the resulting boundary-value problems is presented. Using the derived general solution and the generalized potential theory method, the governing two-dimensional singular integral equation is obtained in terms of a normal stress discontinuity across the inclusion. An analogy between the thermoelectromechanical anticrack problem and its thermomechanical counterpart is exploited. As an illustration, a typical application to a circular rigid disc-inclusion is presented. In this case a complete solution in elementary functions is achievable. Explicit expressions for the thermoelectrostressed state in the inclusion plane are given and discussed from the point of view of material failure.

APPLICATION OF THE METHOD OF BOUNDARY INTEGRAL EQUATIONS FOR NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY IN AXIALLY SYMMETRIC DOMAIN

The article considers the non-stationary initial-boundary problem of thermal conductivity in axially symmetric domain in Minkowski space, formulated as equivalent boundary integral equation. Using the representation of the solution in the form of a Fourier series expansion, the problem is reformulated as an infinite system of two-dimensional singular integral equations regarding expansion coefficients. The paper presents and investigates the explicit form for fundamental solutions used in the integral representation of the solution in the domain and on the border. The obtained results can be used in the construction of efficient numerical boundary element method for estimation of structures behavior under the influence of intense thermal loads in real-time.

Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel

In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.

On 3D symmetrical thermoelastic anticrack problems

A potential theory method is developed to solve a symmetrical thermoelastic problem of a cooling temperature field applied over the faces of a rigid sheet-like inclusion (an anticrack) in an elastic space. The governing boundary two-dimensional (2D) singular integral equations for an arbitrarily shaped anticrack are derived in terms of unknown thermal shear stress jumps. As an illustration, a complete solution expressed in elementary functions to the problemof a circular rigid inclusion subjected to a uniform temperature is presented and interpreted from the point of view of fracture theory.

Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.

A rigid inclusion in an elastic space under the action of a uniform heat flow in the inclusion plane

A solution is presented for the three dimensional static thermoelastic problem of an absolutely rigid inclusion (anticrack) in the case when a uniform heat flow is directed along the inclusion plane. By using the potential method and the Fourier transform technique, the problem is reduced to a system of coupled two-dimensional singular integral equations for the shear stress jumps across the inclusion. As an illustration, a typical application to the circular anticrack is presented. Explicit expressions for the thermal stresses in the inclusion plane are obtained and discussed from the point of view of material failure.

Approximate solution for a class of two-dimensional nonlinear singular integral equations by the contraction mapping method

We obtain solutions for a class of two-dimensional nonlinear singular integral equations with Hilbert kernel using the contraction mapping method and find the rate of convergence of successive approximations to the exact solution.

Interface circular inclusion under mixed conditions of interaction with a piecewise homogeneous transversally isotropic space

We reduce the problem of a circular absolutely rigid inclusion of arbitrary shape that is under conditions of complete adhesion to one transversally isotropic half-space and under conditions of smooth contact with another transversally isotropic half-space to a system of two-dimensional singular integral equations. We obtain solutions of this system in an explicit form, which enables us to determine the stress field and the field of displacements in the vicinity of the inclusion under arbitrary load. Dependences of translational and circular displacements of the inclusion on the resultant loads, principal moments, and elastic properties of the half-spaces are determined. We investigate the asymptotics of stresses in the vicinity of the inclusion and determine the directions of the largest and smallest stress concentration.

Singular integral relations and equations for a piecewise homogeneous transversally isotropic space with interphase defects

Using the constructed discontinuous solution for a piecewise homogeneous transversally isotropic space, we have obtained two-dimensional singular integral relations, which connect the jumps and sums of components of the stress tensor and displacement vector. These relations enable one to reduce the problems of interphase defects of arbitrary nature directly to systems of two-dimensional singular integral equations, whose kernels can be written in explicit form.

Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries

We propose a technique for the analytic investigation of features of contact stresses in the vicinity of the nonstationary moving boundary of a contact region in plane nonstationary contact problems with moving boundaries, which is based on the reduction of a boundary two-dimensional singular integral equation resolving the problem to a system of two one-dimensional singular equations. As tools of research, a method for the reduction of singular integral equations to an equivalent Riemann type problem for piecewise analytic functions and a technique of fractional integro-differentiation are used. It is shown that, on the moving boundary of the contact region, a power singularity, the order of which depends on the velocity of the boundary, takes place.

On the numerical solution of two-dimensional singular integral equation

The mechanical quadrature method described for two-dimensional nonlinear singular integral equation with Hilbert kernel in Banach space. The results are compared with the exact solution of the integral equation.

On the approximate solution of some two‐dimensional singular integral equations

AbstractAn approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.

The Problem of an External Circular Crack Under Asymmetric Loadings

Using the boundary integral equation method, the problem of an external circular crack in a three-dimensional infinite elastic body under asymmetric loadings is investigated. The two-dimensional singular boundary integral equations of the problem were reduced to a system of Abel integral equations by means of Fourier series and hypergeometric functions. The exact solutions of stress intensity factors are obtained for the problem of an external circular crack under asymmetric loadings, which are even more universal than the results obtained by the use of Hankel transform method. The results demonstrate that the boundary integral equation method has great potential as a new analytic method.

Quadrature methods for 2D and 3D problems

In this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules.

Singular integral equations in douglis algebra

In this paper, it is considered for some twodimensional singular integral equations of the hypercomplex functions in he Douglis scnse. In some special cases. the Fredholm conditions and index formula of such equations are obtained.