Numerical solution of certain Cauchy singular integral equations using a collocation scheme

The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.

Rapid Sliding Contact in Three-Dimensional by an Ellipsoidal Die on Transversely Isotropic Half-Spaces With Surfaces on Different Principal Planes

A rigid ellipsoidal die slides on the surfaces of transversely isotropic half-spaces. In one case the material symmetry axis coincides with the half-space surface normal. In the other, the axis lies in the plane of the surface. In both cases sliding proceeds with constant sub-critical speed along a straight path at an arbitrary angle to the principal material axes. A three-dimensional dynamic steady state is considered, i.e., the contact zone surface must conform to the die profile and contact zone traction remains constant in the frame of the die. Exact solutions for contact zone traction are derived in analytic form, as well as formulas for contact zone geometry. Symmetry need not be assumed in the solution process. Anisotropy is found to largely determine zone shape at low sliding speed, but the direction of sliding can become a major influence at higher sliding speeds. Cartesian coordinates are used in the analysis, but introduction of quasi-polar coordinates allows problem reduction to a Cauchy singular integral equation.

Mixed Multi-Layered Model for Fracture Analysis of Functionally Graded Interfacial Zone under Anti-Plane Loading

In this paper, a new mixed multi-layered model is put forward to study the crack problem of the functionally graded interfacial zone between tow homogeneous half-spaces. In the model, the interfacial zone is divided into some sub-layers with the properties of each layer varying in linear and exponential manners alternately. By applying Fourier transform and using the transfer matrix method, the mixed boundary problem of anti-plane fracture can be reduced to a Cauchy singular integral equation, which is solved numerically. Stress intensity factors of some examples are derived. The results show that the present model is effective and accurate and compared with the liner multi-layered model, the present one can save more CPU time in computation.

Analysis of Fretting Problems Using a Numerical Method Based on Cauchy Singular Integral Equation

Numerical analysis by influence function method (IFM) is demonstrated in this study in order to investigate the fretting wear problems on the secondary side of the steam generator, caused by flow induced vibration. Two-dimensional numerical contact model is developed in terms of Cauchy integral equation. The distributions of normal pressures, shear stresses and displacement fields are derived between two contact bodies which have similar elastic properties. The work rate model is adopted to find the wear amounts between two materials. The results are compared with the solutions by finite element analyses, which validates the application of the present method to fretting wear problems.

A problem related to the hall effect in a semiconductor with an electrode of an arbitrary shape

A problem on electric current in a semiconductor film from an electrode of an arbitrary shape is studied in the presence of a magnetic field. This situation describes the Hall effect, which indicates the deflection of electric, current from electric field in a semiconductor. From mathematical standpoint we consider the skew derivative problem for harmonic functions in the exterior of an open arc in a plane. By means of potential theory the problem is reduced to the Cauchy singular integral equation and next to the Fredholm equation of the 2nd kind which is uniquely solvable. The solution of the integral equation can be computed by standard codes by discretization and inversion of the matrix. The uniqueness and existence theorems are formulated.