Finite-part Singular Integral Equations Research Articles

Solving Finite Part Singular Integral Equations Using Orthogonal Polynomials

In this paper, we use the Gauss–Jacobi quadrature formula to get an approximate solution for a finite part singular integral equation. The zeros of Jacobi polynomials and Jacobi functions of the second kind are used to construct a square system of algebraic equations which yield an interpolating function for the regular part of the solution of the singular integral equation. In a special case, another approach consists in converting the finite part singular integral equation to a Fredholm integral equation of the second kind using the analytical solution of a simple finite part singular integral equation. To simplify the calculation of kernel and right-hand side of the obtained Fredholm equation, we use the new polynomials $$J_n(x)$$, which admit a recurrence relation. The equivalent form of the integral equation in the last case is used to investigate the error bound of the approximated solution.

Nonlinear finite-part singular integral equations arising in two-dimensional fluid mechanics

Nonlinear finite-part singular integral equations arising in two-dimensional fluid mechanics

Periodic array of staggered planar cracks in an anisotropic elastic medium

The problem of determining the elastic displacements and stresses in an infinite anisotropic medium containing a periodic array of staggered planar cracks is considered. It is reduced to a system of Hadamard finite-part singular (hypersingular) integral equations with the crack-opening displacements as unknown functions. The integral equations may be solved numerically by a collocation technique. Numerical results for specific cases involving isotropic and transversely isotropic materials are obtained.

Hypersingular integral equations for multiple interacting planar cracks in an elastic layered material under antiplane shear stresses

A system of Hadamard finite-part singular (hypersingular) integral equations is derived for an antiplane elastic problem involving an arbitrary number of arbitrarily-located planar cracks in a tri-layered material. The equations are solved approximately to compute the stress intensity factors for specific cases of the problems.

Systems of finite-part singular integral equations in Lp applied to crack problems

The solution of systems of finite-part singular integral equations defined in L p with many applications in mathematical physics is investigated. A finite-part singular integral representation analysis in L p is proposed, by proving that every system of finite-part singular integral equations is equivalent to an additional system of Fredholm equations. Some finite-part singular integral equation solubility theorems defined in L p are further presented, by applying the index of the finite-part singular integral equations. A fracture mechanics application is finally given, for the determination of the stress intensity factors at the tips of a crack normal to a bimaterial interface.

Coplanar cracks in a finite rectangular anisotropic elastic slab under antiplane shear stresses: A hypersingular integral formulation

An antiplane crack problem concerning an arbitrary number of pairs of coplanar cracks in a finite rectangular anisotropic elastic slab is considered. Using Fourier integral transforms and Fourier series, the problem is reduced to a system of Hadamard finite-part singular (hypersingular) integral equations which can be solved numerically by using a collocation technique. Once these integral equations are solved, the relevant crack energy and stress intensity factors can be readily computed. Numerical results for a specific case of the problem are given.

Arbitrarily-located multiple planar cracks in an elastic slab under antiplane shear stresses: A hypersingular integral formulation

A system of Hadamard finite-part singular (hypersingular) integral equations is derived for an elastic problem concerning an arbitrary number of arbitrarily-located planar cracks in an infinitely long strip subjected to antiplane stresses. These integral equations contain the differences in displacements on opposite faces of the cracks as unknown functions to be solved for. Numerical results for a specific problem are obtained by solving these equations.

Scattering and diffraction of sh waves by multiple planar cracks in an anisotropic half-space: A hypersingular integral formulation

The scattering and diffraction of SH waves by arbitrarily-located multiple planar cracks in an anisotropic half-space is considered. The problem is formulated in terms of a system of Hadamard finite-part singular (hyper-singular) integral equations which can be solved numerically using collocation techniques.

A hypersingular-boundary integral equation method for the solution of an elastic multiple interacting crack problem

The authors formulate the antiplane problem concerning a finite elastic body with an arbitrary number of coplanar cracks in its interior in terms of a system of boundary integral equations coupled with Hadamard finite-part singular integral equations. These integral equations are readily amenable to numerical treatment and, for the purpose of illustrations, they are employed to obtain numerical values of the stress intensity factors for certain specific problems.

Finite-part singular integral representation analysis in [formula omitted] of two-dimensional elasticity problems

A finite-part singular integral representation analysis in P is proposed for the solution of two-dimensional elasticity problems. The method consists of the reduction of the finite-part singular integral equations to the equivalent Fredholm equations. Nöther theorems are generalized by using finite-part singular integral equations, by investigating the necessary and sufficient conditions for the solution of these equations in L p and by defining the index of such types of equations. A two-dimensional fracture mechanics application is given for the determination of the stress intensity factors in the neighbourhood of a straight crack in a bimaterial infinite and Isotropic solid under antiplane shear.

Product integration for finite-part singular integral equations: Numerical asymptotics and convergence acceleration

We consider equations of the form ƒ(x)=g(x)+ƒ b ak(x,t) ƒ(t) (t−x)α dt, a > x > b, for integral α>1 and where the integral is taken in the Hadamard or finite-part sense (Davis and Rabinowitz (1984, pp. 11–13), Hadamard (1952, pp. 133–141), Paget (1981, p.447)). Such equations occur in fracture mechanics, gas radiation and fluid flow (Kaya and Erdogan (1987)). In particular, we study a variety of examples with α=2, k( x, t)=1 and [ a, b]=[0,1]. In order to investigate the dependence of error upon the differentiability of the solution we construct equations whose solutions are x; v for many rational and integral values of v in the interval [-2, 10]. In addition, we compute cases where the solution function is discontinuous, or continuous but not differentiable, at an interior point of the interval. Based solely upon the empirical error analysis, a single extrapolation technique was developed which universally enhanced convergence.

Static stresses in a periodically layered anisotropic elastic composite containing a periodic array of planar cracks

The problem of calculating the static elastic stresses in a periodically layered anisotropic composite containing a periodic array of planar cracks is considered. We formulate the problem in terms of a system of simultaneous finite-part singular integral equations which can be solved numerically using collocation techniques. The solution of the integral equations enables relevant quantities such as the stress intensity factors to be computed. Numerical results are obtained for specific cases of the problem.

A pair of arbitrarily-oriented coplanar cracks in an anisotropic elastic slab

AbstractThe problem of an anisotropic elastic slab containing two arbitrarily-oriented coplanar cracks in its interior is considered. Using a Fourier integral transform technique, we reduce the problem to a system of simultaneous finite-part singular integral equations which can be solved numerically. Once the integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results for specific examples are obtained.

The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics

A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.

On the numerical evaluation of the general type of finite-part singular integrals and integral equations used in fracture mechanics

A new method is established for the numerical solution of the general type of the finite-part integrals and integral equations of the first and second kind, which are frequently encountered in problems of applied mechanics and especially in plane and antiplane elasticity. Many numerical integration rules are properly combined in order to handle numerically finite-part integral equations along the integration interval [−1, 1] and for a large variety of weight functions. In order to solve this type of singular integral equations, we introduce some new formulae, which are the general type of the Plemelej formulae. In this way such an equation may be numerically solved by reduction to a system of linear equations. Finally, two applications to fracture mechanics are given.

On the numerical solution of the finite-part singular integral equations of the first and the second kind used in fracture mechanics

A new technique is proposed for the numerical evaluation of the finite-part singular integral equations of the first and the second kind with general kernels, which are frequently encountered in problems of applied mechanics and especially in plane and antiplane elasticity. Some new formulas are established in order to solve this type of singular integral equations. It can be said, that these formulas are a generalization of the well known Plemelej formulas. A numerical application is given to the determination of the stress field in the neighbourhood of a crack parallel to the free boundary of an elastic isotropic semiplane.