satisfies the condition f ρ ∈ Lloc(]a, b]; R). In regular case where f ρ ∈ L([a, b]; R) for any ρ > 0, problem (1), (2) is investigated in detail (see, e.g., [1]–[4], [9], [10] and the references therein). We are, in the main, interested in the case when f ρ is nonintegrable on [a, b], having singularity at t = a. In this case, the problem (1), (2) is singular one. The methods of investigation ...

Quantum gravity is often expected to solve both the singularity problem and the information-loss problem of black holes. This article presents an example from loop quantum gravity in which the singularity problem is solved in such a way that the information-loss problem is made worse. Quantum effects in this scenario, in contrast to previous non-singular models, do not eliminate the event horiz...

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on lattice ħZn. We allow speed to vanish leading weakly hyperbolic nature equations. Curiously, very much contrary Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] other authors, Cauchy problem in is well-posed ℓ2(ħZn). However, also recover well-posed...

Based on the theory of elasticity, previous analytical solutions concerning a penny-shaped interface crack employ the derivative of the crack surface opening displacements as the primary unknowns, thus leading to singular integral equations with Cauchy-type singularity. The solutions to the resulting integral equations permit only the determination of stress intensity factors and energy release...

Introduction. Hamada [1] proved the following result about the propagation of singularities in the Cauchy problem for an analytic linear partial differential operator. Assume that the initial data are analytic at the point 0 except for singularities along a submanifold T of the initial surface containing 0. Let K\ • • •, K be the characteristic surfaces of the operator emanating from T. Under t...

in this paper, the solution of a moving volterra-type screw dislocation in an orthotropic layer, bonded between two piezoelectric layers is obtained using complex fourier transform. the dislocation solution is then employed as strain nuclei to derive singular integral equations for a medium weakened by multiple moving cracks. these equations, which are classified as, cauchy singular equations, ...

This paper is concerned with finding approximate solution for the singular integral equations. Relating the singular integrals to Cauchy principal-value integrals, we expand the kernel and the density function of singular integral equation by the sum of the chebyshev polynomials of the first, second, third and fourth kinds. Some numerical examples are presented to illustrate the accuracy and ef...