This paper has proposed a model that reflects process of energy dissipation in dynamic oscillatory systems. According to the hereditary theory viscoelasticity, relationship between stress and strain is realized an integral form. At same time, weakly singular kernel heredity simultaneously describes both internal friction deformation aftereffects (creep relaxation). Application developed shown t...

In this paper, a homotopy perturbation method (HPM) and modified homotopy perturbation (MHPM) are proposed to solve singular integral equation with generalized Abel's kernel.

A high-order accurate, explicit kernel-split, panel-based, Fourier–Nyström discretization scheme is developed for integral equations associated with the Helmholtz equation in axially symmetric domains. Extensive incorporation of analytic information about singular integral kernels and on-the-fly computation of nearly singular quadrature rules allow for very high achievable accuracy, also in the...

In this paper, we consider solutions of Fredholm integral equations of the second kind where the kernel functions are asymptotically smooth or products of such functions with highly oscillatory coefficient functions. We present a scheme based on polynomial interpolation to approximate matrices A from the discretization of these integral operators. Our approximation matrix B is obtained by parti...

In this paper we deal with a nonlinear singular integral inequality which arises in the study of partial differential equations. The integral term is non local in time and space and the kernel involved is also singular in both the time and the space variable. The estimates we prove may be used to establish (global) existence and asymptotic behavior results for solutions of the corresponding pro...

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

where D = (−1, 1)× (−1, 1) , f(x, y) is a given Hölder continuous function in D, and φ(x, y) is an unknown function. The equation (1) has applications in the theory of aeroelasticity [1]. Note that the equation without logarithmic singularities was many times considered in different classes of functions. In the literature the solutions of the equation (1) in bounded domains [2, 5, 6, 9] as well...

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integro-differential equation (FIDE) with a weakly singular kernel‎. ‎First‎, ‎a collocation method based on Haar wavelets (HW)‎, ‎Legendre wavelet (LW)‎, ‎Chebyshev wavelets (CHW)‎, ‎second kind Chebyshev wavelets (SKCHW)‎, ‎Cos and Sin wavelets (CASW) and BPFs are presented f...

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebys...

The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel (t − s)−μ with 0 < μ < 1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both th...