Abstract In this work we study the Ulam–Hyers stability of a differential equation. Its proof is based on Banach fixed point theorem in some space continuous functions equipped with norm $\|\cdot \|_{\infty }$ ∥ ⋅ ∞ . Moreover, ge...

‎In this paper‎, ‎a matrix based method is considered for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $s^{beta}(t-s)^{-alpha}G(y(s))$ based on the Tau method‎. ‎In this method‎, ‎a transformation of the independent variable is first introduced in order to obtain a new equation with smoother solution‎. ‎Error analysis of this method is also ...

This article considers weakly singular, singular and hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used to solve problems in science and engineering. For their regularization, an approach based on the theory of distribution and application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, sin...

Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann-Schwinger (LS) equation is considered without invoking the traditional partial-wave decomposition. The singular kernel of the three-dimensional LS equation in coordinate space is regularized by a subtraction technique. The resulting nonsingular integral equation is then solved via the Nystro...

Abstract: A one dimensional heat equation in a semi-infinite medium controlled through a heat source depending on the delayed heat flux at the extremum is studied. By reducing the problem to a delayed Volterra integral equation with a weakly singular kernel, we find conditions on the initial datum and on the source term of the equation to control the asymptotic behaviour of the mean temperatures.

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

We analyze electromagnetic scattering of TM polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative-index dielectric material. Such scattering problems are important for the modelization of, e.g., light propagation in nano-optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integr...

Two issues that arise when solving Stokes flow problems with the hydrodynamical single-layer potential are addressed. First, the resulting boundary integral equation is singular, and second, discretizations lead to dense matrices. We discuss a well-posed modified equation which is equivalent for zero net-flux. Furthermore, we describe a multiscale basis that lead to sparse stiffness matrices. T...

We study the linearized Vlasov equation for a bunched beam subject to an arbitrary wake function. Following Oide and Yokoya, the equation is reduced to an integral equation expressed in angle-action coordinates of the distorted potential well. Numerical solution of the equation as a formal eigenvalue problem leads to difficulties, because of singular eigenmodes from the incoherent spectrum. We ...

Preface In our course we will consider the volume integral equations in the following form) () () () () () (x f dy y u y b y x y x K x u x a Q m = − − + ∫ , 3 ≤ m. Many important classes of the wave scattering problems can be described by equations of this form; for example, this is the case for problems of electromagnetic and acoustic scattering on 3D transparent bodies. The corresponding inte...