Analytical and numerical treatment of a fractional integro-differential equation of volterra-type

An analytical treatment of a fractional generalisation of the free electron laser (FEL) equation is given, whereby a closed form solution is obtained in terms of Kummer functions Φ(a, c; z). Subsequently, Tau method approximation of these functions facilitates the numerical evaluation of the solution of the FEL equation.     

A solution of an integral equation

In this paper an integral equation $$\int\limits_0^\infty {f(u)f\left( {\tfrac{x}{u}} \right)\tfrac{{du}}{u} = g(x)} $$ is studied.     

A certain integral equation of first kind

Kulyab, Tazik. SSR. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 190–193, January–February, 1989.     

Two-dimensional transport equation as Fredholm integral equation

The transport equation has many applications in various fields of science and engineering. In this paper we shown that we can transform a transport equation in two-dimensional case into a Fredholm integral equation of the second kind with a compact integral operator for the angular flux by using the Sumudu transform.     

A logarithmic singular integral equation over multiple intervals

Two quick methods of solution of a singular integral equation of the first kind defined over disjoint multiple intervals involving a logarithmic singularity are explained.     

A stable well-conditioned integral equation for electromagnetism scattering

Finding a formulation for electromagnetic scattering of surfaces which is both well-posed and produces a well-conditioned linear system is still a challenging problem. We here propose one such formulation valid in the high-frequency regime. The mathematical analysis is provided and numerical results on rather complex geometries show the performance of the method.     

A Liapunov functional for a singular integral equation

We consider a scalar integral equation where a∈L²[0,∞), while C(t,s) has a significant singularity, but is convex when t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L²[0,∞) and that x(t)−a(t)→0 pointwise as t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals.     

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.     

A spin integral equation for electromagnetic and acoustic scattering

We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three-dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two-by-two matrices as values, using a spin representation of the fields.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A problem with integral conditions for an elliptic-parabolic equation

Mathematical Notes - A problem with integral conditions for a differential equation of elliptic-parabolic type in a mixed domain consisting of a rectangle and a half-disk is posed. The unique...