Gennadi Vainikko

Periodic Integral and Pseudodifferential Equations with Numerical Approximation

1 Preliminaries.- 2 Single Layer and Double Layer Potentials.- 3 Solution of Boundary Value Problems by Integral Equations.- 4 Singular Integral Equations.- 5 Boundary Integral Operators in Periodic Sobolev Spaces.- 6 Periodic Integral Equations.- 7 Periodic Pseudodifferential Operators.- 8 Trigonometric Interpolation.- 9 Galerkin Method and Fast Solvers.- 10 Trigonometric Collocation.- 11 Integral Equations on an Open Arc.- 12 Quadrature Methods.- 13 Spline Approximation Methods.

Hypersingular Integral Equations and Their Applications

A number of new methods for solving singular and hypersingular integral equations have emerged in recent years. This volume presents some of these new methods along with classical exact, approximate, and numerical methods. The authors explore the analysis of hypersingular integral equations based on the theory of pseudodifferential operators and consider one-, two- and multi-dimensional integral equations. The text also presents the discrete closed vortex frame method and some other numerical methods for solving hypersingular integral equations. The treatment includes applications to problems in areas such as aerodynamics, elasticity, diffraction, and heat and mass transfer.

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

Fast Solvers of the Lippmann-Schwinger Equation

The electromagnetic and acoustic scattering problems for the Helmholtz equation in two and three dimensions are equivalent to the Lippmann-Schwinger equation which is a weakly singular volume integral equation on the support of the scatterer. We propose for the Lippmann-Schwinger equation two discretizations of the optimal accuracy order, accompanied by fast solvers of corresponding systems of linear equations. The first method is of the second order and based on simplest cubatures; the scatterer is allowed to be only piecewise smooth. The second method is of arbitrary order and is based on a fully discrete version of the collocation method with trigonometric test functions; the scatterer is assumed to be smooth on whole space ℝn and of compact support.

Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

In the first part of this paper we study the regularity properties of solutions of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.

A Spline Product Quasi-interpolation Method for Weakly Singular Fredholm Integral Equations

A discrete method of accuracy

Integral Equations with Diagonal and Boundary Singularities of the Kernel

O(h^{m})

A SPLINE COLLOCATION METHOD FOR LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS ∗

is constructed and justified for a class of Fredholm integral equations of the second kind with kernels that may have weak diagonal and boundary singularities. The method is based on (i) improving the boundary behavior of the kernel with the help of a change of variables, and (ii) the product integration using quasi-interpolation by smooth splines of order

Fast solvers of integral and pseudodifferential equations on closed curves

m

Error Estimates for the Cardinal Spline Interpolation

.