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Regularity properties and singularities of solutions to linear and nonlinear weakly singular integral equations, integro-differential equations and systems of Fredholm type integral equations will be studied. Such study is partly realized for linear equations and it is at very starting point for nonlinear equations. The singularity of the kernel causes singularities of the solution (more precisely, of certain derivatives of the solution). This complicates an effective solution of the equation. Our main purpose is to design and justify discretization methods which are adapted to those singularities and are of the same accuracy as if there were no singularity of the solution at all. The reduction of computing costs by the use of multi-grid approach and fully discrete collocation and Galerkin methods is of special interest.We propose also to consider the numerical solution of more general weakly singular integral equations (in particular weakly singular integral equations with operator valued kernels so that the general radiation transfer problems will be included), boundary integral equations and pseudodifferential equations. General periodic pseudodifferential equations, the theory and fast solvers, have been examined in the monograph "Periodic Integral and Pseudodifferential Equations with Numerical Approximation", Springer, Berlin-Tokyo, 2002, by J.Saranen and G.Vainikko. Our purpuse is to extend these results to nonperiodic pseudodifferential equations in an interval. All our numerical algorithms will be verified by numerical experiments. |