Integral Equations: A compendium

Description

Paper, 82 p, Karlstad 2002. Contents. Notion and examples of integral equations (IEs). Fredholm IEs of the first and second kind. Examples of solution to integral equations and ordinary differential equations. Reduction of ODEs to the Volterra IE and proof of the unique solvability using the contraction mapping. Unique solvability of the Fredholm IE of the 2nd kind using the contraction mapping. Neumann series. IEs as linear operator equations. Fundamental properties of completely continuous operators. Elements of the spectral theory of linear operators. Spectrum of a completely continuous operator. Linear operator equations: the Fredholm theory. IEs with degenerate and separable kernels. Hilbert spaces. Self-adjoint operators. Linear operator equations with completely. continuous operators in Hilbert spaces. IEs with symmetric kernels. Hilbert–Schmidt theory. Harmonic functions and Green’s formulas. Boundary value problems. Potentials with logarithmic kernels. Reduction of boundary value problems to integral equations. Functional spaces and Chebyshev polynomials. Solution to integral equations with a logarithmic singulatity of the kernel. Solution to singular integral equations. Matrix representation. Matrix representation of logarithmic integral operators. Galerkin methods and basis of Chebyshev polynomials.