Volume integral equation method in problems of mathematical physics (Q2908550)

These lecture notes are devoted to volume integral equations of the form NEWLINE\[NEWLINE a(x) u(x) + \displaystyle \int_Q {k(x-y) \over |x-y|^m}\, b(y) u(y) dy = f(x) \, , \; x \in Q \, , \tag{1}NEWLINE\]NEWLINE where \(Q\) is a bounded domain in \(I\!\!R^3\) and \(m \leq 3\). Many important classes of wave scattering problems lead to equations of this form. The corresponding integral operator is compact \((m=1)\) in acoustic problems and singular \((m=3)\) in electromagnetic problems. The author gives a survey on the analysis of (1) including results on existence, uniqueness and spectral properties. The second part is concerned with iteration and discretization methods for the numerical solution of (1). In particular, algorithms based on the discrete fast Fourier transform are presented.