The theme of this mathematical research is the investigation of mechanical systems described by systems with memory effects. The memory comes in either by a viscoelastic constitutive equation of the equation of the material, yielding a Volterra type partial differential-integral equation, or by coupling to another system which is solved explicitly, so that its contribution to the model comes in as a set of boundary conditions containing convolution integrals. The main objective is the approximation of the convolution kernels in a way such that the Volterra type conditions are replaced by a higher order differential equation, subject to the constraint that the approximating systems are again dissipative, to ensure convergence. In the frequency domain this means a rational approximation of the Laplace transform of the kernel. In addition to providing numerical tools for handling equations modeling phenomena with memory effects, this work advances the theory of integrodifferential equations in two directions in which the established theory is still weak. The first is the analysis of operator valued kernels and the second is the study of phenomena where the memory enters the boundary conditions.
