Ordinary and partial differential equations are often derived as a first approximation to model a real-world situation, where the state of the system depends not only on the present time, but also on the history of the system. In this situation, a higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations which results in delay Volterra integro-differential equations (denoted DVIDEs). Although DVIDEs serve as indispensable tools for modelling real systems, we still lack efficient and reliable software to approximate the solution of systems of DVIDEs. This thesis is concerned with designing, analyzing and implementing an efficient method to approximate the solution of a general system of neutral Volterra integro-differential equations with time-dependent delay arguments using a continuous Runge-Kutta (CRK) method. We introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. We prove the classic and optimal convergence of the numerical approximation to the exact solution. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides an effective estimate of the global error of the approximate solution, at a modest increase in cost. We have implemented our approach as an experimental Fortran 90 code capable of handling various kinds of DVIDEs with non-vanishing, vanishing, and infinite delay arguments.

Ordinary and partial differential equations are often derived as a first approximation to model a real-world situation, where the state of the system depends not only on the present time, but also on the history of the system. In this situation, a higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations which results in delay Volterra integro-differential equations (denoted DVIDEs). Although DVIDEs serve as indispensable tools for modelling real systems, we still lack efficient and reliable software to approximate the solution of systems of DVIDEs. This thesis is concerned with designing, analyzing and implementing an efficient method to approximate the solution of a general system of neutral Volterra integro-differential equations with time-dependent delay arguments using a continuous Runge-Kutta (CRK) method. We introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. We prove the classic and optimal convergence of the numerical approximation to the exact solution. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides an effective estimate of the global error of the approximate solution, at a modest increase in cost. We have implemented our approach as an experimental Fortran 90 code capable of handling various kinds of DVIDEs with non-vanishing, vanishing, and infinite delay arguments.

The purpose of this paper is to supply a summary of library subroutines and functions for parallel MIMD computers. The subroutines have been developed at the University of Chemnitz during a period of the last five years. In detail, they are concerned with vector operations, inter-processor communication and simple graphic output to workstations. One of the most valuable features is the machine-independence of the communication subroutines proposed in this paper for a hypercube topology of the parallel processors (excepting a kernel of only two primitive system-dependend operations). They were implemented and tested for different hardware and operating systems including transputer, nCube, KSR, PVM. The vector subroutines are optimized by the use of C language and enrolled loops (BLAS1-like). The paper includes hints for using the libraries with both Fortran and C programs.

The purpose of this paper is to supply a summary of library subroutines and functions for parallel MIMD computers. The subroutines have been developed at the University of Chemnitz during a period of the last five years. In detail, they are concerned with vector operations, inter-processor communication and simple graphic output to workstations. One of the most valuable features is the machine-independence of the communication subroutines proposed in this paper for a hypercube topology of the parallel processors (excepting a kernel of only two primitive system-dependend operations). They were implemented and tested for different hardware and operating systems including transputer, nCube, KSR, PVM. The vector subroutines are optimized by the use of C language and enrolled loops (BLAS1-like). The paper includes hints for using the libraries with both Fortran and C programs.