The contents of this thesis are concerned with the implementation of parallel algorithms for solving partial differential equations (POEs) by the Alternative Group EXplicit (AGE) method and an investigation into the numerical inversion of the Laplace transform on the Balance 8000 MIMO system. Parallel computer architectures are introduced with different types of existing parallel computers including the Data-Flow computer and VLSI technology which are described from both the hardware and implementation points of view. The main characteristics of the Sequent parallel computer system at Loughborough University is presented, and performance indicators, i.e., the speed-up and efficiency factors are defined for the measurement of parallelism in the system. Basic ideas of programming such computers are also outlined.....