This report describes the research and development of on-line techniques and their application to problem solving. For convenience, it is organized in several parts. Part 1 lists the published papers describing research problems solved in the course of this effort and summarizes the computational aspects of the multi-dimensional problem (a nonlinear diffusion equation) which received the most attention; the physical significance of this problem and a discussion of the results obtained is given in part 5. Part 2 contains a general description of the on-line system from the user's viewpoint, including some elementary examples of the 'constructive' aspects of on-line computing. In part 3 we have collected the on-line problem solving techniques developed in this program which are of general interest. These involve a blending of numerical and mathematical analysis somewhat different from that generally encountered in conventional computing, being best characterized as emphasizing a global rather than a local approach to problem formulation and solution. Part 4 constitutes a complete user's manual for the present system. It gives a detailed description of each of the basic programs. A general understanding of the on-line system can be obtained from part 2; parts 3 and 4 provide the detailed information needed to actually use this system. (Author).