Abstract: Most of the physical processes arising in nature are modeled by differential equations, either ordinary (example: the spring/mass/damper system) or partial (example: heat diffusion). From the point of view of analog computability, the existence of an effective way to obtain solutions (either exact or approximate) of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon [Journal of Mathematical Physics 20 (1941), 337–354] as a model of the Differential Analyzer and improved by Pour-El [Transactions of the American Mathematical Society 199 (1974), 1–28], Lipshitz and Rubel [Proceedings of the American Mathematical Society 99(2) (1987)], Costa and Graça [Journal of Complexity 19(5) (2003), 644–664] and others. The GPAC is capable of manipulating real-valued data streams. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, which is its fundamental inability to reason about functions of more than one independent variable (the ‘time’ variable), as noted by Rubel [Advances in Applied Mathematics 14(1) (1993), 39–50]. In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. We extend the class of data types using networks with channels which carry information on a general complete metric space X; here we take X=C(R), the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon’s construction (constants, adders, multipliers, integrators) and we add a differential module which has one input and one output. For input u, it outputs the spatial derivative v(t)=∂xu(t). We then define an X-GPAC to be a network built with X-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. Such a framework was considered by Tucker and Zucker [Theoretical Computer Science 371 (2007), 115–146]. We study the properties of these analog systems and their associated operators, and present a characterization of the X-GPAC-generable functions which generalizes Shannon’s results.
Keywords: Analog computation, computable analysis, partial differential equations
Journal: Computability, vol. Pre-press, no. Pre-press, pp. 1-22, 2017