From Graph Transformations to Differential Equations

In a variety of disciplines models are used to predict, measure or explain quantitative properties. Examples include the concentration of a chemical substance produced within a given period, the growth of the size of a population of individuals, the time taken to recover from a communication breakdown in a network, etc. The models such properties arise from are often discrete and structural in nature. Adding information on the time and/or probability of any actions performed, quantitative models can be derived. In the first example above, commonly referred to as kinetic analysis of chemical reactions, a system of differential equations describing the evolution of concentrations is extracted from specifications of individual chemical reactions augmented with reaction rates. Recently, this construction has inspired approaches based on stochastic process specification techniques aiming to extract a continuous, quantitative model of a system from a discrete, structural one. This paper describes a methodology for such an extraction based on stochastic graph transformations. The approach is based on a variant of the construction of critical pairs and has been implemented using the AGG tool and validated for a simple reaction of unimolecular nucleophilic substitution (SN1)

Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples

Classification of integrable equations on quad-graphs. The consistency approach

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed

Formula manipulation in the bond graph modelling and simulation of large mechanical systems

A multibond graph element for a general single moving body is derived. A multibody system can easily be described as an interconnection of these elements. 3-D mechanical systems usually contain dependent inertias having both differential and integral causality. A method is described for the transformation of inertias with differential causality to an integral form, using formula manipulation. The program also helps to find experimentally the optimal choice for the generalized coordinates. The resulting explicit differential equation may be solved using a standard integration routine or simulation program

Classification of discrete equations linearizable by point transformation on a square lattice

We provide a complete set of linearizability conditions for nonlinear partial difference equations de- fined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Mobious transformatio