A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411-1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483]. Here, by choosing a different set of the "collocation points" (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483].The new collocation points lead to wellconditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.
Cellular automata that exhibit soliton behavior are studied. A simple rule-the fast-rule theorem (FRT)-is introduced, which allows immediate calculation of the evolution process. The FRT is suitable for the development of parallel algorithms, for the analysis and prediction of the behavior of the automata, and for hand calculation as well. The FRT consists of first selecting a finite set of sites and then obtaining the next state by simply inverting one bit in these sites, while leaving the remaining bits unchanged. This finite set can be detected by inspection. The distinction between single and multiple particles is made precise and shown to depend not only on the number of consecutive zeros separating particles but also on their position relative to the sites in the finite set mentioned above. Three applications are given. The first is a demonstration of the FRT as an analytical tool and settles the (up to now) open question of stability. The other two applications demonstrate the use of the FR T in the construction of a periodic particle and in obtaining soliton collisions by hand calculations.
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