Integration of Difficult Initial and Boundary Value Problems by Arc-Length Strategy

Abstract

For differential equations exhibiting steep gradients and large curvature of the solution. the arc-length transformation has been proposed. The transformed differential equations can be accurately represented by a small number of equally spaced mesh points. The transformation can be used both for initial and boundary value problems. This transformation proved to be useful for reactor, flame and combustion problems. calculation of dissipative structures and transient behavior of certain autocatalytic reactions. Three sample examples are solved.

Analysis of interface evolution and pattern formation during CVD

“…Therefore, the arc length (say s) is introduced as new independent variable and the original independent variable (say x) becomes a dependent variable. The relation between x and s introduces an additional equation, and the original system equations are transformed and expressed in terms of the arc length s (Bhatia and Hlavacek, 1983;Kubicek, 1976). The problem is solved in terms of the new parameter s, which proves computationally more convenient in the region of steep gradients than the original system.…”

Morphology and Film Growth in CVD Reactions

“…Therefore, the arclength (say s) is introduced as new independent variable and the original independent variable (say x) becomes a dependent variable. The relation between x and s introduces an additional equation and the original system equations are transformed and expressed in terms of the arc-length s (Bhatia and Hlavacek [53], Kubicek [54]). The problem is solved in terms of the new parameter s, which proves computationally more convenient in the region of steep gradients than the original system.…”