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A Simple Algorithm to Solve Non-linear Equations with Multiple Roots of Two Variables Appearing in Autocatalytic Reaction in Open Systems

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Amih SAGAN^{a}, Umpei NAGASHIMA^{a}* and
Yoshihito MORI^{b}

^{a}National Institute of Materials and Chemical Research,

1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan

**e-mail: *

^{b}Department of Chemistry, Faculty of Science, Ochanomizu University,

2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

(Received: January 11, 1999 ; Accepted for publication: February 18, 1999 ; Published on Web: March 23, 1999)

We propose a simple computational algorithm to solve non-linear equations with multiple roots of two variables: F(x,y)=0. The algorithm is effective only where the relationship between two variables x and y is monotonous and continuous (for example, see Figures 1, 3). The algorithm also gives a guideline of automatic adjustment of the notch width in the Euler method.

The efficiency of the algorithm was demonstrated by analyzing two typical examples and an autocatalytic reaction in open systems.

**Keywords:** Auto-catalytic Reaction, Non-linear Equation, Multiple Roots, Differential Equation, Euler Method

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