BZ reaction3 - ON A MATHEMATICAL MODEL FOR THE...

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57 ON A MATHEMATICAL MODEL FOR THE BELOUSOV-ZHABOTINSKII (BZ) REACTION Min Kyung Kwon, McNair Scholar Dr. V.S. Manoranjan, Faculty Mentor Department of Mathematics ABSTRACT Many real world problems can be modeled using ordinary differential equations (ODEs). An example is the Belousov-Zhabotinskii (BZ) reaction, where malonic acid is oxidized by acid bromate in the presence of ferroin. Many researchers have studied the BZ reaction and its ability to generate nonlinear waves. In this paper, we analyze a mathematical model that describes the BZ reaction in order to understand the nonlinear phenomena associated with it. INTRODUCTION Many real world problems can be modeled and analyzed using ordinary differential equations (ODEs). An example is the Belousov-Zhabotinskii (BZ) chemical reaction, in which malonic acid is oxidized by acid bromate in the presence of ferroin. The Belousov-Zhabotinskii reaction was first noted when Belousov discovered the oscillation of a chemical reaction (Belousov, 1958). However, his study was not proven or published until it was rediscovered by Zhabotinskii (1964). When Zhabotinskii published his finding, Belousov was given credit for his original discovery. Tyson (1976) discusses the BZ reaction in great detail. The chemical reactions that are part of the BZ reaction can be described as: . 5 2 4 2 ) ( 4 2 2 2 3 2 2 2 2 3 2 2 4 3 2 2 2 4 2 3 3 2 2 3 + + + + + + + + + + + + + + + + + + + + + + + + + + + H CO HCOOH Br Ce O H COOH BrCH Ce H HOBr BrO HBrO O H HBrO Ce H HBrO BrO Ce HOBr H Br HBrO HOBr HBrO H Br BrO (1) Now equations (1) can be simplified into . 2 2 2 2 hY Z P A X Z X X A P Y X P X Y A + + + + + + (2)
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58 by replacing A with [ 3 BrO ], P with [ HOBr ], X with [ 2 HBrO ], Y with [ Br ], and Z with [ 4 + Ce ]. Tyson re-formulates equations (2) into a mathematical form as given below. (3) This system is described only with respect to X= [ 2 HBrO ] , Y = [ Br ], and Z= [ 4 + Ce ] , and the left hand sides of the equations express the rates of change of the chemical substances in the system. One can seen the terms on the right hand side of the equations (3) are nonlinear. Therefore, these equations represent a specific example of a nonlinear phenomenon in chemical studies. The work presented here focuses on a non-dimensioned variation of the mathematical model given by (3). As noted above, a detailed background on the BZ reaction and on the various mathematical concepts used in this study can be obtained from Tyson (1976; see also, Field and Noyes, 1974). PROBLEM The BZ reaction model as examined here is as follows: v ruv u u dt du 1 ) 1 ( ε + = (4) buv v dt dv = 2 where u and v represent the concentrations as [ 2 HBrO ], and [ Br ], b, r , 1 , and 2 are positive constants. Here u and v are unknown variables and they need to be solved and interpreted. The above model is a nonlinear system of ODEs and the independent variable of the system is t , the time. It is also a coupled system with the coupling coming via the interaction terms ruv and buv . Therefore, when solving, both nonlinear ODEs should be solved simultaneously.
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BZ reaction3 - ON A MATHEMATICAL MODEL FOR THE...

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