rober - ODE - Problem ROBER II-10-1 10 Problem ROBER 10.1...

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Unformatted text preview: ODE - Problem ROBER II-10-1 10 Problem ROBER 10.1 General information The problem consists of a sti system of 3 non-linear ordinary di erential equations. It was proposed by H.H. Robertson in 1966 [ Rob66 ]. The name ROBER was given by Hairer & Wanner [ HW96 ]. The INdAM-Bari Test Set group contributed this problem to the test set. The software part of the problem is in the le rober.f available at [ MI03 ]. 10.2 Mathematical description of the problem The problem is of the form dy dt = f ( y ) ; y (0) = y ; with y 2 IR 3 ; t 2 [0 ; T ] ; The function f is de ned by f ( y ) = @ : 04 y 1 + 10 4 y 2 y 3 : 04 y 1 10 4 y 2 y 3 3 10 7 y 2 2 3 10 7 y 2 2 1 A (II.10.1) The initial vector y is given by (1 ; ; 0) T . 10.3 Origin of the problem The ROBER problem describes the kinetics of an autocatalytic reaction given by Robertson (1966) [ Rob66 ]. The structure of the reactions is given in Table II.10.1 , where k 1 ; k 2 ; k 3 are the rate constants and A , B and C are the chemical species involved. Under some idealized conditions [ Aik85 ] and the 1. A k 1 ! B 2. B + B k 2 ! C + B 3. B + C k 3 ! A + C Table II.10.1: Reaction scheme for problem ROBER assumption that the mass action law is applied for the rate functions, the following mathematical model consisting of a set of three ODEs can be set up @ y 1 y 2 y 3 1 A = @ k 1 y 1 + k 3 y 2 y 3 k 1 y 1 k 2 y 2 2 k 3 y 2 y 3 k 2 y 2 2 1 A ; (II.10.2) with ( y 1 (0) ; y 2 (0) ; y 3 (0)) T = ( y 01...
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rober - ODE - Problem ROBER II-10-1 10 Problem ROBER 10.1...

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