Bounds on the burning-rate eigenvalue

When the problem is formulated in a well-posed manner, upper and lower bounds may be established rigorously for A. This has been accomplished by Johnson and Nachbar [21], [29] for the formulation based on the ignition temperature. The analysis employs a form of equation (46) obtained by defining g = ch/d and by treating t as the independent variable and g as the dependent variable. From equation (46) we see that 

By substituting equation (50) into the integral on the right-hand side of equation (49), it is found that 

Since a , (1 — t), and g are all nonnegative, it is clear that by substituting an upper bound for g x) into equation (51) a lower bound for the integral is obtained, whence the resulting value of A given by equation (51) will be an upper bound. On the other hand, substituting this same upper 

Hence t may be used for the upper bound of g in equations (51) and (54). Equation (51) then implies that 

An upper bound for (i) is substituted into equation (54) in order to obtain a lower bound for A. If this value of A and the upper bound for (t) are substituted into the right-hand side of equation (60), then, since the integral in equation (60) will clearly assume a lower bound, the right-hand side of equation (60) will assume an upper bound, and hence a new upper bound for g (x) will be given by equation (60). The fact that the resulting new upper bound for (t) is lower than the previous upper bound requires proof by mathematical induction.  

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