The quenching problem for nonlinear parabolic differential equations

A numerical method for semilinear singular parabolic quenching problems

“…2 and 3, by using subroutines from the IMSL Library to compute the critical lengths corresponding to different values of b. In particular for b = 0, we obtain the result of Acker and Walter [1], Two graphs are also given to illustrate other results. Our numerical method is very stable since it involves mainly Newton's method of finding a root, and numerical integrations.…”

“…Results on the behavior of (1.3) subject to (1.2) with a = a* were given by Levine and Montgomery [22], The case when the solution is driven by the boundary condition with the forcing term of the differential equation being identically zero was studied by Levine [19]. Higher space dimensional problems were investigated by Walter [27], Acker and Walter [1,2], and Levine and Lieberman [21]. Related phenomena for hyperbolic equations were given by Chang and Levine [12], and Levine [19].…”

“…Critical lengths. By modifying the proofs of the corresponding theorems of Acker and Walter [1,2] for the special case b = 0, the following theorem can be established. 1) is a solution of the nonlinear singular twopoint boundary-value problem (1.5); furthermore, u < v for 0 < x < a, 0 < t < Ta.…”

“…2, we use the method of Acker and Walter [1,2] to establish that there exists a unique critical length a*. We then use a Sturm-Liouville problem to find a lower bound for the problem (1.4) and (1.2), and to obtain an estimate for a* as well as an upper bound for the time when quenching occurs.…”

“…Acker and Walter [1,2] studied the phenomena of quenching for the equations, uxx -m* = -/("). 0 <x<a, 0 <t<Ta, (…”