A numerical method for semilinear singular parabolic quenching problems

For the problem given by ${u_{xx}} + b{u_x}/x - {u_t} = - {\left ( {1 - u} \right )^{ - 1}}$ for $0 < x < a, \\ 0 < t < {T_a} \le \infty , u\left ( {x, 0} \right ) = 0 = u\left ( {0, t} \right ) = u\left ( {a, t} \right )$, where $b$ is a constant less than one, a lower bound of $u$ is used to estimate the critical length $a$ beyond which quenching occurs, and an upper bound for the time when quenching happens. An upper bound of $u$, given by the minimal solution of its steady state, is constructed by using a modified Picard method with the construction of the appropriate Green’s function. To determine the critical length numerically, it is shown that for a given length $a$, all iterates attain their maximum values at the same $x$-coordinate; the largest interval for existence of the minimal solution corresponds to the critical length for the parabolic problem. As illustrations of the numerical method, the critical lengths corresponding to four given values of $b$ are computed.