Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations—An application to the blow-up problems of partial differential equations

On the finite difference approximation for a parabolic blow-up problem

“…Recently a substantial generalization was made on the convergence of the blow-up time by Abia and others [3]. A different approach was proposed by Hirota and Ozawa [9].…”

Numerical Solution of a Nonlocal Problem Modelling Ohmic Heating of Foods

“…the blow-up time, useful for the point of view of applications. Similar works that investigate the convergence of a numerical solution during blow-up have been done, for a parabolic problem, in [2,3,11]. In our case, for example, for f being an increasing function a discontinuity in the initial data may cause blow-up of the solution [15].…”

On the convergence of numerical blow-up time for a second order nonlinear ordinary differential equation

“…Although the possibility of T (τ ) = ∞ is not excluded by this definition only, we can prove that it is finite if the time mesh ∆t n is given by (5) and satisfies the assumptions (A1)-(A4). In fact, we have the following Theorem 2.…”

“…See, for instance, [1][2][3][4][5][6][7]. Numerical estimation of the blow-up is important, but the error bound for the blow-up time is not known, except for a scalar ordinary differential equation of first order in [2].…”

“…The condition (5) implies that ∆t n = τ is chosen while u n is small, and that smaller ∆t n is chosen if u n becomes large. In practice,…”