We study numerical approximations of solutions of a heat equation with nonlinear boundary conditions which produce blow-up of the solutions. By a semidiscretization using a finite difference scheme in the space variable we get a system of ordinary differential equations which is an approximation of the original problem. We obtain sufficient conditions which guarantee the blow-up solution of this system in a finite time. We also show that this blow-up time converges to the theoretical one when the mesh size goes to zero. We present some numerical results to illustrate certain point of our work.
In this paper, we study the quenching behavior of semidiscretizations of the heat equation with nonlinear boundary conditions. We obtain some conditions under which the positive solution of the semidiscrete problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate. Finally we give some numerical results to illustrate our analysis.
In this paper, we study a numerical approximation of the following problem ut = uxx, vt = vxx, 0 < x < 1, 0 < t < T ; ux(0, t) = u −m (0, t) + v −p (0, t), vx(0, t) = u −q (0, t) + v −n (0, t) and ux(1, t) = vx(1, t) = 0, 0 < t < T, where m, p, q and n are parameters. We prove that the solution of a semidiscrete form of above problem quenches in a finite time only at first node of the mesh. We show that the time derivative of the solution blows up at quenching node. Some conditions under which the non-simultaneous or simultaneous quenching occurs for the solution of the semidiscrete problem are obtained. We establish the convergence of the quenching time. Finally, some numerical results to illustrate our analysis are given.
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