Abstract
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.
Highlights
In this paper, we study the behavior of a semidiscrete approximation of the following heat equation involving nonlinear boundary flux conditions : ut = uxx, x ∈ (0, 1), t ∈ (0, T ), ux(0, t) = up(0, t), ux(1, t) = uq(1, t), u(x, 0) = u0(x), x ∈ [0, 1], t ∈ (0, T ), (1)where p, q are positive constants
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
Summary
In (Ozalp & Selcusk, 2015) Ozalp and Selcuk show that under certain conditions, any positive solution of the problem (1) must blow up in a finite time and the blow-up point occurs only at the boundary x = 1. By using a theorem of Ushijima (Ushijima, 2000) and under certain conditions we show that any positive solution of semidiscrete scheme of (1) blows up in a finite time and the semidiscrete blow-up time converges to the theoretical one when the mesh size goes to zero. When Th is finite, we affirm that the solution Uh blows up in a finite time and the time Th is called the blow-up time of the solution Uh
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