No abstract

In this paper, we study numerical approximations for positive solutions of a semilinear heat equations, $u_{t}=u_{xx}+u^{p}$, in a bounded interval $(0,1)$, with a nonlinear flux boundary condition at the boundary $u_{x}(0,t)=0$, $u_{x}(1,t)=-u^{-q}(1,t)$. By a semi-discretization using finite difference method, we get a system of ordinary differential equations which is expected to be an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches or blows up in a finite time and estimate its semidiscrete blow-up and quenching time. We also estimate the semidiscrete blow-up and quenching rate. Finally, we give some numerical results to illustrate our analysis.

In this paper, we study a localized nonlinear reaction diffusion equation. We investigate interactions among the localized and local sources, nonlinear diffusion with the zero boundary value condition to establish the blow-up solution and estimate the numerical approximation for the following initialboundary value problem: We find some conditions under which the solution of a discrete form of the above problem blows up in a finite time and a numerical method is proposed for estimating its numerical blow-up time. We also prove the convergence of the numerical blow-up time to the theoretical one. Finally, we give some numerical results to illustrate our analysis.

In this paper, we study the numerical approximation for the following initial-boundary value problem v_t=v_{xx}+v^q\int_{0}^{t}v^p(x,s)ds, x\in(0,1), t\in(0,T) v(0,t)=0, v_x(1,t)=0, t\in(0,T) v(x,0)=v_0(x)>0}, x\in(0,1) where q>1, p>0. Under some assumptions, it is  shown that the solution of a semi-discrete form of this problem blows up in the finite time and estimate its semi-discrete blow-up time. We also prove that the semi-discrete blows-up time converges to the real one when the mesh size goes to zero. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.

No abstract