Abstract
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.
Highlights
Consider the following problem∫t vt = vxx + vq vp(x, s)ds, x ∈ (0, 1), t ∈ (0, T ), (1)v(0, t) = 0, vx(1, t) = 0, t ∈ (0, T ), (2)v(x, 0) = v0(x) > 0, x ∈ (0, 1), (3)which models the temperature distribution of a large number of physical phenomenon from physics, chemistry and biology
They have considered a scheme as the one given in (4)-(6). They have shown that the semi-discrete solution blows up in the finite time and its blow-up time goes to the real one when the mesh size tends to zero
In order to obtain the convergence of the semi-discrete blow-up time, we firstly prove the following theorem about the convergence of the semi-discrete scheme
Summary
Which models the temperature distribution of a large number of physical phenomenon from physics, chemistry and biology. They have considered a scheme as the one given in (4)-(6) They have shown that the semi-discrete solution blows up in the finite time and its blow-up time goes to the real one when the mesh size tends to zero. We show that under some assumptions, the solution of the semi-discrete problem defined in (4)-(6) blows up in a finite time and estimate its semi-discrete blow-up time. We show that, under some additional hypothesis, the semi-discrete blow-up time goes to the real one when the mesh size goes to zero. In the last section we report on some numerical experiments to illustrate our analysis
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