NUMERICAL QUENCHING FOR A SLOW DIFFUSION SYSTEM COUPLED AT THE BOUNDARY

P. A. T. Diabaté, A. Coulibaly, K. B. Edja, and A. K. Touré

  DOI:  https://doi.org/10.37418/amsj.9.11.100

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This paper concerns the study of a numerical approximation for the following problem, $u_t=u_{xx}$, $v_t=v_{xx}$, $0<x<1$, $0<t<T$; $u_x(0,t)=(u^{-p_1}v^{-q_1})(0,t)$, $v_x(0,t)=(u^{-p_2}v^{-q_2})(0,t)$ and $u_x(1,t)=v_x(1,t)=0$, \linebreak $0<t<T$, with $p_1$, $q_1$, $p_2$ and $q_2$ real parameters. We show that the solution of the semidiscrete scheme, obtained by the finite differences method quenches in a finite time only at first node of the mesh. We also prove that the time derivative of the solution blows up at quenching node and establish some conditions under which occurs the non-simultaneous or simultaneous quenching for the solution of the semidiscrete problem. After show the convergence of the quenching time, we finally present some numerical results to illustrate certain point of our work.

Keywords: Numerical quenching, non-simultaneous quenching, Newton filtration equations, boundary singularities, quenching time.