A Method of Estimating the Blow-up Time and Blow-up Rate of the Solution of the System of Ordinary Differential Equations : An Application to the Blow-up Problems of Partial Differential Equations

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  • HIROTA Chiaki
    Faculty of Systems Science and Technology Akita Prefectural University
  • OZAWA Kazufumi
    Faculty of Systems Science and Technology Akita Prefectural University

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Other Title
  • 常微分方程式系の解の爆発時刻および爆発レートの推定法 : 偏微分方程式の爆発問題への応用
  • ジョウ ビブン ホウテイシキケイ ノ カイ ノ バクハツ ジコク オヨビ バクハツ レート ノ スイテイホウ ヘンビブン ホウテイシキ ノ バクハツ モンダイ エ ノ オウヨウ

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A numerical method is proposed for estimating the blow-up time and blow-up rate of the solution of the system of ordinary differential equations (ODEs), whose solution has a pole at a finite time, that is, the blow-up time. The main idea is to transform the ODE system into a tractable form by Moriguti's technique, and to generate a linearly convergent sequence to the blow-up time. The sequence is accelerated by the Aitken Δ^2 method. The present method is applied to the problem of finding the blow-up time of the solution of partial differential equations (PDEs), by discretizing the PDEs in space. Numerical experiments on the two PDEs, the semilinear reaction-diffusion equation and the heat equation with a nonlinear boundary condition, show the validity of the present method.

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