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 我們考慮對於半線性熱方程和其非負初始質以及Dirichlet邊界條件的有限差分近似. 我們知道,雖然半線性熱方程在有限時間爆炸是可以有方法和使用適當的時間網格去證明,但在數值爆炸集合方面卻不一定會與真實的一致. 而在這篇文章中, 我們的目的是使用不同的時間網格來尋找數值爆炸集合.
 We consider the finite difference approximation for the semilinear heat equationu_t = u_xx+f(u) (0 < t; 0 < x < 1) with nonnegative initial data u(0, x) = u_0(x) (0 < x < 1) and the Dirichlet boundary condition u(t, 0) = u(t, 1) = 0 (t > 0). It is known that although the finite-time blow-up for the semilinear heat equation can be reproduced by a scheme with adaptively-defined time mesh, the numerical blow-upsets do not always coincide with that of the real one. In this paper, we are aimed to investigate the numerical blow-up sets with respect to different time meshes.
 1 Introduction 22 Explicit scheme 53 Implicit scheme 114 Conclusion 155 References 16
 [1] C.-H. Cho, On the Computation of the Numerical Blow-up Time. Method Partial Dif-ference. To appear in Japan J. Indust. Appl. Math.[2] C.-H. Cho, S. Hamada and H. Okamoto, On the Finite Di erence Approximation for aParabolic Blow-up Problem. Japan J. Appl. Math. 24 (2007), 475-498.[3] C.-H. Cho and H. Okamoto, Further Remerks on Asymptotic Behavior of the NumericalSolutions of the Parabolic Blow-up Problems. Meth. Appl. Anal. 14 (2007), 213-226.[4] Nakagawa, T., Blowing up of a Finite Di erence Solution to ut = uxx+u2. Appl. Math.Optim. 2 (1976), 337-350.[5] Y.-G. Chen, Asymptotic Behaviours of Blowing-up Solutions for Finite Di erence Ana-logue of ut = uxx + u1+ . J. Fac. Sci., Univ. Tokyo 33 (1986), 541-574.[6] P. Groisman, Totally Discrete Explicit and Semi-implicit Euler methods for a Blow-upProblem in Several Space Dimensions. Computing, 76 (2006), 325-352.[7] C.-F. Chang, A Finite Di erence Scheme for Blow-up Solutions of the ConvectiveReaction-di usion Eequations. Math Thesis.[8] T.-F. Chen, Levine H. A., and Sacks P. E.: Analysis of a convective reaction-di usionequation. Nonlinear Anal. Theor. Meth. Appl. 12, (1988), 1394-1370.[9] K. Deng and H.A Levine, The Role of Critical Exponents in Blow-up Theorems: TheSequal, J. Math. Anal. Appl., 243 (2000), pp. 85-126.[10] A. Friedman and B. BcLeod, Blow-up of Positive Solutions of Semilinear Heat Equa-tions, Indiana Univ. Math. J., 34 (1985), pp. 425-447.[11] S. Ito, On Blow-up of Positive Solutions of Semilinear Parabolic Equations, J. Fac. Sci.Univ. Tokyo, Sect. IA, 37 (1990), pp. 527-536[12] L.Abia, J.C. Lopez-Marcos, J. Martnez, The Euler method in the numerical integrationof reaction-di usion problem with blow-up, Appl. Numer. Math., 38 (2001) 287-313.17
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