【作者】 邓钧；

【导师】 徐大；

【作者基本信息】 湖南师范大学 ， 基础数学， 2007， 硕士

【摘要】 在记忆材料的热传导、多孔粘弹性介质的压缩、原子反应、动力学等问题中,常常碰到抛物型偏积分微分方程,对于该方程的数值求解,国外的V.thomee,W.Mclean,Ch.Lubich,L.Wahlbin,Sanz-Serna,E.G.Yanik,G.Fairweather,国内的陈传淼、黄元清、徐大等做了大量的研究,他们采用了有限元方法、谱配置方法及样条配置方法,但用Lubich的拉普拉斯变换数值逆离散却很少涉及.本文考虑一类带弱奇异核抛物型偏积分微分方程时间、空间全离散格式,采用Lubich的拉普拉斯变换数值逆离散等方法进行数值计算,主要结果如下:(1)给出偏积分微分方程空间x方向用有限差分法离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算。(2)给出偏积分微分方程空间x方向用线性有限元离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算.以上两种方法计算结果精度较高,并且计算也比较简便.更多还原

【Abstract】 The integro-differential equation of parabolic type often occurs in application such as heat conduction in material with memory, comression of poro-viscoelastic media , nuclear reactor dynamics, ect. theres arelotsof documents of V. thomee, W. Mclean, Ch. Lubich, L. Wahlbin, Sanz-Serna, E. G. Yanik, G. Fairweather in overseas and Chuan-miao Chen、Yuan-qing Huang、Da-Xu in home. A lot of them use FEM ;Spectral collocation methods;Spline collocation methods.but few of them use numerical inversion for the laplace transform of Lubich.We study a partial integro-differential equations of parabolic type with a weakly singular kernel, which use numerical inversion for the laplace transform of Lubich for numerical calculation, main results follows:(1) Give a kind of fully discrete scheme of a partial integro-differential equations which use finite difference method discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.(2) Give a kind of fully discrete scheme of a partial integro-differential equations which use linear finite element discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.Calculating result of two methods is accuratly higher, and calculate is simpler also.更多还原