Numerical Solution Of A Kind Of Differential Equation

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.