The Global Behavior Of Second Order Discrete Difference Scheme For A Partial Integro-differential Equation With A Weakly Singular Kernel

The integro-differential equation of parabolic type often occurs in applications such as heat conduction in material with memory, compression ofporo-viscoelastic media, population dynamics, nuclear reactor, dynamics, etc..There are lots of papers of V. Thomee [1,5,7,16,17,18,19,20,21,22,23,24,31], Stig. Larsson[19], W. Mclean[5,17,20,24], Ch. Lubich[18], J. C. Ldpez-Marcos [14], J. M. Sanz-Serna [6], G.Fairweather[3,15], L. Wahlbin [1,17,19], I. H. Sloan [7,18,22,23], Yanping Lin [31] in overseas and Chuan-miao Chen [1,35], Yun-qingHuang [2], Da Xu [8,9,10,11,12,13], Tao Tang [33], Qiya Hu[34], Tie Zhang [45] in home. A lot of them use FEM[1,5,10,13,16,31,35,39]; Spline collocation methods[3,15]; Spectral collocationmethods[25]. But a few of them make the global behavior of discretization byfinite difference scheme.We study a partial intcgro-deffcrential equation of parabolic type witha weakly singular kernel, using second order discrete difference scheme, andderive stabilities and error estimate respectively.Main results follows:(1)Given the scheme and its global stability of second order spatial semidiscretization difference method for the equation;(2)Given the global error estimate of second order spatial semi-discretizationdifference scheme for the equation;(3)Given the full discretization scheme of finite difference method for theequation.