A Second Order Fully 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 of poro-viscoelastic media ,nuclear reactor dynamics , etc. . There are lots of documents of V.thomee([l , 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.L6pez-Marcos([3]) , J.M.Sanz-Serna([6]) , G.Fairweather([l4 , 15]) , L.Wahlbin([l , 17 , 19]) , I.H.Sloan([7 , 18 , 22 , 23]) , Yanping Lin([31]) in overseas and Chuan-miao Chen([l , 35]), Yun-qing Huang([2]) , Da Xu ([8 , 9 , 10 , 11 , 12 , 13]) , Tao Tang([33]) , Qiya Hu([34]) , Zhang Tie[39] in home. A lot of them use FEM([1 , 5 , 10 , 13 , 16 , 31 , 35 , 39]) ; Spectral collocation methods([33]) ; Spline collocation methods([15]) . but a few of them use finite difference scheme([3 , 6]) .We study a partial integro-differential equations of parabolic type with a weakly singular kernel, which using second order fully discrete difference scheme.derived stabilities anh error estimeted respectively.Main results follows:(l)Given the stability,error estimate and numerical experiments for the linear equation of second order backward difference and Crank-nicolson scheme(2)Given the stability,error estimate and numerical experiments for the linear equation of a O(k^3/2 + h⁴) high accuracy scheme.(3)Given the stability,error estimate and numerical experiments for the nonlinear equation of second order backward difference(4)Given the weighted of second order convolution quadrature.