Orthogonal Spline Collocation For A Partial Integro-differential Equation With A Weakly Singular Kernel

The integro-differential equation of parabolic type often occurs in application such as heat conduction in material with memory, compression of poro-viscoelastic media, population dynamics, nuclear reactor dynamics, etc. there are lots of documents of V. Thomée[7. 19, 22, 27, 28, 29], W. Mclean[27, 28, 29], Ch. Lubich[19], L. Wahlbin[7, 22, 29], Graeme Fairweather [4, 11, 12, 13, 31, 43, 44]in overseas and Chuanmiao Chen[7], Tang Tao[35], Chuanju Xu[20], Sun Zhizhong [34], Xu Da[37-42]in home. A lot of them use FEM; finite difference methods ; spectral collocation methods; spline collocation methods. But a few of them make global behavior of full discretization by orthogonal spline collocation methods.We study a partial integrol-differential equations of parabolic type with a weakly singular kernel, using orthogonal spline collocation methods derived stabilities and error estimated respectively.Main results as follows:(1)Given the stability,error estimate of time semi-discretization Euler methods for the linear equation;(2)Given the stability and error estimate of full discretization for the based on the orthogonal spline methods for the linear equation.(3)Numerical experiments.