Numerical Solution Of Partial Integro-differential Equation (Orthogonal Spline Collocation Method And The Proposed Wavelet Method)

The integro-differential equation of parabolic type often occurs in applica-tion such as Engineer application, porous viscoelastic of Known fluctuations questions, vibration problem,Dynamic population,etc.In the numerical solu-tion of such problems, there are lots of documents of V. Thomee[7,19,22, 27,28,29], Ch. Lubich[19],W. Mclean[27,28,29],Graeme Fairweather[4,11, 12,13,31,43,44] L. Wahlbin[7,22,29] in overseas and Chuanmiao Chen[7], Chuanju Xu[20], Tang Tao[35], Xu Da[37-42], Sun Zhizhong [34]in home.A lot of them use FEM;spectral collocation methods; finite difference methods ; spline collocation methods. But a few of them make global behavior of full discretization by orthogonal spline collocation methods.Firstly,We study a partial integrol-differential equations of parabolic type with a weakly singular kernel,making time semi-discrete, time and space all discrete,using orthogonal spline collocation methods derived stabilities and error estimated respectively.Main results as follows:(1) Gives the corresponding time semi-discrete form of stability and error estimates.(2)Given the stability and error estimate of full discretization for the based on the orthogonal spline methods for the linear equation.Secondly, using the quasi-wavelet method to solve a partial integro-differential equation with a weakly singular kernel,we solve the spatial derivatives by us-ing the quasi-wavelet,the time derivative using the first-order derivative.The quasi-wavelet method can well describe the local fast-changing characteristics of the function.