Superconvergent Numerical Algorithms Of Integral Equations And Their Compact Operators

This master thesis mainly focuses on superconvergent numerical algorithm of integralequation and its compact operators, they are priorities and hot spots of computationalmathematics that have strong background in application in mathematics, physics and en-gineering. At present, the study focused on numerical solution of di?erential equations andoperators in domestic, while the numerical results of the integral equation and the compactoperators are very little, which is a new hot spot of the fast algorithm in computationalmathematics. Our purpose is choosing low complexity polynomial as the basement of fi-nite dimensional subspace, and constructing framework of the multi-projection algorithmand the iterative projection algorithm for solving problem that the traditional projec-tion method does not have the super-convergence. We apply the algorithm to Galerkinmethod, iterative Galerkin method, and Collocation method for fredholm integral equa-tions of the second kind, When the smoothness of the integral kernel function meets certainconditions, accuracy of solution of mult-projection method and iterative mult-projectionmethod could reach three times and four times than the projection method, and reducingthe computational complexity ,while we requires a super-convergence of numerical solu-tion of the contradictions. Meanwhile, for the convergence order of numerical solution , wechoose proper numerical integration formula for constructing fully discrete case of polyno-mial multi-iterative projection algorithm and its theoretical framework, then we analyzefully discrete solutions and iterative approximation solution with the super-convergence.We choose proper polynomial space as the projection space in multi-projection algo-rithm for eigenvalue problem of compact integral operators which di?er from the standardprojection space of piecewise polynomials, because the basement of polynomial space iseasy to construct, and reduce the computational complexity . On the other hand, becausethe polynomial space is poor ?exibility, theoretical analysis is relatively di?cult. Then, weselect the polynomial space to try to establish a multi-projection method approximationframework. so, we have received three times and four times than standard approximatesolution. Approximate solution has improved the numerical accuracy of scholar Rekha P.Kulkani of India in 2003, while reducing the complexity of computation.