High Precision Algorithm For Multidimensional Weakly Singular Integrals And Integral Equations

Since the 19 th century,many problems in mathematics,physics,and engineering have been modeled by different kinds of singular integrals,singular integral operators and singular integral equations.The pioneering work of Volterra and Fredholm in the late 19 th century and early 20 th century dominated the development of analysis in the20 th century.Inspired by Fredholm’s work,Hilbert defined the Hilbert space,which provides a powerful tool for future theoretical analysis.Integral equations were developed vigorously in the 20 th century,and now many problems in science and engineering can be described by integral equations or integro-differential equations.For example,many problems,including oil and gas exploration,medical scanning,material detection and parameter identification problems,can be solved by seeking the density of an object through sound waves,radiations,and so on.The heat transfer problem with memory materials can be reduced to solving a Volterra integro-differential equation.Many partial differential equations with initial and boundary conditions can be transformed into boundary integral equations of the first or second kind by a direct or indirect method.From the point of view of computational mathematics,it is more difficult to deal with an integral equation than a differential equation: firstly,the discrete matrix is a full matrix,and the complexity of calculating a full matrix is the cubic order of the number of unknowns;secondly,each element of a full matrix is obtained by calculating an integral,so the computational complexity of generating a discrete matrix may exceed the problem itself.The problems studied in this paper are multidimensional problems or singular kernel problems,which increase the difficulty and complexity of solving these problems.Many effective numerical methods lose their advantages when extending from one dimension to multi-dimension.Therefore,in order to propose efficient numerical algorithms,the following four aspects are studied in this dissertation.1.We first study the numerical quadrature method for weakly singular integral of product-type,which only has singularity at end-points.According to quadrature formulas,we also construct the asymptotic expansions of errors.Furthermore,we give the quadrature formulas of two-dimensional and multidimensional weakly singular integrals of product-type with parameters and their asymptotic expansions of errors,respectively.Then,the extrapolation and splitting extrapolation algorithms are constructed according to the errors expansions to speed up the convergence.The algorithm improves the accuracy and convergence order of the numerical solution by eliminating the lower order terms in the errors expansions one by one.Different from the single-step expansions,the errors asymptotic expansions derived in this paper are multi-step,which can be discretized in each direction,and then achieving the purpose of improving the accuracy by linear combination.It can be seen that this algorithm is a highly parallel algorithm,which can effectively solve the problem of high computational complexity caused by high dimension.2.We propose a numerical method for solving two-dimensional nonlinear Volterra integral equation.The Nystr?m method can avoid the calculation of integral and reduce the computational complexity;the extrapolation method can improve the error accuracy and convergence order of the numerical solution.Our method combines the advantage of Nystr?m method and extrapolation algorithm.We first generalize the Gronwall inequality,which plays a key role in proving the existence and uniqueness of the solution of the equation.In this method,the integral in the equation is replaced by the quadrature formula,and the numerical solution of each point is calculated by iteration method.The accuracy and convergence order of numerical solution can be further improved by using the extrapolation method.In order to analyze the existence and uniqueness of the solution of the discrete equation,we further generalize the two-dimensional discrete Gronwall inequality.At the same time,the convergence and stability of the method are also given.The numerical results are in good agreement with the theoretical analysis.3.The numerical method for solving multidimensional Volterra integral equations with weakly singular kernels is studied.Based on the important application of Bernstein polynomials in the function approximation theory,we first generalize Bernstein polynomials from one dimension to dimension,and use them to construct a set of basis functions to approximate unknown functions.Then,we use the quadrature and extrapolation method proposed in chapter 2 to approximate the weakly singular integral in the equation.In addition,we further generalize the multidimensional Gronwall inequality,and prove the existence and uniqueness of the solution of the discrete equation.The convergence analysis of numerical method is also proved in this dissertation.The numerical examples show that this method is an effective numerical method.4.Furthermore,a numerical method for solving integro-differential equation of fractional order is given.It is difficult to analyze the existence and uniqueness of the solution of fractional order equation directly.Therefore,we transform a fractional integrodifferential equation into a Volterra integral equation of the second kind,and then use the analysis result of chapter 3 to prove the existence and uniqueness of the solution.After the transformation,there is no need to derive any derivative of the unknown function,which can expand the scope of application of the method,and we can solve them by the discrete collocation method.Before that,we prove that “discrete collocation method”is the so-called “iterative Nystr?m method”.The convergence of the method is analyzed under the framework of Nystr?m theory.