Research On Solving Integral Equations Based On Barycentric Interpolation Collocation Methods

Multi-dimensional integral equations have been widely applied in mathematical models of many engineering problems.For example,double integral equations can model the bridged problem of composites fracture mechanics,multi-dimensional Fredholm integral equations can describe the electromagnetic scattering problem,multi-dimensional weakly singular integral equations can characterize the electromagnetic casting process.These integral equations often need to be solved by numerical methods due to the restriction of dimension effect and singularity.Barycentric interpolation is an improved variant of classical Lagrange interpolation,which has the advantages of simple formula,excellent numerical stability,high accuracy,good adaptability of nodes.This dissertation solves several kinds of multidimensional integral equations by barycentric interpolation collocation methods based on barycentric Lagrange interpolation and barycentric rational interpolation.The main contents are as follows:1.The barycentric interpolation collocation methods for solving multidimensional linear integral equations are proposed based on the multidimensional barycentric interpolations.The multidimensional linear integral equation is discretized into the corresponding system of linear algebraic equations by the multidimensional barycentric interpolation formulas and the multidimensional Gauss-Legendre quadrature formula.The existence,uniqueness,and error estimation of the approximate solution can be analyzed under the framework of collectively compact convergence theory by introducing a linear numerical integral operator.During calculation,the generation of discrete matrices does not require calculating the multiple integral,which is easy to be generalized to solve a system of multi-dimensional linear Fredholm integral equations.Numerical results manifest that the proposed methods are numerical methods with high precision and good stability.2.The barycentric interpolation collocation methods for solving(a system of)multidimensional nonlinear integral equations are established based on the multidimensional barycentric interpolations.The multidimensional nonlinear Fredholm integral equation is discretized by combining the multidimensional barycentric interpolation formulas with the multidimensional Gauss-Legendre quadrature rule.The obtained system of nonlinear algebraic equations is solved by direct iterative method or Newton iterative method.The existence and uniqueness of the approximate solution are analyzed by introducing a nonlinear numerical integral operator.Furthermore,error estimation is derived.During calculation,the multidimensional integral operation is transformed into a function value operation by a numerical integral rule,which is more convenient for calculation.Numerical results show the reliability and effectiveness of the proposed methods.3.The transformed barycentric interpolation collocation methods for solving a multidimensional fractional Volterra integral equation with nonsmooth kernel or solution are presented based on smoothing transformations and the multidimensional barycentric interpolations.Due to the singularity,it is difficult to obtain a highly accurate numerical solution by using the barycentric interpolation collocation methods directly.By introducing a smoothing transformation,the original equation is transformed into a new form,so that the solution of the transformed equation has better regularity.Then,the transformed equation can be solved by the multidimensional barycentric interpolation formulas together with the multidimensional Gauss-Jacobi quadrature formula.Furthermore,the approximate solution of the original equation is obtained by the corresponding inverse transformation,and its error estimate is analyzed.The smoothing transformation not only improves the smoothness of solution,but also guarantees the high accuracy and excellent numerically stability of the barycentric interpolation collocation methods.Since the provided smoothing transformation and the corresponding inverse transformation are both continuous functions,the whole algorithm has a strong feasibility.Numerical results manifest that the proposed methods can achieve a highly accurate numerical solution of the nonsmooth multidimensional fractional Volterra integral equation.4.The barycentric interpolation collocation methods for solving a kind of timedependent multidimensional fractional evolution equations are built based on the multidimensional barycentric interpolations and their differential matrices.This kind of equation is a partial integro-differential equation with a weakly singular kernel,whose solution has weak singularity at the initial time.Therefore,it is difficult for the numerical technique concerning time variable on the uniform mesh to reach the desired convergence order.In the temporal direction,a generalized Crank-Nicolson difference scheme and the trapezoid product integral formula on the graded mesh is employed to discretize the differential term and the integral term,respectively.Moreover the derived time semi-discrete scheme is unconditionally stable and second order convergent.In the spatial direction,the undetermined solution and its derivatives of multidimensional fractional evolution equation are approximated by the multidimensional barycentric interpolations and their differential matrices,respectively.Then the fully discrete schemes are obtained.Furthermore,the error estimation of the numerical solution is given.Numerical results are consistent with the theoretical analyses.