Second-order Volterra integro-differential equation is solved by the linear barycentric rational collocation method. Following the barycentric interpolation method of Lagrange polynomial and Chebyshev polynomial, the matrix form of the collocation method is obtained from the discrete Volterra integro-differential equation. With the help of the convergence rate of the linear barycentric rational interpolation, the convergence rate of linear barycentric rational collocation method for solving Volterra integro-differential equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.

The linear barycentric rational collocation method for solving heat conduction equation is presented. The matrix form of discrete heat conduction equation by collocation method is also obtained. With the help of convergence rate of the barycentric interpolation, the convergence rate of linear barycentric rational collocation method for solving heat conduction equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.

Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational method, the unknown function was approximated by the barycentric rational function. For the biharmonic equation, we change the biharmonic equation into the two Poisson equations by depression of order. The linear equations of discrete the biharmonic equation was changed into matrix form. For the basis of barycentric rational function, we present the convergence rate of linear barycentric rational method for biharmonic equation by depression of order. At last, several numerical examples are provided to validate the theoretical analysis. KEYWORDS barycentric rational function, barycentric rational method, biharmonic equation, depression of order, matrix form where Ω = [a, b] × [c, d] and f (x, y), g(x, y), h(x, y) are consistent on Ω. Biharmonic equation is widely used in electrostatics, mechanical engineering and theoretical physics and so on. There are lots of methods to solve biharmonic equation such as finite difference method, finite element method, boundary element method and spectral method and so on.

No abstract

The linear barycentric rational collocation method for beam force vibration equation is considered. The discrete beam force vibration equation is changed into the matrix forms. With the help of convergence rate of barycentric rational interpolation, both the convergence rates of space and time can be obtained at the same time. At last, some numerical examples are given to validate our theorem.

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernelcot((x-s)/2)is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.

No abstract

No abstract