Highly Efficient Collocation Method For Solving A Class Of Differential Equations

Barycentric rational interpolation has been widely used in approximation theory and solving differential equations in recent decades. Many practical problems, such as diffusion problems, signal processing, fluctuation and vibration problems often involve in computation of differential equation. As is known to all, finite difference method, finite element method and spectrum method are often used to solve differential equations. As for these three methods, they have different advantages and disadvantages. Adopting barycentric rational interpolation collocation method to solve differential equations has the advantage of simple calculation formula, high precision, numerical stability and good adaptability to calculate nodal etc. The work of this paper includes discussion of the selection of interpolation points and barycentric weights. These points mainly involve in Gauss-Legendre points and Gauss-Legendre-Lobatto points. We compare the interpolation error effect of different weights with same interpolation points and different interpolation points with same weights. We compute the vibration equation and telegraph equation with initial and boundary conditions using barycentric rational interpolation collocation method. When solving the vibration equation, we explain why the Gauss-Legendre points are not suitable for this method. For the calculation of telegraph equation, we use barycentric rational interpolation collocation method. For the nodes of time domain and space domain, we approximate the unknown functions with rational interpolation. And we discrete the initial boundary value conditions with the same method. Then, we obtain the algebraic equations of differential equation with additional method. Numerical experiments show that this method has the advantage of well error effect and low computation complexity under the action of interpolation nodes and compound rights.