Birkhoff Collocation Methods For Fourth-order Integro-differential Equations

The paper focuses on the Birkhoff collocation methods for solving fourth-order differential and integro-differential equations.Firstly,we consider the Birkhoff collocation method for the clamping rod model.The Birkhoff interpolation basis functions which satisfy the boundary conditions are constructed.We then obtain algebraic equations with stable condition numbers.The numerical results indicate that the Birkhoff method is of high accuracy and effectiveness by comparing with a kind of Legendre collocation method.Moreover,the time-dependent example shows that the method is stable in long-time calculation.Secondly,we consider the Birkhoff collocation method for the suspension bridge model and the hinge beam model.The Birkhoff interpolation basis functions satisfying the other kind of boundary conditions are constructed.The resulting algebraic equations are solved via an iterative algorithm.Numerical examples show again that the Birkhoff iteration coefficient matrix has stable condition number.The highly accurate numerical results indicate that the Birkhoff collocation method is effective.