Spline Method with Five Collocation Points for Solving Systems of High-Index Linear Differential Algebraic Equations

In this paper, we introduce a numerical method for solving systems of high-index differential algebraic equations. This method is based on approximating the exact solution by spline polynomial of degree eight with five collocation points to find the numerical solution in each step. The study shows that the method when applied to linear differential-algebraic systems with index equal one is stable and convergent of order 8, while it is stable and convergent of order 9-u for index equal u . Numerical experiments for four test examples and comparisons with other available results are given to illustrate the applicability and efficiency of the presented method

Numerical Method for Finding Single and Multiple Integrals

In this paper, an iterative numerical method for obtaining approximate values of definite single, double and triple integrals will be illustrated. This method depends on approximating the single integral function by spline polynomial of fifth degree, while Gauss Legendre points as well as spline polynomials are used for finding multiple integrals. The study shows that when the method are applied to single integrals is convergent of order sixth, as well as when applied to triple integrals is convergent of order sixth for three Gauss Legendre points or greater. Errors estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method.