A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method

作者：Dehghan Mehdi^*; Aryanmehr S; Eslahchi M R

来源：Journal of Computational and Applied Mathematics, 2013, 244: 125-139.

DOI：10.1016/j.cam.2012.11.013

• Birkhoff interpolation
• Initial value problems
• Special functions
• Quadrature rules
• Error analysis

In this paper a class of Birkhoff-type interpolation problems on arbitrary nodal points is studied. The explicit representation (characterization), the uniqueness and the error function are explicitly given. Furthermore, we apply the obtained Birkhoff-type interpolation method to find: (i) the numerical solution of high order initial-value problems (IVPs) and the corresponding error of this approximation, (ii) the approximation of some special functions with their explicit error functions, and (iii) new interpolatory type quadrature formulae of precision degree at least m+n - 1 and m+kn - 1(m, n, k is an element of N, n, k >= 2). Numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results. The results reveal that the new method is effective, simple and accurate.

• 出版日期2013-5-15