Numerical Analysis of Ordinary Differential Equations and Its Applications

The main topics of this study are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Moreover, some relations between integration and differentiation are made clear. The volume is self-contained, and should be of interest to specialists in the field of real functions, and to students, since only the basics of mathematical analysis and vector spaces are required Limiting Formulas of Eight-Stage Explicit Runge-Kutta Method of Order Seven / H. Ono -- A Series of Collocation Runge-Kutta Methods / T. Mitsui and H. Sugiura -- Fourth Order P-Stable Block Method for Solving the Differential Equation y" = f(x,y) / K. Ozawa -- Two-Point Hermite-Birkhoff Quadratures and Its Applications to Numerical Solution of ODE / C. Suzuki -- Improved SOR-like Method with Orderings for Non-Symmetric Linear Equations Derived from Singular Perturbation Problems / E. Ishiwata and Y. Muroya -- Analysis of the Milne Device for the Finite Correction Mode of the Adams PC Methods I / M. Fuji -- A New Algorithm for Differential-Algebraic Equations Based on HIDM / T. Watanabe and G. Gnudi -- Semi-Explicit Methods for Differential-Algebraic Systems of Index 1 and Index 2 / H. Shintani -- Computational Challenges in the Solution of Nonlinear Oscillatory Multibody Dynamics Systems / J. Yen and L. Petzold