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Journal ArticleDOI

Numerical Integration of Ordinary Differential Equations

W. E. Milne1
01 Nov 1926-American Mathematical Monthly (Informa UK Limited)-Vol. 33, Iss: 9, pp 455-460
TL;DR: In this article, the integration of Ordinary Differential Equations (ODE) has been studied in the context of algebraic geometry, and it has been shown that it is possible to integrate ODE
Abstract: (1926). Numerical Integration of Ordinary Differential Equations. The American Mathematical Monthly: Vol. 33, No. 9, pp. 455-460.
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Journal ArticleDOI
TL;DR: In this paper, a new numerical scheme has been developed for the newly established fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives.
Abstract: Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. A new numerical scheme has been developed, in this paper, for the newly established fractional differentiation. We present in general the error analysis. The new numerical scheme was applied to solve linear and non-linear fractional differential equations. We do not need a predictor-corrector to have an efficient algorithm, in this method. The comparison of approximate and exact solutions leaves no doubt believing that, the new numerical scheme is very efficient and converges toward exact solution very rapidly.

330 citations

Journal ArticleDOI
TL;DR: In this article, several finite element in time algorithms are developed based on Hermitian expansions of increasing order which are then compared with Pade approximations and alternative Norsett formulas.

80 citations

Journal ArticleDOI
TL;DR: In this article, a single-step implicit block method involving one hybrid point with the introduction of a third derivative is proposed to solve nonlinear differential equation of a circular sector oscillator.
Abstract: Solving nonlinear differential equation of a circular sector oscillator is of a scientific importance. Thus, to solve such equations, a single- step implicit block method involving one hybrid point with the introduction of a third derivative is proposed. To derive this method, the approximate basis solution is interpolated at {xn, xn + 3/5} while its second and third derivatives are collocated at all points {xn, xn + 3/5, xn + 1}on the integrated interval of approximation. Numerical results are presented in the form of table and graphs for the variation of different physical parameters. The study reveals that the proposed hybrid block method is zero stable, which proves that it is convergent beside a significant interval of absolute stability, thus making it suitable for solving stiff ODEs.

55 citations

Journal ArticleDOI
TL;DR: The unified theory developed by Henrici for one step methods for one-step methods is extended to the more general case where the order of the system of difference equations can exceed the ordered system of ordinal equations.
Abstract: The unified theory developed by Henrici [5] for one-step methods is extended to the more general case where the order of the system of difference equations can exceed the order of the system of ord...

53 citations

Journal ArticleDOI
TL;DR: The development of numerical solution techniques from the identification of a problem to the never-final preparation of automatic codes for the solution of classes of similar problems is examined.
Abstract: Two decades ago many people thought that there was very little left to occupy the researcher in the numerical initial value problem for ordinary differential equations; but there has since been an enormous increase in the effort devoted to this problem and related software. Is everything finally done so we can turn to more difficult areas? This paper examines the development of numerical solution techniques from the identification of a problem to the never-final preparation of automatic codes for the solution of classes of similar problems. It discusses a number of ODE problems that are only a part of the way along this path of development. These problems include ones with highly oscillatory solutions, ones with frequent discontinuities, and ones in which very little accuracy is needed (and which should be integrated at low cost). There is a spectrum of problems, from ones that can now be solved using very robust software to ones that we still do not know how to solve effectively. The paper then surveys s...

51 citations