Numerical Algorithms for Ordinary Differential Equations and Application

AbstractThere are many numerical algorithms for solving the fractional-order ordinary differential equations (FODEs). They are usually very different in nature, and it is difficult to compare their performances. To solve this problem, a set of five benchmark problems of different categories of FODEs with known analytical solution are designed and proposed, they can be used as benchmark problems for testing the numerical algorithms. A Simulink block diagram scheme is used for solving these benchmark problems, with computing errors and the running times reported.

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Numerical Algorithms for Direct Solution of Fourth Order Ordinary Differential Equations

Journal of the Nigerian Society of Physical Sciences ◽

10.46481/jnsps.2020.100 ◽

2020 ◽

pp. 218-227

Author(s):

John Kuboye ◽

O. R. Elusakin ◽

O. F. Quadri

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Fourth Order ◽

Initial Value Problems ◽

Gaussian Elimination ◽

Numerical Algorithms ◽

Step Length ◽

Implicit Scheme ◽

Initial Value ◽

Fourth Derivative

This paper examines the derivation of hybrid numerical algorithms with step length(k) of five for solving fourth order initial value problems of ordinary differential equations directly. In developing the methods, interpolation and collocation techniques are considered. Approximated power series is used as interpolating polynomial and its fourth derivative as the collocating equation. These equations are solved using Gaussian-elimination approach in finding the unknown variables aj, j=0,...,10 which are substituted into basis function to give continuous implicit scheme. The discrete schemes and its derivatives that form the block are obtainedby evaluating continuous implicit scheme at non-interpolating points. The developed methods are of order seven and the results generated when the methods were applied to fourth order initial value problems compared favourably with existing methods.order initial value problems compared favourably with existing methods.

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Numerical Solutions of Fuzzy Differential Equations by Taylor Method

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2002-0006 ◽

2002 ◽

Vol 2 (2) ◽

pp. 113-124 ◽

Cited By ~ 66

Author(s):

Saeid Abbasbandy ◽

Tofigh Allah Viranloo

Keyword(s):

Differential Equations ◽

Error Analysis ◽

Ordinary Differential Equations ◽

Numerical Solutions ◽

Numerical Algorithms ◽

Cauchy Problems ◽

Linear And Nonlinear ◽

Taylor Method

Abstract In this paper, numerical algorithms for solving “fuzzy ordinary differential equations” are considered. A scheme based on the Taylor method of order p is discussed in detail and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy Cauchy problems.

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Ordinary differential equations

10.1093/oso/9780198708636.003.0011 ◽

2018 ◽

Author(s):

Joseph F. Boudreau ◽

Eric S. Swanson

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Software Design ◽

Multiple Scales ◽

Numerical Algorithms ◽

Object Oriented ◽

Step Size ◽

Derivative System ◽

Adaptive Step ◽

Classical And Quantum Mechanics

This chapter surveys the ordinary differential equations (ODEs) that occur in classical and quantum mechanics, and describes both numerical algorithms and appropriate software design for solving them. Systems of ordinary differential equations, together with a few constants of integration, can in most cases be regarded as a means of defining a function (the “solution”). In this chapter, we develop an object-oriented architecture that applies integrators of the Runge-Kutta family to create these functions. Together with an automatic derivative system for generating partial derivatives from functions of one or more variables, the differential equation solver becomes a powerful tool for solving a variety of few-body problems in classical Hamiltonian systems. This chapter presents a blend of numerical algorithms, physics, and computing techniques. The phenomenon of energy drift is discussed and used to motivate symplectic solvers. Techniques such as adaptive step size and possible problems with stability and multiple scales are also discussed.

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