A unified computer program for the solution of ordinary differential equations

1987 ◽  
Vol 9 (4) ◽  
pp. 222-224 ◽  
Author(s):  
Enayat Mahajerin
SIMULATION ◽  
1965 ◽  
Vol 4 (5) ◽  
pp. 317-323 ◽  
Author(s):  
Joseph L. Hammond

State variable techniques are reviewed and applied to analog computer programming. The concise rep resentation for ordinary differential equations made possible by this technique is used to formulate a gen eral program for all such equations. It is shown that an analog computer program based on state variables will not have redundant integrators. The fact that the use of state variables facilitates the choice of variables internal to an analog com puter program is illustrated by two techniques, namely, (1) a technique for avoiding derivatives of the forcing function in programming a large class of ordinary differential equations, and (2) a technique for simulating certain systems in such a way that the effect of each characteristic root is placed in evi dence.


Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.


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