CONDITIONAL LINEARIZABILITY OF FOURTH-ORDER SEMI-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

We constructed a new fourth order four-stage diagonally implicit Runge-Kutta (DIRK) method which is specially designed for the integrations of linear ordinary differential equations (LODEs). The method is obtained based on theButcher’s error equations. In the derivation, the error norm is minimized so that the free parameters chosen are obtained from the minimized error norm. Row simplifying assumption is also used so that the number of equations forthe method can be reduced and simplified. A set of test problems are used to validate the method and numerical results show that the new method is more efficient in terms of accuracy compared to the existing method.

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Two non-elementary integral functions defined using central factorial powers

Researches in Mathematics ◽

10.15421/241512 ◽

2015 ◽

Vol 23 ◽

pp. 98

Author(s):

T.P. Goy

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Fourth Order ◽

Variable Coefficients ◽

Elementary Functions ◽

Order Linear ◽

Linear Ordinary Differential Equations

We study two new real-valued non-elementary functions generated by central factorial powers. Graphs of such functions are plotted and some of their properties are proved. It is also shown that new integral functions are solutions of fourth order linear ordinary differential equations with variable coefficients.

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TECHNIQUE FOR SOLVING SOME TYPE OF THE SPECTRAL LINEAR ORDINARY DIFFERENTIAL EQUATIONS OF SECOND ORDER

10.31530/17009 ◽

2018 ◽

Author(s):

Aryan Mohammed ◽

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Second Order ◽

Linear Ordinary Differential Equations

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Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

Information and Control Systems ◽

10.31799/1684-8853-2018-6-24-34 ◽

2018 ◽

pp. 24-34

Author(s):

V. F. Edneral ◽

O. D. Timofeevskaya

Keyword(s):

Normal Form ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Periodic Solutions ◽

Fourth Order ◽

Numerical Solutions ◽

Numerical Calculations ◽

Computational Time ◽

Resonant Normal Form ◽

Normal Form Method

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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