Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

作者：Ramos J I^*

来源：Applied Mathematics and Computation, 2010, 215(12): 4304-4319.

DOI：10.1016/j.amc.2009.12.057

• Third-order nonlinear ordinary differential equations
• Jerk dynamics
• Order reduction
• Linstedt-Poincare method
• Artificial parameter

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.

• 出版日期2010-2-15