Interpolation perturbation method for solving nonlinear problems
The interpolation perturbation method [cf. \textit{Yiwu Yuan}, Mechanics and Practice, Vol. 12, No. 1 (1990); in Chinese] is designed to produce \(O(\varepsilon^2)\) approximations to the solution of first-order regularly perturbed ordinary differential equations, with better global properties than does the usual perturbation method. Given a system (i) \(V'(t)= f(\varepsilon, t,V(t))\), (ii) \(V(t_0)= c\), \(0\leq\varepsilon< \varepsilon_0\), suppose that \(V_0(t)\) is the solution of (i), (ii) when \(\varepsilon= 0\). The author introduces an interpolation function \(y(t)\) which satisfies (iii) \((1- y)= (y- g)/g= K\), \(g= f(\varepsilon, t,V)/f(0,t,V)\). Replacing \(V\) by \(V_0\) in (iii) one solves for \(K\), \(y\) and truncates these solutions to obtain \(K_*\), \(y_*\) of first order in \(\varepsilon\). Next, one solves (iii) for \(V_*= V+ O(\varepsilon^2)\) in terms of \(K_*\), \(y_*\). The author presents three test problems in which the new method is applied and compared with the exact solution and solutions using the regular perturbation method or the method of scales. The new method is shown to produce better accuracy over a wider range of the \(\varepsilon\) parameter than do the usual methods.
scientific article; zbMATH DE number 1159632