Analytical Approximate Solutions To Fractional-Power Nonlinear Oscillators

Analytical (approximate) solutions can supply explicit expressions of the solution and allow the direct discussion of the influence of parameters and initial conditions on the solution. The analytical approximate methods are important methods. The perturbation method is one of the most commonly used analytical techniques for solving nonlinear oscillations with a small parameter. However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. In many cases, one can apply the harmonic balance method to determine analytical approximate periods and periodic solutions to the conservative single-degree-of-freedom nonlinear oscillation systems with odd nonlinearity, even these analytical approximations are valid for larger amplitude. However, applying the method of harmonic balance to construct higher-order approximate analytical solutions is very difficult, since it requires analytical solution of algebraic equation(s) with very complex nonlinearity. Newton-harmonic balance method is an improved HB method. The most interesting features of the Newton-harmonic balance method are that it can be used to construct analytical periods and periodic solutions with high accuracy, and requires solution of simple linear algebraic equations only instead of nonlinear algebraic equations without analytical solution. In this paper, analytical approximations to the fractional power nonlinear oscillators are constructed.Consider the following nonlinear equation where p = n m is a fraction,such as p = 3/4,2/3,1/3...By introducing an independent variableτ=ωt, and Eq.(1) can be rewritten aswhere a prime represents derivative with respect toτ, and ? =ω2m. The new independent variable is chosen in such a way that the solution to Eq.(2) is a periodic function ofτof period 2π. The corresponding period of the nonlinear oscillation is given by T =2πω. Here, both the periodic solution u (τ) and frequencyωdepend on A.1. m is oddEquation (2) can be rewritten as Following the single term HB method, we first set u1 (τ)= Acosτwhich satisfies the initial conditions in Eq.(3). Substituting Eq.(4) into Eq. (3), and setting the coefficient of cosτto zero. Solving the equation in unknown ? yields the first analytical approximation ?1 of ? . Hence we get the initial approximation of Eq.(3) asWe can obtain further approximate solutions to Eq.(3) by combining Newton's method with the harmonic balance method. The periodic solution u (τ)and 2m power of frequencyωof Eq.(3) can be expressed aswhere ?u k is the correction part to be determined later, they are periodic functions ofτof period 2π.Substituting Eq.(6) into Eq.(3), and linearizing the resulting equation with respect to the correction terms ?u k and ?? k yieldThe HB method will again be applied to solve Eq.(7) for ?u kand ?? k. The ?u k(τ) in Eq.(7) can be set asSubstituting Eq.(8) into Eq.(7), and setting the coefficient of cosτ,cos3τ, ,cos(2 k+ 1)τto zeros .We can get the linear algebra equations with unknowns x1 , x2 , , xk and ?? k, from which we can obtain the ( k + 1)th analytical approximate periods and periodic solutions.2 m is even For this case, Eq. (2) can be rewritten as Following the single term HB method, we first set u1 (τ)= Acosτ(10)which satisfies the initial conditions in Eq.(9). Substituting Eq. (10) into Eq.(9), and setting the constant term to zero. Solving the equation in unknown ? , one gets the first analytical approximation ?1 of ? . Hence we get the initial approximation to Eq. (9) asWe can achieve further approximate solutions by combing Newton's method and the harmonic balance method. The periodic solution and 2m power of frequencyωof Eq. (2) can be expressed aswhere ?u k is the correction part to be determined later, they are periodic functions ofτof period 2π.Substituting Eq.(12) into Eq.(9),and linearizing the resulting equation with respect to the correction terms ?u k and ?? k yieldThe HB method will again be applied to solve Eq.(13) for ?u kand ?? k. The ?u k(τ) in Eq.(13) can be set asSubstituting Eq.(14) into Eq.(13), and setting the constant term and the coefficient of cos2τ,cos4τ, ,cos2kτto zeros, we can get the linear algebra equations with unknowns x1 , x2 , , xk and ?? k, from which we can obtain the ( k + 1)th analytical approximate periods and periodic solutions.It should be clear that the procedures of constructing approximate solutions to fractional power nonlinear oscillators mentioned above are very simple. These analytical approximations are valid for all fractions p < 1 and near to...