Sinusoidal Time-Dependent Power Series Solution to Modified Duffing Equation

Duffing equation, , has a standard well-known exact solution [1]. Approximate solutions to this equation also are available [2]. Reference [3] [4] introduces a sinusoidal time-dependent Power Series solution. Applying this method successfully we investigate the approximate solution of the modified Duffing equations, and 5. Symbolic manipulative utilities of a Computer Algebra System (CAS) specifically Mathematica [5] extensively is used investigating the results.


Introduction
There are numerous scientific and engineering phenomena encountering oscillations. The simplest oscillations are described by a generic linear differential , where ξ(t) is the time-dependent quantity of interest. For instance, ξ(t) could describe the oscillation of the electric charge, q(t), in an LC series electric circuit, or it may describe oscillating coordinate, x(t), of  . Solving these ODEs is a prime interest.
Aside searching for their exact solutions [1] various approximation methods have been sought for [2]. Method introduced in [

A Brief Review of Exact and Power Series Solution of Duffing Equation
To establish the basis, we begin with the standard Duffing equation, Its exact solution with two initial conditions, ( ) 1 0 where, 2 u a t = , with a 2 and k subject to, In Equation (2) Having established the fundamentals now we cross-examine the accuracy of the Power Series approximation Method outlined in [3] [4].
The fundamental underlying concept of the approximation method proposed in [3] stems from the fact that the Duffing equation describes oscillations. As ( ) x t  , in Duffing equation warrants the conservativeness of energy of the system leading to a unique value for the frequency of the polynomial. As is true in general, the accuracy of the series method is being controlled by the order of the polynomial-for the case on hand the order depends implicitly to the coefficient of the nonlinear term, ϵ.
As is outlined in [3] [4] a Power Series solves the Equation (1), x t a n t A change of variable, Noticing significant changes; e.g. Equation (6) For the chosen initial conditions Equation (7) gives the remaining coefficients yielding the solution of Equation Equation (8) for the generic case, Utilizing the time independence of energy and implied initial conditions yields an equation conducive the needed, ω, in Equation (5).
The maximum potential energy occurs at t=0 when the particle is at maximum distance from the equilibrium, i.e.
( ) The maximum kinetic energy occurs when the particle passes the equilibrium, Equating these two gives the ω.
x max max With these outlines we craft a Mathematica program leading the quantity of interest. For objectively chosen numeric values for initial displacement, A, stiffness, ϵ, we investigate the accuracy of the Power Series Method vs. the exact solution; these are given in Sect. 3.

Solutions of Duffing Equation: Power Series, Exact, and Numeric
For the chosen value of A and ϵ the number of terms, N, in Equation (5)  . By trial and error and the CPU run-time limitations, we set N = 28. As we pointed out because other than the first term only the even powers of sin(ωt) are contributing, evaluation of x(t), Equation (5), requires determination of fourteen expansion coefficients. Recursive relationship between the coefficients, Equation (7), makes the symbolic expressions massive. For instance, a [7], is, Evaluation of the kinetic energy, Equation (8) and their intersection is shown in Figure 1. Figure 1 helps guesstimating the abscissa of the intersection.
Applying the FindRoot utility of Mathematica at about the guesstimated abscissa gives the exact root value of Equation (9), ω = 0.7992207. Reference [4] uses eighty terms, e.g. N = 80 evaluating ω. In our case, N = 28, yields an acceptable value.
To further our investigation, we compare the expansion coefficients of our approach vs. [4]. Utilizing the recurrence relationship between the coefficients given by Equation (7) along with the initial conditions outlined in the text we tabulate the first fourteen expansion coefficients of the series in Equation (5). Table 1 is our values that ought to be compared to the "corrected notations" of

H. Sarafian American Journal of Computational Mathematics
Having the values of expansion coefficients and ω we plot the t-dependent position x(t) given by Equation (5). This is shown in Figure 2 in Black.
Duffing equation, Equation (1) as discussed has an exact solution [1]. The   , Figure 2 displays this function in Green. Although our numeric computation applying Power Series includes "only" 14-terms its accuracy is as close as the exact solution. We further our investigation applying Mathematica NDSolve solving Equation (1) numerically.
NDSolve uses one of the common numeric methods of Implicit Runge Kutta or StiffnessSwitching which is a combination of the explicit and implicit methods solving the equation. The numeric solution perfectly matches the exact solution.
The Green curve is the exact solution, it indistinguishably overlaps with the numeric solution. The black curve is the Power Series solution.
Despite the fact that the number of terms in Equation (5) is limited to fourteen the agreement between the exact/numeric and approximate results are quite acceptable. As shown in Figure 2, within the long range of the t-axis the positive amplitude of the approximated method overlaps the exact amplitude and overestimates the negative ones. This can be rectified if a greater number of terms could be included in Equation (5). On the other hand, the period of all three methods is in very good agreements.
As shown in Figure 2, the oscillation has a period of about four. Applying the value of ω the period is ( ) ( )

Nonlinear Quartic Term
In Plotting the kinetic and potential energies vs. ω their intersection abscissa according to Equation (9) yields ω. This is shown in Figure 3. Figure 3 assists evaluating the exact abscissa of the intersection; ω = 0.510225. Utilizing ω, we tabulate the expansion coefficients; these are given in Table 2.
Numeric values of Table 2 are based on utilizing the recurrence relationship between the coefficients given by Equation (7). Figure 4 (Black). Applying NDSolve of Mathematica, as in the previous section we display the numeric solution in Green.

Plot of Equation (5) is shown in
For chosen parameters the results are in good agreement; noting analytic solution for the case on hand is not available.

Nonlinear Quintic Term
In this subsection similar to Sect. 3.2 we investigate the validity of the Power Series Method for solving a modified Duffing equation with ϵx 5 (t). We consider, . With trial and error, we set the number of the terms, N, in Equation (5) to 20. The maximum potential energy for the case on hand is ( ) We skip displaying the kinetic and potential energies vs. ω as is similar to Figure 1 and

Conclusions and Suggestions
We applied a Computer Algebra System (CAS), Mathematica examining its applicability to solving Duffing equation utilizing the Power Series Method. We have shown with Mathematica we are able producing the numeric results of [4]. Moreover, we extend the calculation to symbolic domain; this is not included in [4]. Applying Mathematica we then successfully explore its applicability to solving modified Duffing equations embodying nonlinear quartic and quintic terms.
For Duffing equation computation power of a laptop with a double processor prevents increasing the number of the terms evaluating Equation (5) [8]. Readers may also find [9] resourceful addressing related issues.