A NEW PERTURBATION ALGORITHM WITH BETTER CONVERGENCE PROPERTIES : MULTIPLE SCALES LINDSTEDT POINCARE METHOD

A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and variable amplitude cases. Three problems are solved: Linear damped vibration equation, classical duffing equation and damped cubic nonlinear equation. Results of Multiple Scales, new method and numerical solutions are contrasted. The proposed new method produces better results for strong nonlinearities. KeywordsPerturbation Methods, Lindstedt Poincare method, Multiple Scales method, Numerical Solutions


INTRODUCTION
Perturbation methods are well established and used for over a century to determine approximate analytical solutions for mathematical models.Algebraic equations, integrals, differential equations, difference equations and integro-differential equations can be solved approximately with these techniques.The direct expansion method (pedestrian expansion) does not produce physically valid solutions for most of the cases and depending on the nature of the equation, many different perturbation techniques such as Lindstedt-Poincare technique, Renormalization method, Method of Multiple Scales, Averaging methods, Method of Matched Asymptotic Expansions etc. are developed within time.
One of the deficiencies in applying perturbation methods is that a small parameter is needed in the equations or the small parameter should be introduced artificially to the equations.Nevertheless, the problem solved is a weak nonlinear problem and it becomes hard to obtain an approximate solution valid for strongly nonlinear systems.
There have been a number of attempts recently to validate perturbation solutions for strongly nonlinear systems also.Hu and Xiong [1] contrasted two different approaches of Lindstedt-Poincare methods using the duffing equation.First, they solved the equation with classical method and then they made a slight modification in the expansions.Instead of expanding the transformation frequency, they expanded the natural frequency and obtained solutions with excellent convergence properties for the duffing equation.The time histories of solutions agree with the numerical solutions for arbitrarily large perturbation parameters.In a similar paper, the approximate and exact frequencies are contrasted for the duffing equation [2].The case of vanishing restoring force was also treated for the same equation [3].The periods obtained are contrasted with the exact period with good convergence properties for large parameters.
While a complete review of the attempts to validate perturbation solutions for strongly nonlinear oscillators is beyond the scope of this work, a partial list will be given.Among the many developed methods, Linearized perturbation method [4][5][6], parameter expanding method [7,8], new time transformations as modifications of Lindstedt-Poincare method [9][10][11], iteration methods [12,13] are some examples.
In this work, Multiple Scales method is modified by incorporating the time transformation of Lindstedt Poincare method.One of the major advantages of Multiple Scales method over the Lindstedt Poincare method is that transient solutions can be found using the former whereas it is impossible to retrieve such solutions using the latter.However, by expanding the natural frequency in modified Lindstedt Poincare method as in [1][2][3], convergent solutions for large parameters are possible.Combining both methods would augment the advantages of them in a single method so that transient solutions with improved convergence properties can be retrieved.The method will be outlined in the next section and will be applied to three models in the subsequent sections.Comparisons of both methods with numerical results will be displayed.For the models considered, the new method is in good agreement with the numerical simulations.The method may find applications in a variety of problems.

MULTIPLE SCALES LINDSTEDT POINCARE (MSLP) METHOD
The outline of the method and the guidelines will be given in this section.Consider the nonlinear equation where ε is the perturbation parameter.The essential steps in the method are 1) Make the time transformation as in Lindstedt-Poincare method.
(2) where prime denotes differentiation with respect to new time variable τ.
2) Define the fast and slow time scales 3) The transformed time derivatives and dependent variable are expanded as in classical Multiple Scales method 4), ( 5) and (7) 7) The equations at each order are separated O(1): O(ε 2 ): The solution at the first order is cc ) where cc stands for complex conjugates of the preceding terms.8) The problem at the second and third order is that one has two mechanisms for eliminating secular terms.Either select D 1 A or ω 1 in equation (10) and select D 2 A or ω 2 in equation (11).One should first try D 1 A=0 and solve ω 1 .If ω 1 is a real number then the selection is fine.If ω 1 is complex, this choice will provide unphysical solutions as also pointed by Nayfeh [14].In that case, select ω 1 =0 and eliminate secularities by choosing D 1 A. The same reasoning is also valid at the next order of approximation.If ω 2 is real when D 2 A=0, the selection is fine.If not, then select ω 2 =0 and choose D 2 A to eliminate secularities.The algorithm will be applied to three different problems in the forthcoming sections.

3.LINEAR DAMPED OSCILLATOR
Consider the linear damped oscillator 0 u 2 u u 2 0 The new method will be applied to this equation.First, the time transformation is made where prime denotes differentiation with respect to new time variable τ.Fast and slow time scales are T 0 = τ = ωt, The time derivatives, dependent variable and frequency are expanded Note that instead of ω 2 , ω 0 2 is expanded and substituted.The equations at each order are O(ε): The solution at the first order is cc (24) Substituting the solution to (22) and rearranging yields cc (26) As outlined, one first tries D 1 A=0 and solve ω 1 =2µωi.Since ω 1 is complex, this is not a suitable choice.Therefore, the appropriate selection is Substituting the polar form to ( 27), separating real and imaginary parts yield (29) Since the right hand side of ( 25) is annihilated, u 1 can be selected as zero u 1 =0 (30) At the last order, substitute (24), ( 27) and (30) to the right hand side of (23) and eliminate the secular terms 0 , ω 2 is a real number, so this choice is appropriate D 2 A=0, ω 2 = -µ 2 (32) From (32) A and hence a and β does not depend on T 2 .Equation (29) reads (34) Using (34), ( 33), ( 30), ( 28), ( 24), ( 18) and ( 16), the final solution in terms of original time variable is ( ) In fact, this solution is the exact solution of the original equation.Application of classical multiple scales produces the following solution As can be verified easily, the frequency is a Taylor expansion of the exact frequency.Solution (36) is valid only when εµ<<ω 0 whereas one does not have this restriction for (35) obtained by MSLP method.

DUFFING EQUATION Duffing equation will be solved by classical Multiple Scales (MS) and Multiple
Scales Lindstedt Poincare (MSLP) method.Solutions will be contrasted with the numerical simulations.Consider the equation 0 u u u

Multiple Scales (MS) Method
Fast and slow time scales are and substituting the expansion (41) into the original equation and initial conditions yields after separation O(1): O(ε): O(ε 2 ): In terms of real amplitude and phase, the first order solution is (47) Applying the initial conditions yield a(0)=a 0 , β(0)=0 (48) Equation ( 45) is substituted into (43) and secular terms are eliminated The polar form is substituted to above with the following results The solution at order ε is where The initial conditions at this order imply At the last order, equations ( 51) and ( 45) are inserted into (44) and secular terms are eliminated If ( 46), ( 50), ( 52) and (53) are used above, one finally has The final solution is [ ] where Invoking the well known perturbation criteria that correction terms are much smaller than the leading terms for valid solutions requires and substituting the expansions (63) into (59) and initial conditions (38) yield after separation O(1): O(ε): O(ε 2 ): Equation ( 67) is substituted into (65) and secular terms are eliminated 0 which is suitable because ω 1 is real.The solution at order ε is The initial conditions at this order imply At the last order, equations ( 71) and ( 67) are inserted into (66) and secular terms are eliminated After algebraic manipulations, equation (73 The frequency is which is exactly the same frequency given by Hu and Xiong [1] obtained by modified Lindstedt Poincare method.The final solution in terms of this frequency is [ ] For valid solutions, the criterion is

Comparisons with the Numerical Solutions
Instead of criterion (58) in the classical method, convergence criterion is given in (78) in the new method.The major difference is that while ε tends to infinity, criterion (58) would not be valid whereas (78) is valid for arbitrarily large ε.To test, one takes the limit Therefore, the correction term in solution (77) is always a small quantity compared to the leading term which assures convergence.This cannot be said for solution (56).ω 0 being a constant in the denominator makes the correction term much larger than the leading term for large ε.
To verify the results, time histories of both solutions are contrasted with the numerical solutions obtained by directly integrating the original Duffing equation.In all simulations ω 0 =1 and a 0 =1 are selected.In Figure 1, results are compared for ε=1.Although numerical and MSLP solutions are indistinguishable, MS solution is in reasonable agreement.This is because the convergence criterion (58) is still valid.In Figure 2, ε=10 and criterion (58) ceases to be valid and a qualitative and quantitative difference between the results can readily be seen.Note that numerical and MSLP solutions are still indistinguishable.Finally, in Figure 3, ε=100 is selected.For this strongly nonlinear case, MSLP and numerical solutions have an excellent agreement, while MS solutions are not realistic at all.

DAMPED CUBIC NONLINEAR SYSTEM
In the previous section, a problem with constant amplitude solution was treated.For these problems, modified Lindstedt Poincare method also produces results convergent for large ε as was presented by Hu and Xiong [1].For variable amplitude solutions, however, Lindstedt Poincare method does not produce realistic solutions.The aim in this section is to show that the new method proposed here is effective in producing convergent solutions for strongly nonlinear variable amplitude problems.Consider the equation 0

Multiple Scales (MS) Method
Similar expansions are selected as in Section 4.1.The equations at each order are O(1): O(ε): O(ε 2 ): Applying the initial conditions yield a(0)=a 0 , β(0)=0 (87) Solution ( 86) is substituted into (84) and secular terms are eliminated (88) The polar form is substituted to above with the following results The solution at order ε is where The initial conditions at this order imply At the last order, equations (90) and ( 86) are inserted into (85) and secular terms are eliminated If ( 88), ( 89), ( 91) and (92) are used above, one finally has ( ) The final solution is [ ] O(ε): O(ε 2 ): The solution at the first order is ) T cos( a cc Ae u 0 iT 0 (102) Applying the initial conditions yield a(0)=a 0 , β(0)=0 (103) The first order solution is substituted into (100) and secular terms are eliminated 0 which is suitable because ω 1 is real.The solution at order ε is The initial conditions at this order imply At the last order, equations ( 106) and ( 102) are inserted into (101) and secular terms are eliminated , ω 2 will be a complex number which is not a suitable option.Therefore ω 2 = 0 (109) is selected.After algebraic manipulations, equation (108) yields where ( ) For valid solutions, the criterion is

Comparisons with the Numerical Solutions
In criteria (97) and (113), amplitude is a function of time.For sufficiently large t, convergence is assured because the amplitude decays in time.Letting ε tending to infinity would not spoil both convergence criteria also since increasing ε increases the amplitude decay rate.Instead, for strongly nonlinear systems, one should inspect the system behavior for large α.Adjusting α, nonlinearity can be increased without increasing the damping in the system.For large α and small times, criterion (97) can not be satisfied.
Instead of criterion (97) in the classical method, convergence criterion is given in (113) in the new method.In case of large α, since ω is a function of α, it serves as a stabilizing factor in the denominator.The criterion is largest for t=0 which means a=a 0 .For this choice 1 24 Therefore, for large α, agreement with numerical solutions is expected in MSLP method.Time histories of both solutions are contrasted with the numerical solutions obtained by directly integrating the original damped equation with cubic nonlinearity in Figures 4-7.In all simulations ω 0 =1, a 0 =1, µ=1 and ε=0.1 are selected.In Figure 4, results are compared for α=1.Since criteria (97) and (113) are both satisfied, both approximate solutions are indistinguishable from the numerical ones.With increasing α in Figure 5 (i.e.α=10), separation starts, MSLP being in better agreement with the numerical solutions.For α=50 in Figure 6, the qualitative behavior can also be distinguished for numerical and MS solutions but a good agreement can be seen with MSLP and numerical results.Same qualitative and quantitative behavior can be observed for α=100 in Figure 7.

CONCLUDING REMARKS A new perturbation technique combining the Multiple Scales and Lindstedt
Poincare method is proposed for the first time.Instead of the transformation frequency, the natural frequency is expanded in a perturbation series.This choice leads to solutions with good convergence properties in Lindstedt Poincare method [1].The new method is applied to three models: 1) Damped linear oscillator, 2) Duffing Equation, 3) Damped cubic nonlinear equation.In the first case, even the exact solution is retrieved by the method.In the second and third cases, the new method produced solutions with good agreement with the numerical solutions for strongly nonlinear problems.The classical Multiple Scales method however failed to produce close solutions with the numerical ones for strongly nonlinear problems.A further study would be to apply this new technique to partial differential equations.The nonlinearities arising in partial differential equations are classified using a suitable operator notation and general solution algorithms were developed for the models previously [15][16][17].
44)The solution at the first order is cc e Applying the initial conditions yield a(0)=a 0 , β(0)=0 both at the left and right hand sides.Frequency is solved