Solution of Quadratic Nonlinear Problems with Multiple Scales Lindstedt-poincare Method

A recently developed perturbation algorithm namely the multiple scales Lindstedt-Poincare method (MSLP) is employed to solve the mathematical models. Three different models with quadratic nonlinearities are considered. Approximate solutions are obtained with classical multiple scales method (MS) and the MSLP method and they are compared with the numerical solutions. It is shown that MSLP solutions are better than the MS solutions for the strongly nonlinear case of the considered models.


INTRODUCTION
Perturbation theories have been widely used to obtain approximate analytical solutions of linear and nonlinear physical problems.Although the methods provide acceptable solutions for weakly nonlinear problems, the solutions do not represent the physics for the strongly nonlinear cases.Recently, for solution of the strongly nonlinear problems, a new perturbation method was developed by Pakdemirli et al. [1].This method named multiple scales Lindstedt-Poincare method (MSLP) combines the classical multiple scales method and the Lindstedt-Poincare method.Pakdemirli and Karahan [2] and Pakdemirli et al. [3] applied the method to many strongly nonlinear problems and obtained good results compatible with the numerical solutions.
This new method has not been tested for problems with strong quadratic nonlinearities.In this study, three different quadratic nonlinear problems are solved by MSLP and MS method.The approximate solutions are contrasted with the numerical solutions.For weak nonlinearities, all three methods yield similar solutions.As the nonlinearity is increased, the solutions deviate from each other, with MSLP yielding better approximate solutions in contrast to the numerical solutions.

QUADRATIC NONLINEAR MODEL I
Consider the below problem with a quadratic nonlinearity (2)

1. Multiple Scales Method (MS)
First, the problem is solved with the classical method.Solutions are assumed to be of the form; (3) where T0=t is the usual fast time scale, T1=εt and T2=ε 2 t are the slow time scales.Time derivatives are defined as where 3) and (4) are substituted into the original equation, the following equations are obtained at each order of ε; (7) At order 1, the solution may be expressed as where cc stands for the complex conjugates of the preceding terms and The first order solution is obtained in terms of real amplitude and phase (10) Applying the initial conditions yields 0 (0) 0 (0) aa    (11) Equation ( 8) is substituted into (6) and secular terms are eliminated 1 2

( )
The solution at order ε is This solution may be represented in terms of real amplitude and phase where Applying the initial conditions yields Equation ( 8) and ( 13) are inserted into (7) and secular terms are eliminated, If (5), (15), ( 16) are inserted into (17), one finally has with Dn= / n T  and substituting the expansions (25) into the original equation, the following equations are obtained at each order of approximation; The first order solution is  29) is substituted into (27) and secular terms are eliminated (31) If D1A=0 is selected, a=a(T2), β= β (T2) and ω1=0.Since ω1 is not complex, this choice is admissible.The solution at order ε is In terms of real amplitude and phase, the solution is Applying the initial conditions yields 2 0 (0) 0 (0) 3 Equations ( 29) and ( 32) are inserted into (28) and secular terms are eliminated D1B=0 can be assumed.If D2A=0 is selected, ω2 comes out to be real and this is again an admissible choice.After algebraic calculations, Equation (35) yields The frequency is The final solution in terms of this frequency is For valid solutions, the perturbation criteria is In this section, the approximate solutions are contrasted with the numerical solutions for the quadratic nonlinear model considered.In Figure 1, results are compared for ε=2.The agreement between MSLP and numerical solutions is better than MS solution and the amplitude values of the MS solution yield higher errors.The positive amplitudes agree with numerical and MSLP cases whereas the positive amplitudes introduce errors in case of MS solutions.For negative amplitude values, the error is less for MSLP.ε=3 is selected in Figure 2 and MSLP and numerical solutions agree well for positive values of amplitudes whereas, the error is smaller for negative amplitudes for MSLP solutions.For ε=4, in Figure 3, the same trend is observed with more amplification.

QUADRATIC NONLINEAR MODEL II
Consider the below problem with a quadratic nonlinearity (41)

Multiple Scales Method (MS)
First, the problem is solved with the classical method.Solutions are assumed to be of the form; (42) where T0=t is the usual fast time scale, T1=εt and T2=ε 2 t are the slow time scales.Time derivatives are defined as where 42) and ( 43) are substituted into the original equation, the following equations are obtained at each order of ε; where cc stands for the complex conjugates of the preceding terms and The first order solution is obtained in terms of real amplitude and phase

( )
The solution at order ε is This solution may be represented in terms of real amplitude and phase where Applying the initial conditions yields Equation ( 47) and ( 52) are inserted into (46) and secular terms are eliminated, If ( 45), ( 54), (55) are inserted into (56), one finally has The final solution is

Multiple Scales Lindstedt-Poincare Method (MSLP)
The time transformation t   is applied to the model (59) where prime denotes derivative with respect to the new variable  .The time scales in this method are slightly different from classical multiple scales with Dn= / n T  and substituting the expansions (63) into the original equation, the following equations are obtained at each order of approximation; The first order solution is  67) is substituted into (65) and secular terms are eliminated (69) If D1A=0 is selected, a=a(T2), β= β (T2) and ω1=0.Since ω1 is not complex, this choice is admissible.The solution at order ε is In terms of real amplitude and phase, the solution is Applying the initial conditions yields Equations ( 67) and ( 70) are inserted into (66) and secular terms are eliminated D1B=0 can be assumed.If D2A=0 is selected, ω2 comes out to be real and this is again an admissible choice.After algebraic calculations, Equation (73) yields The frequency is The final solution in terms of this frequency is For valid solutions, the perturbation criteria is In this section, the approximate solutions are contrasted with the numerical solutions for the quadratic nonlinear model considered.In Figure 4, results are compared for ε=2.The agreement between MSLP and numerical solutions is better than MS solution and the amplitude values of the MS solution yield higher errors.ε=3 is selected in Figure 5 and ε=4 is selected in Figure 6.In Figures 5 and 6, the same trend is observed with more amplification.

QUADRATIC NONLINERITY WITH DAMPING
A damping term is added to the previous model (79)

Multiple Scales Method (MS)
First, the problem is solved with the classical method.Solutions are assumed to be of the form; (80) where T0=t is the usual fast time scale, T1=εt and T2=ε 2 t are the slow time scales.Time derivatives are defined as where 80) and ( 81) are substituted into the original equation, the following equations are obtained at each order of ε; (85) where cc stands for the complex conjugates of the preceding terms and The first order solution is obtained in terms of real amplitude and phase (87) Applying the initial conditions yields 0 (0) 0 (0) aa 85) is substituted into (83) and secular terms are eliminated ( ) , ( ) This solution may be represented in terms of real amplitude and phase where Applying the initial conditions yields Equation ( 85) and ( 90) are inserted into (84) and secular terms are eliminated, If ( 82), ( 92), ( 93) are inserted into (94), one finally has T T a a e b b e a T The final solution is

Multiple Scales Lindstedt-Poincare Method (MSLP)
The time transformation t   is applied to the model (97) where prime denotes derivative with respect to the new variable  .The time scales in this method are slightly different from classical multiple scales ) ) In terms of real amplitude and phase, the solution is Applying the initial conditions yields Equations ( 105) and ( 108) are inserted into (104) and secular terms are eliminated For valid solutions, the perturbation criteria is The approximate solutions are contrasted with the numerical solutions for the quadratic nonlinear model with damping.In Figure 7, results are compared for ε=2.The agreement between MSLP and numerical solutions is better than MS solution and the amplitude values of the MS solution yield higher errors.ε=3 is selected in Figure 8 and ε=4 is selected in Figure 9.In summary, the MSLP solutions are better compared to the MS solutions.

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Comparison of numerical solutions and approximate analytical solutions (MS and MSLP) for 0 0 2, 1, a       into the original equation, the following equations are obtained at each order of approximation; (T2) and ω1=0.Since ω1 is not complex, this choice is admissible.The solution at order ε is