Mthemticl nd Computtionl Applictions, Vol., No., pp. 37-5, 5 http://dx.doi.org/.99/mc-5- SOLUTION OF QUADRATIC NONLINEAR PROBLEMS WITH MULTIPLE SCALES LINDSTEDT-POINCARE METHOD Mehmet Pkdemirli nd Gözde Srı Applied Mthemtics nd Computtion Center, Cell Byr University, Murdiye, 454 Mnis, Turkey. mpk@cbu.edu.tr, gozde.deger@cbu.edu.tr Abstrct- A recently developed perturbtion lgorithm nmely the multiple scles Lindste-Poincre method (MSLP is employed to solve the mthemticl models. Three different models with qudrtic nonlinerities re considered. Approximte solutions re obtined with clssicl multiple scles method (MS nd the MSLP method nd they re compred with the numericl solutions. It is shown tht MSLP solutions re better thn the MS solutions for the strongly nonliner cse of the considered models. Keywords- perturbtion methods; numericl solutions; systems with qudrtic nonlinerities. INTRODUCTION Perturbtion theories hve been widely used to obtin pproximte nlyticl solutions of liner nd nonliner physicl problems. Although the methods provide cceptble solutions for wekly nonliner problems, the solutions do not represent the physics for the strongly nonliner cses. Recently, for solution of the strongly nonliner problems, new perturbtion method ws developed by Pkdemirli et l. []. This method nmed multiple scles Lindste-Poincre method (MSLP combines the clssicl multiple scles method nd the Lindste-Poincre method. Pkdemirli nd Krhn [] nd Pkdemirli et l. [3] pplied the method to mny strongly nonliner problems nd obtined good results comptible with the numericl solutions. This new method hs not been tested for problems with strong qudrtic nonlinerities. In this study, three different qudrtic nonliner problems re solved by MSLP nd MS method. The pproximte solutions re contrsted with the numericl solutions. For wek nonlinerities, ll three methods yield similr solutions. As the nonlinerity is incresed, the solutions devite from ech other, with MSLP yielding better pproximte solutions in contrst to the numericl solutions.. QUADRATIC NONLINEAR MODEL I Consider the below problem with qudrtic nonlinerity u u uu ( with initil conditions u( u( (
38 M. Pkdemirli nd G. Srı.. Multiple Scles Method (MS First, the problem is solved with the clssicl method. Solutions re ssumed to be of the form; u u( T, T, T u( T, T, T u( T, T, T (3 where T=t is the usul fst time scle, T=εt nd T=ε t re the slow time scles. Time derivtives re defined s d d (4 D D D, D DD ( D DD D T. where n / n If (3 nd (4 re substituted into the originl eqution, the following equtions re obtined t ech order of ε; O( D u u u( Du ( O D u u D D u u D u u ( ( D u D u ( (5 (6 O D u u D D u ( D D D u u ( D u D u u D u (7 At order, the solution my be expressed s it u Ae cc where cc stnds for the complex conjugtes of the preceding terms nd i A e The first order solution is obtined in terms of rel mplitude nd phse u ( T, T cos( t ( T, T ( ( ( ( Eqution (8 is substituted into (6 nd seculr terms re eliminted DA A A( T ( The solution t order ε is it i it (3 u Be A e cc 3 This solution my be represented in terms of rel mplitude nd phse (4 sin( sin( 6 u b T T where i B ibe (6 b ( ( 3 Eqution (8 nd (3 re inserted into (7 nd seculr terms re eliminted, i D B i D A A A (7 3 If (5, (5, (6 re inserted into (7, one finlly hs, b,, T 3 4 The finl solution is (8 (9 (5 (8
Solution of Qudrtic Nonliner Problems with MSLP 39 u cos(( t (sin( t sin(( t 4 6 4 For vlid solutions, the correction term should be much smller thn the leding term. For the problem, this criterion yields 6 ( (9.. Multiple Scles Lindste-Poincre Method (MSLP Detils of the method ws presented in the previous literture [-3]. The time trnsformtion t is pplied to the model u u uu ( where prime denotes derivtive with respect to the new vrible. The time scles in this method re slightly different from clssicl multiple scles T, T, T ( Expressing the time derivtives d d (3 D D D D D D D D D ( with Dn= / nd substituting the expnsions Tn u u( T, T, T u( T, T, T u( T, T, T (4 (5 into the originl eqution, the following equtions re obtined t ech order of pproximtion; O( D u u, u( Du ( (6 O D u u D D u u u D u u ( ( D u D u ( (7 O D u u D D u ( D D D u u u ( u ( D u D u u D u The first order solution is it u Ae cc cos( T (9 (, ( (3 Eqution (9 is substituted into (7 nd seculr terms re eliminted i DA A (3 If DA= is selected, =(T, β= β (T nd ω=. Since ω is not complex, this choice is dmissible. The solution t order ε is i (3 3 it it i u Be A e cc B ibe In terms of rel mplitude nd phse, the solution is (33 sin( sin( u b T T 6 (34 b ( ( 3 Equtions (9 nd (3 re inserted into (8 nd seculr terms re eliminted (8
4 M. Pkdemirli nd G. Srı i D B i D A A A A (35 3 DB= cn be ssumed. If DA= is selected, ω comes out to be rel nd this is gin n dmissible choice. After lgebric clcultions, Eqution (35 yields (36, b,, 3 The frequency is (37 The finl solution in terms of this frequency is u cos( t (sin( t sin( t O( 6 For vlid solutions, the perturbtion criteri is 6 (38 (39.3. Comprisons with the Numericl Solutions Figure. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for,,
Solution of Qudrtic Nonliner Problems with MSLP 4 Figure. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 3,, Figure 3. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 4,, In this section, the pproximte solutions re contrsted with the numericl solutions for the qudrtic nonliner model considered. In Figure, results re compred for ε=. The greement between MSLP nd numericl solutions is better thn MS solution nd the mplitude vlues of the MS solution yield higher errors. The positive mplitudes gree with numericl nd MSLP cses wheres the positive mplitudes introduce errors in cse of MS solutions. For negtive mplitude vlues, the error is less for MSLP. ε=3 is selected in Figure nd MSLP nd numericl solutions gree well for positive vlues of mplitudes wheres, the error is smller for negtive mplitudes for MSLP solutions. For ε=4, in Figure 3, the sme trend is observed with more mplifiction.
4 M. Pkdemirli nd G. Srı 3. QUADRATIC NONLINEAR MODEL II Consider the below problem with qudrtic nonlinerity u u u (4 with initil conditions u( u( (4 3.. Multiple Scles Method (MS First, the problem is solved with the clssicl method. Solutions re ssumed to be of the form; u u( T, T, T u( T, T, T u( T, T, T (4 where T=t is the usul fst time scle, T=εt nd T=ε t re the slow time scles. Time derivtives re defined s d d (43 D D D D D D ( D DD, where Dn / T. n If (4 nd (43 re substituted into the originl eqution, the following equtions re obtined t ech order of ε; O( D u u u( Du ( (44 O D u u D D u u u ( ( D u D u ( (45 At order, the solution my be expressed s O D u u D D u ( D D D u u u (46 it u Ae cc where cc stnds for the complex conjugtes of the preceding terms nd i A e The first order solution is obtined in terms of rel mplitude nd phse u ( T, T cos( T ( T, T (49 ( ( (5 Eqution (47 is substituted into (45 nd seculr terms re eliminted DA A A( T (5 The solution t order ε is i T i T u Be cc ( A e cc AA 3 (5 This solution my be represented in terms of rel mplitude nd phse u bcos( T cos( T 6 (53 where i (54 B ibe ( b( (55 3 (47 (48
Solution of Qudrtic Nonliner Problems with MSLP 43 Eqution (47 nd (5 re inserted into (46 nd seculr terms re eliminted, i D B i D A A A 3 If (45, (54, (55 re inserted into (56, one finlly hs 5, b,, 3 T 3 The finl solution is 5 u cos(( 3 t ( cos( t 3 (56 5 (cos(( 3 t 3 O( 6 3.. Multiple Scles Lindste-Poincre Method (MSLP The time trnsformtion t is pplied to the model u u u (59 where prime denotes derivtive with respect to the new vrible. The time scles in this method re slightly different from clssicl multiple scles T, T, T (6 Expressing the time derivtives d d (6 D D D D D D ( D DD with Dn= / T nd substituting the expnsions n u u( T, T, T u( T, T, T u( T, T, T (6 (63 into the originl eqution, the following equtions re obtined t ech order of pproximtion; O( D u u, u( Du ( (64 O D u u D D u u u u ( ( D u D u ( (65 The first order solution is O D u u D D u ( D D D u u u u u (66 it u Ae cc cos( T (67 (, ( (68 Eqution (67 is substituted into (65 nd seculr terms re eliminted i DA A (69 If DA= is selected, =(T, β= β (T nd ω=. Since ω is not complex, this choice is dmissible. The solution t order ε is it it i u Be cc ( A e cc AA B ibe 3 (7 In terms of rel mplitude nd phse, the solution is u b cos( T cos(t 6 (7 (57 (58
44 M. Pkdemirli nd G. Srı ( b( 3 (7 Equtions (67 nd (7 re inserted into (66 nd seculr terms re eliminted i D B i D A A (4 (73 A A 3 DB= cn be ssumed. If DA= is selected, ω comes out to be rel nd this is gin n dmissible choice. After lgebric clcultions, Eqution (73 yields 5, b (74,, 3 6 The frequency is 4 (75 3 The finl solution in terms of this frequency is u cos( t ( cos( t (cos( t 3 O( 3 6 For vlid solutions, the perturbtion criteri is 3 (76 (77 3.3. Comprisons with the Numericl Solutions Figure 4. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for,,
Solution of Qudrtic Nonliner Problems with MSLP 45 Figure 5. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 3,, Figure 6. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 4,, In this section, the pproximte solutions re contrsted with the numericl solutions for the qudrtic nonliner model considered. In Figure 4, results re compred for ε=. The greement between MSLP nd numericl solutions is better thn MS solution nd the mplitude vlues of the MS solution yield higher errors. ε=3 is selected in Figure 5 nd ε=4 is selected in Figure 6. In Figures 5 nd 6, the sme trend is observed with more mplifiction.
46 M. Pkdemirli nd G. Srı 4. QUADRATIC NONLINERITY WITH DAMPING A dmping term is dded to the previous model u u u u (78 with initil conditions u( u( (79 4.. Multiple Scles Method (MS First, the problem is solved with the clssicl method. Solutions re ssumed to be of the form; u u( T, T, T u( T, T, T u( T, T, T (8 where T=t is the usul fst time scle, T=εt nd T=ε t re the slow time scles. Time derivtives re defined s d d (8 D D D D D D ( D DD, where Dn / T. n If (8 nd (8 re substituted into the originl eqution, the following equtions re obtined t ech order of ε; O( D u u u( Du ( (8 O D u u D D u D u u u ( ( D u D u ( (83 At order, the solution my be expressed s O D u u D D u ( D D D u ( D u D u u u (84 it u Ae cc where cc stnds for the complex conjugtes of the preceding terms nd i A e The first order solution is obtined in terms of rel mplitude nd phse u ( T, T cos( T ( T, T (87 ( ( (88 Eqution (85 is substituted into (83 nd seculr terms re eliminted T D A A ( T e, ( T (89 The solution t order ε is i T i T u Be cc ( A e cc AA 3 (9 This solution my be represented in terms of rel mplitude nd phse u bcos( T cos( T 6 (9 where i (9 B ibe (85 (86 3 (93 Arc b 9 ( tn(, (
Solution of Qudrtic Nonliner Problems with MSLP 47 Eqution (85 nd (9 re inserted into (84 nd seculr terms re eliminted, i D B D A i D A i B D A (4 A A 3 If (8, (9, (93 re inserted into (94, one finlly hs T T 5 e, b be,, ( T 3 The finl solution is t 5 t u e cos(( ( 3 e t (94 3 ( cos( tn( t e t Arc 9 5 (cos(( ( 3 ( 6 t t e 3 e t O 4.. Multiple Scles Lindste-Poincre Method (MSLP The time trnsformtion t is pplied to the model u u u u (97 where prime denotes derivtive with respect to the new vrible. The time scles in this method re slightly different from clssicl multiple scles T, T, T (98 Expressing the time derivtives d d (99 D D D D D D ( D DD with Dn= / T nd substituting the expnsions n u u( T, T, T u( T, T, T u( T, T, T ( ( into the originl eqution, the following equtions re obtined t ech order of pproximtion; O( D u u, u( Du ( ( O D u u D D u u u D u u ( ( D u D u ( (3 O D u u D D u ( D D D u u u u u ( D u D u (4 The first order solution is it u Ae cc cos( T (5 (, ( (6 Eqution (5 is substituted into (3 nd seculr terms re eliminted i D A A ia (7 T If D A A is selected, ( T e, β= β (T nd ω=. Since ω is not complex, this choice is dmissible. The solution t order ε is it it i u Be cc ( A e cc AA B ibe 3 (8 (95 (96
48 M. Pkdemirli nd G. Srı In terms of rel mplitude nd phse, the solution is u b cos( T cos(t 6 (9 3 ( Arc tn(, b( 9 ( Equtions (5 nd (8 re inserted into (4 nd seculr terms re eliminted i D B i D A A ( ( A A 3 DB B cn be ssumed. If DA= is selected, ω comes out to be rel nd this is gin n dmissible choice. After lgebric clcultions, Eqution ( yields 3 5 e b e T t,, Arctn(,, 9 6 The frequency is ( ( 3 The finl solution in terms of this frequency is t t 3 u e cos( t ( e cos( t tn ( 9 t (cos( 3 ( e t O 6 For vlid solutions, the perturbtion criteri is 6 4.3. Comprisons with the Numericl Solutions ( (3 (4 (5 Figure 7. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for,,,.
Solution of Qudrtic Nonliner Problems with MSLP 49 Figure 8. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 3,,,. Figure 9. Comprison of numericl solutions nd pproximte nlyticl solutions (MS nd MSLP for 4,,,. The pproximte solutions re contrsted with the numericl solutions for the qudrtic nonliner model with dmping. In Figure 7, results re compred for ε=. The greement between MSLP nd numericl solutions is better thn MS solution nd the mplitude vlues of the MS solution yield higher errors. ε=3 is selected in Figure 8 nd ε=4 is selected in Figure 9. In summry, the MSLP solutions re better compred to the MS solutions.
5 M. Pkdemirli nd G. Srı 5. CONCLUDING REMARKS A recently developed perturbtion technique nmely the multiple scles Lindste Poincre method (MSLP combining the multiple scles nd Lindste Poincre methods is pplied to three different mthemticl models with qudrtic nonlinerities. In conclusion, the MSLP method yields closer solutions to the numericl ones compred to those of MS for strong qudrtic nonlinerities. 6. REFERENCES. M. Pkdemirli, M.M.F. Krhn, H. Boycı, A new perturbtion lgorithm with better convergence properties: Multiple Scles Lindste Poincre Method. Mthemticl nd Computtionl Applictions 4(, 3-44, 9.. M. Pkdemirli nd M. M. F. Krhn, A new perturbtion solution for systems with strong qudrtic nd cubic nonlinerities, Mthemticl Methods in the Applied Sciences 3, 74 7,. 3. M. Pkdemirli, M. M. F. Krhn nd H. Boycı, Forced Vibrtions of Strongly Nonliner Systems with Multiple Scles Lindste Poicre Method, Mthemticl nd Computtionl Applictions 6(4, 879-889,.