American Journal of Computational Mathematics, 2013, 3, 185194 http://dx.doi.org/10.4236/ajcm.2013.33027 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Stochastic Oscillators with Quadratic Nonlinearity Using WHEP and HPM Methods Amnah S. AlJohani1,2 1Department of Applied Mathematics, College of Science, Northern Borders University, Arar, Saudi Arabia 2College of Home Economics, Northern Borders University, Arar, Saudi Arabia Email: xxwhitelinnetxx@hotmail.com Received May 2, 2013; revised June 8, 2013; accepted July 2, 2013 Copyright © 2013 Amnah S. AlJohani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, quadratic nonlinear oscillators under stochastic excitation are consid ered. The WienerHermite expansion with perturbation (WHEP) method and the homotopy perturbation method (HPM) are used and compared. Different approximation orders are considered and statistical moments are computed in the two methods. The two methods show efficiency in estimating the sto chastic response of the nonlinear differential equation s. Keywords: Nonlinear Stochastic Differential Equations; W ienerHermite Expansion; W HEP Tech ni q u e; Homotopy Perturbation Method 1. Introduction Quadrate oscillation arises through many applied models in applied sciences and engineering when studying oscil latory systems [1]. These systems can be exposed to a lot of uncertainties through the external forces, the damping coefficient, the frequency and/or the initial or boundary conditions. These input uncertainties cause the output solution process to be also uncertain. For most of the cases, getting the probability density function (p.d.f.) of the solution process may be impossible. So, developing approximate techniques through which approximate sta tistical moments can be obtained, is an important and necessary work. Since Meecham and his coworkers [2] developed a theory of turbulence involving a truncated WienerHer mite expansion (WHE) of the velocity field, many au thors studied problems concerning turbulence [38]. A lot of general applications in fluid mechanics were also stud ied in [911]. Scattering problems attracted the WHE applications through many authors [1216]. The nonlin ear oscillators were considered as an opened area for the applications of WHE as can be found in [1723]. There are a lot of applications in boundary value problems [24 , 25] and generally in different mathematical studies [26 29]. The WHE properties and description of its usage are given in [30]. In HPM technique [3134], the response of nonlinear differential equations can be obtained analytically as a series solution. The basic idea of the homotopy method is to deform continuously a simple problem (and easy to solve) into the difficult problem under study [35]. The HPM method is a special case of homotopy analysis method (HAM) propounded by Liao in 1992 [36]. The HAM was systematically described in Liao’s book in 2003 [37] and was applied by many authors in [3841]. The HAM method possesses auxiliary parameters and functions which can control the convergence of the ob tained series solution . The stochastic oscillator with cubic nonlinearity (Duffing oscillator) was considered in [17,42]. The nonlinear term is due to the restoring nonlinear force. In some applica tions, the restoring force is quadratic and it is required to estimate the response in this case. The main goal of this paper is to consider the quadratic nonlinear oscillator under stochastic excitation. The WHEP and HPM meth ods are used and compared. This paper is organized as follows. The problem for mulation is outlined in Section 2. The WHEP technique is described and applied to the stochastic quadratic oscil lator in Section 3. The HPM is outlined in Section 4 and applied also to the quadratic oscillator. A comparison between the two methods is shown in Section 5. 2. Problem Formulation In this section, the following quadratic nonlinear oscilla tory equation is cons idered: C opyright © 2013 SciRes. AJCM
A. S. ALJOHANI 186 222 ;2 ;,0, twxwxwxFtt T (1) under stochastic excitation ; t with deterministic initial condition s 00 0,0 xx x , where w: frequency of oscillation, : damping coefficient, : deterministic nonlinearity scale, ,,p : a triple probability space with as the sample space, is a algebra on events in and P is a probability measure. 3. WHEP Technique The application of the WHE aims at finding a truncated series solution to the solution process of differential eq ua tions. The truncated series composes of two major parts; the first is the Gaussian part which consists of the first two terms, while the rest of the series constitute the non Gaussian part. In nonlinear cases, there exists always dif ficulties of solving the resultant set of deterministic inte grodifferential equations got from the applications of a set of comprehensive averages on the stochastic integro differential equation obtained after the direct application of WHE. Many authors introduced different methods to face these obstacles. Among them, the WHEP technique was introduced in [22] using the perturbation technique to solve perturbed nonlinear problems. The WHE method utilizes the WienerHermite poly nomials which are the elements of a complete set of sta tistically orthog onal random functions [30]. The Wiener Hermite polynomial 12 ,, , i i tt t satisfies the fol lowing recurrence relation: 12 2 11 1212 1 1 2 1 ,, ,,, , ,,,, 2 i ii ii ii iiiim i m i tttHtttH t Htt ttti (2) where 01211 12121 2 32111 12312312321 3 431222 12341234123413 242314 1,, ,, (, , ),()(), (, , ,), ,,,,, HHtntH ttHtHttt HtttH ttHtHtttHttt ttttHtttHtH ttttH ttttH tttt (3) in which n(t) is the white noise with the following statis tical properties 12 12 0, ,EntEnt ntt t (4) where . is the Dirac delta function and E denotes the ensemble average operator. The WienerHermite set is a statistically or thogonal set, i.e. 0 ij EHHi j. (5) The average of almost all H functions vanishes, par ticularly, 0 for 1. i EH i (6) Due to the completeness of the WienerHermite set, any random function ;Gt can be expanded as 0112 2 11112 121 33 123123123 ;;d +;, ;,,,, ddd GtGtGttHttGtttHt ttt GttttHtttttt 2 ,dd (7) where the first two terms are the Gaussian part of G(t; ω). The rest of the terms in the expansion represent the nonGau ssi an part of G(t; ω). The average of G(t; ω) is 0 ; GEG tGt (8) The cov ariance of ;Gt is 112 2 11112 1212 33 3 3 123123132231123 Cov;,;;; ;,d +2;,,,dd 2;,,,,,,,,,,,ddd GG GtGEGttG GttGttG tttGtt tt GttttG tttGtttG tttttt (9) Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 187 ;Gt The variance of is dd (10) The WHE method can be elementary used in solving stochastic differential equations by expanding the solu tion process as well as the stochastic input processes via the WHE. The resultant equation is more complex than the original one due to being a stochastic integrodiffer ential equation. Taking a set of ensemble averages to gether with using the statistical properties of the WH po 22 212 11 1212 2 333 12312 312313212 3 33 123231123 Var;;;d +2;,dd 2;,,ddd2,,,,,,d 2,,,,,, ddd G GtEGttGtttGtt ttt GtttttttGttttGtttt ttt GttttGtttt ttt . lynomials, a set of deterministic integrodifferential equations are obtained in the deterministic kernels ;,0,1,2, i Gt i . To obtain an approximate solu tions for these deterministic kernels, one can use pertur bation theory in the case of having a perturbed system depending on, say, . Expanding the kernels as a power series of , another set of simpler iterative equations in the kernel series components are obtained. This is the main algorithm of the WHEP technique. The technique ied to several nonlinear stochastic equations; see [20,22,23,25]. The WHEP technique can be applied on linear or nonlinear perturbedsystems described by ordinary or partial difrential equations. The solution can be modi fied in the sense that additional parts of the Wiener was successfully appl fe Hermite expansion can always be taken into considera tions and the required order of approximatio ways be made. It can be even run through a package if it is coded in some sort of symbolic languages. CaseStudy ns can al The quadr atic n onlinear oscillatory problem, Equation (1 ) under stochastic excitation ; t with deterministic initial conditions is solved using WHEP technique. The solution process takes the following form: ;xt x 011 111 12312312 3 ;d dd ;,,,, ddd txtt Htt tt xttttHtt tttt 22 12 1212 33 +; ,,xtttHtt (11) Applying the WHEP technique, the following equa tions in the deterministic kernels are obtained: 2 00 1 Lx tw 2 2 2 22 2 11 1212 33 0 123132123 ;d+2 ;,dd ;,,,,, ddd xtwxt ttwxt tttt xttttxttttttt Ft 2 3 212312 3 + 2;,, dddwxtt ttttt 3 (13) 3 3 1232311 23 ;,,,,, dddx ttttx ttttttt (12) 101 12 22 112122 122 3 22 2122 231232 23 1 223231231 ,2,4; ;,d +4,;,d +8;,;,,dd 4;,;,,dd, Lx ttwxtxttwx ttxtttt w xttxttttwxtttxtttttt wxtt txtt ttttFtt Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 188 2021122 2 1212121323 3 13 1313 31323 32313 31233 33 3 256165 562 ,,2,,,,4;,;, d 2; ;,,d2; ;,,d2; ;,,d 2;,,;,,dd4;, Lxtttwxtxtttxttxttxtttxtttt xtt xtttttxttxtttttxtt xttttt xttttxtttttt xtt 3 56156 56 33 33 2565615615626556 33 33 156562 56165562 56 35 ,;,,dd 3;,,;,,dd2;,,;,,dd 3;,, ;,,dd;,, ;,,dd 2;, tt xtttttt xttttxtttttt xttttxtttttt xttttxtttttt xttttxtttttt xttt 333 6256156265561 5612 ,;,, dd;,,;,,dd,,txtt ttttxtt t txttt tttFttt 2 (14) (15) Let us take the simple case of evaluating the only Gaussian part (first order approximation) of the solution process of the previous case study, mainly 1 d 333 0333 2 123132231123 132 23 2323 41 432441 4234424314 23 42 4 ,,,,,,,,, 2,,,,,,, 4 ;,;,,d4;,;,,d4;,;,,d 4;, ; Lxtt ttLxtt ttLxttttwxtxtt ttxtt ttxtt xtttxttttt xtttxttttt xtttxttttt xtttxtt 23 23 134434124434214 232 323 413244 423144 431244 333 123 1322 ,,d4 ;,;,,d4 ;,;,,d 4;, ;,,d4;,;,,d4;,;,,d ,,, ,,,,, tttxtt txtt tttxtt txtt t tt xtttxttttt xtttxttttt xtttxttttt Gtttt GttttGtt 31 ,tt 231 ,,t t 1 01 11 ;; txtxttHt t. (16) In this case, the governing equations are 0 22 001 211 ;dLx twx txtttG t (17) 1 , (18) The ensemble average is (19) an t (20 It has to be noticed that all the previous equations are deterministic linear ones in the general form 1011 2 11 ,2 ,Lx ttwxtxttG tt 0 xtxt d the variance is 2 1 211 ;d xtxtt ) 22 wx wxFt wit ns 0,0 tio h deterministic initial condi 00 xx x . 0102 0 d t txtxhtsFs s (21) In which we have 2 2 22 122 22 1esin1, 1 11 ee 21 21 1ee , 1 wt mt qt mt qt htw t w t t where , 2w 22 1, 1mwwqww . When adding the first term in the nonGaussian part (the second approx imation) of the solutio n pro inly cess of the previous case study, ma d 011 111 22 1212 12 ;; ;,,dd xtxtxttHtt It has the general solution tt tHtttt , (22) Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 189 the governing equations become 0 (23) 1 , (24) 22 001 211 2 2 21212 (); d 2;,dd Lxtwxtxttt wxtttttGt 101 2 11 121 22122 ,2 , 4;;,d Lx ttwxtx tt wxttxtt ttGtt , 212 02 2 ,, 2,, Lxt tt wxt xttt 1 1 1212 22 2 132331 , 4;,;,d,, x ttxtt 2 tt txttttGtt t (25) The ensemble average is still got by Equation (19) 2 while the variance is got as 22 12 211 121 ;d2;,dd xtxtttxtt ttt (26) The WHEP technique uses the following expansion for its deterministic kernels as corrections made under each approximation order. 23 012 3,0,1,2,3,. iiiii xt xxxxi .(27) Example: Let us take ;e;, 0. t Ftq t 3 (28) in the previous casestudy and then solving using the WHEP technique. The following results are obtained, see Figures 13. 4. The Homotopy Perturbation Method (HPM) In this technique, a parameter 0,1p is embedded in a homotopy function ,: 0,1 vrp which sat isfies (a) (b) (c) (d) (e) (f) Figure 1. (a) The first order aximation of the mean at ε correction for different correcti; (b) The first order ap proximation of the mean at ε2 correction for different correction levels; (c) The first order approximation of the mean at ε3 correction; (d) The first order approximation of the mean at ε, ε2, ε3 correction; (e) The first order approximation of the mean at ε, ε2, ε3 correction; (f) The first order approximation of the mean at ε, ε2, ε3 correction. ppro on levels Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 190 (a) (b) (c) (d) Figure 2. (a) The first order approximation of the variance at ε correction for different correction levels; (b) The first order approximation of the variance at ε2 Correction for different correction levels; (c) The first order approximation of the vari ance at ε3 correction for different correction levels; (d) The first order approximation of the variance at. ε, ε2, ε3 correction. (a) (b) Figure 3. (a) The first order approximation of the variance at ε, ε2, ε3 correction; (b) The first order approximation of the variance at ε, ε2, ε3 correction. 0 ,1HvppLvLupAvf r 0 (29) where is an initial approximation to the solution of the equ 0 u ation 0,Aufrr (30) with boundary conditions ,0, u Bu r n (31) in which A is a nonlinear differential operator which can be decompose into a linear operator L and a nonlinear operator N, B is a boundary operator, f(r) is a known analytic function and is the boundary of . The homotopy introducesontinuously deformution for the case of p = 0, , to the case of p = 1, inalation (30)motopyethod which is to deform continuously a simple problem (and easy to solve) into the difficult problem under study [35]. The basic assumption of the HPM method is that the solution of the original Equation (29) can be expanded as a power series in p as: a c Lv , wh sic id ed sol Equ m 00Lu ich is the orig ea the ho 0Avf r . This is the ba of 2 pv3 01 2 3 vv pvpv (32) Now, setting p = 1, the approximate solution of Equa tion (23) is obt ai ned a s: 0123 1 lim p uvvvvv (33) The rate of convergence of the method depends greatly on the initial approximation The idea of the imbeddedrameter can be utilized to solve nonlinear problems by imbedding th is parameter to the problem and then forcing it to be unity in the ob tained approximate solution if converge can be assured. A simple technique enables the extension of th applica bility of the perturbation ods from small valued ap 0 u. pa e meth Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 191 plications to ge n er a l ones. Example Considering the same previous example of SubSect 3.1.1, one can get the following results w.r.t. homo ati ion topy perturbon: 22 xLx wx , 2 2Lxxwxwx , 2 Nx x , ;fr Ft . The homotopy function takes the following form: 0 ,1 0HvppLvLupAvfr or equivalently, 22 0;0LvLup Lt . (34) Lett 23 01 2 3 pvpv , substituting in Equation (34) and equating the equal powers of p in both sides of the equation, one can get the following results: 1) Lv Ly, in which one may consider the fol 0 uwv F ing lo 2) vv pv 00 wing simple solution: 00 ,0,0.vyyxyx 0000 22 1001 ;,0LvFtLvwvv 2 2,00,00Lvwvv vv . 1 0,00v . 3) 20 122 4) 22 310233 2,00,00Lvw vvvvv . 4103124 4 2,00,Lvvvvvvv 00 . 5) The approximate solution is 0123 ;limxtv vvvv 1 p which can be considered to any approximation order. On can notice that the algorithm of the solution is straigh t a lot of flexibilities can be made. For any choices in guessing the initial approximation together with its initial condition zero initial conditions, we can choose e t forward and tha example, we have ms. For 0 v0 which leads to: 5 012345 ;xtxv vv vvv ;dht sF s 22 1 00 213 0 ;d 2;;d tt t swht sv ss whtsvs vss (35) Figures 47 are obtained for 0.5 : [42]. 5. Comparisons between WHEP and HPM Methods Figure [8] shows comparisons between the WHEP and HPM methods for different values of the nonlinearity strength, . As the nonlinearity strength increases, the deviation between the two methods is also increasing. (a) (b) e mean for different correction levels; (b) The first and second levels. Figure 4. (a) The first and second order approximation of th order approximation of the variance at for different correction (a) (b) Figure 5. (a) The third order approximation of the mean for different correction levels; (b) The third order approximation of the variance for different correction levels. Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI 192 (a) (b) Figure 6. (a) A comparison between first, second order and th first, second order and the, third or de r oariance at ε = 0e n f the v.1. third order of the mean at ε = 0.1; (b) Comparison betwee (a) (b) (c) (d) Figure 7. (a) A comparison between first, second order and the third order of the mean at ε = 0.3; (b) A comparison between first, second order and the, third order of the variance at ε = 0.3; (c) A comparison between first, second order and the o third order of the mean at ε = 0.7; (d) A comparison between first, second order and the third order of the variance at ε = 0.7. (a) (b) (c) (d) Copyright © 2013 SciRes. AJCM
A. S. ALJOHANI Copyright © 2013 SciRes. AJCM 193 (f) (e) Figure 8. (a) A comparison between homotopy perturbation and WienerHermite of the mean at ε = 0.1; (b) A comparison between homotopy perturbation and WienerHermite of the variance at ε = 0.1; (c) A comparison between homotopy per turbation and WienerHermite of the mean at ε = 0.3; (d) A comparison between homotopy perturbation and Wie nerHermite of the variance at ε = 0.3; (e) A comparison between homotopy pe rturbation and WienerHermite of the mean at ε = 0.7; (f) A comparison betw ee n homotopy perturbation and WienerHermite of the variance at ε = 0.7. This is due to the convergence condition of the WHEP technique which depends on . For small values of , e the WHEP technique converbut after a certain val of ges u it will diverge. The M is more accurate for HP higher values of . The HPM has advantages when used in solving differential equations with large nonlinearities. 6. Conclusion The quadratic nonlinear oscillator with stochastic excita tion is considered. The solution was obtained using the WHEP technique with different orders and different num ber of corrections. The HPM is used also with different approximations. The WHEP technique is more efficient but it converges only for certain limit of the nonlinearity strength. The HPM is more difficult in the stochastic differential equations but it is more preferable for high values of the nonlinearity stth. The two methods are shown to be efficient in estimating the stochastic re sponse of the quadratic nonlinear oscillators. REFERENCES [1] A. Nayfeh, “Problems in Perturbation,” John Wiley, New York, 1993. [2] S. Crow and G. Canavan, “Relationship between a Wie nerHermite Expansion and Energy Cascade,” Journal of Fluid Mechanics, Vol. o. 2, 1970, pp. 387403. doi:10.1017/S0022112070000654 er reng an 41, N [3] P. Saffman, “Application of WienerHermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow,” Physics of Fluids, Vol. 12, No. 9, 1969, pp. 17861798. doi:10.1063/1.1692743 [4] W. Kahan and A. Siegel, “CameronMartinWiener Me thod in Turbulence and in Burger’s Model: General For mulae and Application to Late Decay,” Journal of Fluid Mechanics, Vol. 41, No. 3, pp. 593618. [6] H. Hogge and W. Meecham, “WienerHermite Expansion Applied to Decaying Isotropic Turbulence Using a Re normalized TimeDependent Base,” Journal of Fluid of Mechanics, Vol. 85, No. 2, 1978, pp. 325347. doi:10.1017/S002211207800066X l. 46, No. 4, 1979, pp. 13581359. doi:10.1143/JPSJ.46.1358 [7] M. Doi and T. Imamura, “An Exact Gaussian Solution for TwoDimensional Incompressible Inviscid Turbulent Flow,” Journal of the Physical Society of Japan, Vo [8] R. Kambe, M. Doi and T. Imamura, “Turbulent Flows Near Flat Plates,” Journal of the Physical Society of Ja pan, Vol. 49, No. 2, 1980, pp. 763778. doi:10.1143/JPSJ.49.763 [9] A. J. Chorin, “Gaussian Fields and Random Flow,” Jour nal of Fluid of Mechanics, Vol. 63, No. 1, 1974, pp. 21 32. doi:10.1017/S0022112074000991 [10] Y. Kayanuma, “Stochastic Theory for NonAdiabatic Crossing with Fluctuating,” Jour nal of the Physical Society of Japan, Vol. 54, No. 5, 1985, pp. 20372046. doi:10.1143/JPSJ.54.2037 OffDiagonal Coupling [11] M. Joelson and A. Ramamonjiarisoa, “Random Fields of Water Surface Waves Using WienerHermite Functional Series Expansions,” Journal of Fluid of Mechanics, Vol. 496, 2003, pp. 313334. doi:10.1017/S002211200300644X [12] C. Eftimiu, “FirstOrder WienerHermite Expansion in the Electromagnetic Scattering by Conducting Rough Sur faces,” Radio Science, Vol. 23, No. 5, 1988, pp. 769779. doi:10.1029/RS023i005p00769 [13] N. J. Gaol, “Scattering of a TM Plane Wave from Peri odic Random Surfaces,” Waves Random Media, Vol. 9, No. 11, 1999, pp. 5367. [14] Y. Tamura and J. Nakayama, “Enhanced Scattering from a Thin Film with OneDimensional Disorder,” Waves in Random and Complex Media, Vol. 15, No. 2, 2005, pp. 269295. [15] Y. Tamura and J. Nakaya ma, “TE Plane Wave Reection and Transmission from OneDimensional Random Slab,” IEICE Transactions on Electronics, Vol. E88C, No. 4, fl , 1970 [5] J. Wang and S. Shu, “WienerHermite Expansion and the Inertial Subrange of a Homogeneous Isotropic Turbu lence,” Physics of Fluids, Vol. 17, No. 6, 1974, pp. 1130 1134. 2005, pp. 713720. [16] N. Skaropoulos and D. Chrissoulidis, “Rigorous Applica tion of the Stochastic Functional Method to Plane Wave
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