 American Journal of Computational Mathematics, 2013, 3, 185-194 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. Al-Johani1,2 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. Al-Johani. 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 Wiener-Hermite 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 iener-Hermite 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 . 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 co-workers  developed a theory of turbulence involving a truncated Wiener-Her- mite expansion (WHE) of the velocity field, many au- thors studied problems concerning turbulence [3-8]. A lot of general applications in fluid mechanics were also stud- ied in [9-11]. Scattering problems attracted the WHE applications through many authors [12-16]. The nonlin- ear oscillators were considered as an opened area for the applications of WHE as can be found in [17-23]. 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 . In HPM technique [31-34], 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 . The HPM method is a special case of homotopy analysis method (HAM) propounded by Liao in 1992 . The HAM was systematically described in Liao’s book in 2003  and was applied by many authors in [38-41]. 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: Copyright © 2013 SciRes. AJCM A. S. AL-JOHANI 186  222;2 ;,0,xtwxwxwxFtt   T (1) under stochastic excitation ;Ft with deterministic initial condition s 000,0xxx 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- gro-differential 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  using the perturbation technique to solve perturbed nonlinear problems. The WHE method utilizes the Wiener-Hermite poly- nomials which are the elements of a complete set of sta- tistically orthog onal random functions . The Wiener- Hermite polynomial 12,, ,iiHtt t satisfies the fol- lowing recurrence relation: 12 2111212 1-1 21,, ,,, ,,,,, 2iiiiiiiiiiim imiHtttHtttH tHtt ttti (2) where       0121112121 23211112312312321 343122212341234123413 2423141,, ,,(, , ),()(),(, , ,), ,,,,,HHtntH ttHtHtttHtttH ttHtHtttHtttHttttHtttHtH ttttH ttttH tttt  (3) in which n(t) is the white noise with the following statis-tical properties   12 120, ,EntEnt ntt t (4) where . is the Dirac delta function and E denotes the ensemble average operator. The Wiener-Hermite set is a statistically or thogonal set, i.e.  0ijEHHi j. (5) The average of almost all H functions vanishes, par- ticularly, 0 for 1.iEH i (6) Due to the completeness of the Wiener-Hermite set, any random function ;Gt can be expanded as 0112 211112 12133123123123;;d +;,;,,,, dddGtGtGttHttGtttHt tttGttttHtttttt    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 non-Gau 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 211112 121233 3 3123123132231123Cov;,;;;;,d +2;,,,dd2;,,,,,,,,,,,dddGGGtGEGttGGttGttG tttGtt ttGttttG tttGtttG tttttt        (9) Copyright © 2013 SciRes. AJCM A. S. AL-JOHANI 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 integro-differ- ential equation. Taking a set of ensemble averages to- gether with using the statistical properties of the WH po 2221211 1212233312312 312313212 333123231123Var;;;d +2;,dd2;,,ddd2,,,,,,d2,,,,,, dddGGtEGttGtttGtt tttGtttttttGttttGtttt tttGttttGtttt ttt             .   lynomials, a set of deterministic integro-differential equations are obtained in the deterministic kernels ;,0,1,2,iGt 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 feHermite expansion can always be taken into considera- tions and the required order of approximatioways be made. It can be even run through a package if it is coded in some sort of symbolic languages. Case-Study ns can al- The quadr atic n onlinear oscillatory problem, Equation (1 ) under stochastic excitation ;Ft with deterministic initial conditions is solved using WHEP technique. The solution process takes the following form: ;xt x01111112312312 3;ddd;,,,, dddtxtt HttttxttttHtt tttt  2212 121233+;,,xtttHtt   (11) Applying the WHEP technique, the following equa- tions in the deterministic kernels are obtained: 200 1Lx tw2 2222 211 121233 0123132123;d+2 ;,dd;,,,,, dddxtwxt ttwxt ttttxttttxttttttt Ft    23212312 3+2;,, dddwxtt ttttt 3 (13) 331232311 23;,,,,, dddx ttttx ttttttt  (12) 101 1222112122122 3222122 23123223 1223231231,2,4; ;,d+4,;,d +8;,;,,dd 4;,;,,dd,Lx ttwxtxttwx ttxttttw xttxttttwxtttxttttttwxtt txtt ttttFtt      Copyright © 2013 SciRes. AJCM A. S. AL-JOHANI 188 202112221212121323 313 131331323 32313 3123333 3256165 562,,2,,,,4;,;, d2; ;,,d2; ;,,d2; ;,,d2;,,;,,dd4;,Lxtttwxtxtttxttxttxtttxttttxtt xtttttxttxtttttxtt xtttttxttttxtttttt xtt    356156 5633 33256561561562655633 33156562 56165562 5635,;,,dd3;,,;,,dd2;,,;,,dd3;,, ;,,dd;,, ;,,dd2;,tt xttttttxttttxtttttt xttttxttttttxttttxtttttt xttttxttttttxttt         3336256156265561 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 1d333 03332123132231123 13223 232341 432441 42344243142342 4,,,,,,,,, 2,,,,,,,4 ;,;,,d4;,;,,d4;,;,,d4;, ;Lxtt ttLxtt ttLxttttwxtxtt ttxtt ttxttxtttxttttt xtttxttttt xtttxtttttxtttxtt    23 23134434124434214232 323413244 423144 431244333123 1322,,d4 ;,;,,d4 ;,;,,d4;, ;,,d4;,;,,d4;,;,,d,,, ,,,,,tttxtt txtt tttxtt txtt t ttxtttxttttt xtttxttttt xtttxtttttGtttt GttttGtt     31,tt 231,,t t10111;;xtxtxttHtt. (16) In this case, the governing equations are 022001211;dLx twx txtttG t    (17) 1, (18) The ensemble average is (19) ant (20It has to be noticed that all the previous equations are deterministic linear ones in the general form 1011211,2 ,Lx ttwxtxttGtt 0xtxt d the variance is 21211;dxtxtt ) 22xwx wxFt witns  0,0tio h deterministic initial condi- 00xxx x. 0102 0dtxtxtxhtsFs s (21) In which we have 2222122221esin1,111ee21 211ee,1wtmt qtmt qthtw twtt   where ,2w221, 1mwwqww  . When adding the first term in the non-Gaussian part (the second approx imation) of the solutio n proinly cess of the previous case study, mad011111221212 12;;;,,ddxtxtxttHttIt has the general solution xtt tHtttt , (22) Copyright © 2013 SciRes. AJCM A. S. AL-JOHANI 189the governing equations become 0 (23) 1, (24)  220012112221212(); d2;,ddLxtwxtxtttwxtttttGt 10121112122122,2 ,4;;,dLx ttwxtx ttwxttxtt ttGtt,212022,,2,,Lxt ttwxt xttt1 1121222 2132331,4;,;,d,,x ttxtt2xtt txttttGttt (25) The ensemble average is still got by Equation (19) 2while the variance is got as 2212211 121;d2;,ddxtxtttxtt ttt   (26) The WHEP technique uses the following expansion for its deterministic kernels as corrections made under each approximation order. 23012 3,0,1,2,3,.iiiiixt xxxxi  .(27) Example: Let us take ;e;, 0.tFtq t  3 (28) in the previous case-study and then solving using the WHEP technique. The following results are obtained, see Figures 1-3. 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 levelsCopyright © 2013 SciRes. AJCM A. S. AL-JOHANI 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 0u ation 0,Aufrr (30) with boundary conditions ,0,uBu rn (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 . 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 cLv, whsic ided sol Equ m00Luich is the origea the ho0Avf r. This is the ba of2pv301 2 3vv pvpv  (32) Now, setting p = 1, the approximate solution of Equa-tion (23) is obt ai ned a s: 01231limpuvvvvv (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- 0u. paemethCopyright © 2013 SciRes. AJCM A. S. AL-JOHANI 191plications to ge n er a l ones. Example Considering the same previous example of Sub-Sect3.1.1, one can get the following results w.r.t. homoatiion topy perturbon:  22AxLx wx , 22Lxxwxwx  , 2Nx x, ;fr Ft. The homotopy function takes the following form:  0,1 0HvppLvLupAvfr  or equivalently, 220;0LvLup Lt . (34) Lett 2301 2 3pvpv , 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- 0uwv Fing lo2) vv pv00wing simple solution:  00,0,0.vyyxyx 0000221001;,0LvFtLvwvv  22,00,00Lvwvv vv . 10,00v. 3) 201224) 223102332,00,00Lvw vvvvv . 4103124 42,00,Lvvvvvvv00 . 5) The approximate solution is 0123;limxtv vvvv1p which can be considered to any approximation order. Oncan notice that the algorithm of the solution is straight a lot of flexibilities can be made. For any choices in guessing the initial approximation together with its initial conditionzero initial conditions, we can choose e t forward and thaexample, we have ms. For 0v0 which leads to: 5 012345;xtxv vv vvv;dht sF s 221002130;d2;;dtttswht sv sswhtsvs vss   (35) Figures 4-7 are obtained for 0.5: . 5. Comparisons between WHEP and HPM Methods Figure  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 thorder 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. AL-JOHANI 192 (a) (b) Figure 6. (a) A comparison between first, second order and thfirst, 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. AL-JOHANI Copyright © 2013 SciRes. AJCM 193 (f) (e) Figure 8. (a) A comparison between homotopy perturbation and Wiener-Hermite of the mean at ε = 0.1; (b) A comparison between homotopy perturbation and Wiener-Hermite of the variance at ε = 0.1; (c) A comparison between homotopy per- turbation and Wiener-Hermite of the mean at ε = 0.3; (d) A comparison between homotopy perturbation and Wie- ner-Hermite of the variance at ε = 0.3; (e) A comparison between homotopy pe rturbation and Wiener-Hermite of the mean at ε = 0.7; (f) A comparison betw ee n homotopy perturbation and Wiener-Hermite 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 valof ges u it will diverge. The M is more accurate for HPhigher 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. 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