Local Galerkin Method for the Approximate Solutions to General FPK Equations

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1 Vol.3 No. Mar. 999 JOURNAL OF SHANGHAI UNIVERSITY Local Galerkin Method for the Approimate Solutions to General FPK Equations Er Guokang (Civil Engineering Institute, Southwest Jiaotong University ; Faculty of Science and Technology, University of Macao ) Abstract In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and etended for more general problems of stochastic differential equations (SDE), therefore the Fokker-Planck-Kolmogorov (FPK) equation is epressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of ~-correlated ecitations, the eistence of muhiplicative ecitations, and the dimension of SDE or FPK equation. Eamples are given and numerical results are provided for comparing with known eact solution to show the effectiveness of the method. Key words stochastic differential equations, probability density function, FPK equation, approimate PDF solution, local Galerkin method Introduction The stochastic differential equations (SDE) are used in many areas of science and engineering. The main purpose for solving problems with SDE is to obtain the probability density function(pdf) of the state variables governed by the SDE because many other statistical analysis are based upon PDF. Usually, the PDF of stochastic process vector is governed by the Fokker- Planck-Kolmogorov (FPK) equation. However, the so- lution to the multi-dimensional FPK equation have trou- bled many researchers in various areas for almost half a century. For the two-dimensional FPK equation which corresponds to two-dimensional stochastic differential equation, it is difficult with any available method to ob- tain desirable approimate solutions to highly nonlinear SDE or the SDE with multiplicative random ecitation. For three- or higher-dimensional problems, it seems that there is even no method available previously for rea- sonable approimate PDF solutions ecept the equivalent Received Jun. 23, 998 The paper partially supported by the Foundation of the Research Committee of the University of Macao (No. 685/96/UM, 628/ 97/UM) Er Guokang, Ph.D., Asso. Prof., Civil Engineering Institute, Southwest Jiaotong University. Currently, visiting scholar of University of Macao, Faculty of Science and Technology, University of Macao, Macao 300 stochastic linearization method which is actually only suitable for weakly nonlinear SDE without multiplicative ecitation, or stochastic average method which is only suitable for weakly nonlinear systems with weak ecitations. Moreover, it is known that most of practical problems are governed by multi-dimensional SDE. A lot of work can be found for the eact or approimate solutions to the FPK equations or SDE. Various methods were proposed in the past decades, such as equivalent stochastic linearization method [], stochastic average method [2], stochastic perturbation method [3, NonGaussian Hermite polynomial closure method E43, equivalent nonlinear system method 53, maimum entropy method E63, and multi-gaussian closure method Iv's], etc., but the methods have one or more of the following restrictions: () It is only suitable for weakly nonlinear system without multiplicative ecitations, e.g., equivalent stochastic linearization method, perturbation method, stochastic average method and so on. (2) The PDF model does not satisfy the probability theory, i.e., negative PDF value may be resulted, e.g., Hermite Polynomial closure method. (3) It is only suitable for two-dimensional problems or the nonlinear random vibrations of single-degree-of-freedom systems, e.g., all other methods ecept equivalent stochastic linearization method, Hermite polynomial closure method and stochastic average method. (4) Highly nonlinear algebraic equations are resulted, which is also a tedious problem, e.g., maimum entropy method

2 26 Journal of Shanghai University with which multi-dimensional integrations are also needed, and multi-gaussian closure method. Recently, a new method was briefly reported by the author [93. With this method, the above four problems were attempted to be solved simultaneously and the only assumption is that the PDF solution to the FPK equation eists. In this paper, the method is presented in detail and etended for more general problems of SDE, therefore the FPK equation is epressed in general form with no limitation on the degree of nonlinearity, the type of g-correlated ecitations, the eistence of multiplicative ecitations, and the dimension of the FPK or SDE equation. Eamples are given and numerical results are provided for comparing with known eact solution to show the effectiveness of the method. 2 Local Galerkin Method It is assumed that the PDF solution to SDE is governed by the following general FPK equation: 3 p(x,t) at + L(X,t)p(X,t) = 0, () where X E ~', L~ (X, t) is a linear differential operator with respect to X, and p(x, t ) is the PDF of X, with p(x,t)>o and lim p(x,t)=o. -~+~ The approimate solution of equation () is assumed to be -fi (X,t;a) = c ep. ('t;~), (2) where Q, (X, t ; a ) is an n degree polynomial in l, 2,"",,, c is normalization constant, a is an un known parameter vector, a E ~Np and Np is the total number of unknown parameters. Substituting Eq. (2) into Eq. () leads to ~aq, R(X,t;a) = k~- t + D,(X,t;a)]'p (X,t;a), (3) which is the residual error of Eq. () due to substitution of --p(x,t;a) for p(x,t). D,(X,t;a) depends on the detailed form of the FPK equation. For eample, for the following Ito's SDE d dii = fi(x,t) + gij(x t)wj(t), i =,2,'", n~ ; j =,2,"', m, (4) where Wj(t) is Gaussian white noise with E[W~(t) Wk(t + v)] : Sjka(v),( j,k :,2,"",m), and d?(" ) is the Dirac delta-function, it can be shown that D.(X,t;a) = f~ 8 O. (a Gij q5 On + a j 2 8 i a i aeq. aq. aq. ag o a Q, + G G0 ) + O X i O Xj 0 XjOXj 0 X i 3 Xj a f~ l 02 Gij 3 i 2 a i O j" (5) where Go(X,t) = Sl~gil(X,t)gjs(X,t). (6) Because p ( X, t ; a ) is only an approimation of the solution to Eq. (), therefore, R(X,t;a)~0. In fact, it is difficult to make R (X, t ; a ) = 0 eactly or approimately because of the difficulties in formulating the governing equations for parameters a and seeking solutions for them. In the following analysis, we get away from the difficulties by solving the problem in another way. Eq. (3) can also be epressed as where R(X,t;a) = r(x,t;a)-p (X,t;a), (7) r(x,t;a) Q, + D,(X,t;a) (8) at which can be considered as local residual error of Eq. ( ) if p (X, t ) is substituted with p (X, t ; a ). Since p (X, t ; a ) :J:0 in general, the only possibility for R(X,t;a)=O is r(x,t;a)=o. However, r(x, t ; a ) :# 0 in general because usually p ( X, t ; a ) :/: p (X, t). In the following analysis, a set of special basic functions which span g~ are introduced to make the projection of local residual error r (X, t ; a ) on jvp vanish. Therefore, this method may be called the local Galerkin method to identify the difference from the ordinary Galerkin method by making the projection of the residual error of original operator equation vanish on fi- nite-dimensional space. Suppose that the basic functions which span ~Np are H k (X, t),k =,2, "-', Np and r (X, t ; a )" H k (X, t ) is integrable in ~gnp, then, according to the above idea of the local Galerkin method ~.r(x,t;a)gk(,t)dx = O, k =,2,'",Np. The problem hereafter lies in the determination of basic (9)

3 Vol.3 No. Mar. 999 Er G.K. : Local Galerkin Method for the Approimate function Hk (X, t ). The Hk (X, t ) can be selected to be Xkl a2""~f(x,t), k k being kt,k2,'",k. =0,, 2,"', Np and k = kl + k2 + "'" + k~, if F (X, t ) can make r(x,t;a)h k(x,t) integrable in ~Np. After that, the problem turns out to be the determination of function F(X, t ). It is obvious that the integrable con- dition can be fulfilled if F(X, t) is selected to be Gaus- sian PDF. Numerical eperience showed that the Gaus- sian PDF resulted from Gaussian closure method, or its modified form is an efficient and powerful substitute for F(X, t) which can lead to much desirable approimate PDF solution to Eq. (). 3 Eamples Eample Consider the following second order lin- ear system with both eternal and multiplicative random ecitations: X + cl[ + Wl(t)].~+ c2[ + W2(t)]X = W3(t), (0) which can be epressed in the form of following two di- mensional system : 2= Y, () ~'= _ c[ + W(t)]Y - c2[ + W2(t)]X + W3(t), (2) where C~0,C2~0, Wl(t), W2(t) and W3(t) are independent Gaussian white noises. In the case of 28 (3) c2s22 ~- c the eact stationary PDF solutions are obtainable to be [m ] and p() = ff-/-'(c~ - /2)(/3 + 2) -(3-/2), (4) ~UnF(8 - ) - ~)) y2/c2) (3-/2) = ZVc c r( -- (/3 + (5) where _P is the gamma function, /3 = $33/( 2 C 2 S22 ), 8 -- Cl/( czs2z ) 4- /2 and 8>. For ct = 0., Cz =, Sll -- See = S3s =, the condition (3) is fulfilled. If the stationary PDF from Gaussian closure is utilized as function F(X, t ), the e- valuated approimate PDFs of X and Y for n = 2 and 6 are shown and compared with eact solutions in Figs. and 2. It is known from Figs. and 2 that the results are much improved as n increases from 2 to 6, and the curves for the case of n = 6 almost coincide with those of eact solutions. In order to show the tail behavior of the PDFs, the logarithmic PDFs are shown in Figs. 3 and 4. It is apparent in Figs. 3 and 4 that the solutions are even much improved in the tails as n increases from 2 to 6. The presented results in this eample validate the proposed mothod for the nonlinear stochastic systems with both eternal and multiplicative ecitations. Eample 2 j O.9~,Q oo7 \ 08, 06. -::o, Fig. The PDFs of X, for eample Z , jiii Eact... n=6 0 ~ 9 2 Y Fig.2 The PDFs of Y, for eample Consider the following two-degree-of- freedom nonlinear stochastic system: Yt + ~- al(sll YI + 2a2S2 Y2) + 2a3YL + 4a4y3 + 6asY5 = Wl(t), (6) Y2 + lat[2( - a2)s2 "~rl + $22 ~rr2] + 2a6 Y2 + 4aTy32 +6a8YSz = W2(t), (7) where a I, a2, "'", a8 are some constants, and W i ( t ), (i =,2), is Gaussian white noise. By setting YI =

4 28 Journal of Shanghai University Xl, ~Zl = X2, Y2 = X3 and ~z 2 = X4, the system can also be epressed by the following four-dimensional non- linear stochastic system: I, }., -2 X... n=6 \'~... n=2 ',~ Fig. 3 The logarithmic PDFs of Y, for eample d" / ~' " / -5 Eac~ \ Y...?/=2 Fig.4 The logarithmic PDFs of Y, for eample R = X2, (8) ff~2 = -- 2al(SllX2 + 2a2S2X4) -- 2a3X, - 4a4X ~ - 6asX 5, + Wl(t), (9) "~73 = X4, (20) X4 : -- ~-a2( -- a2)s2x 2 + $22X4] - For this system, obtainable to be []']2] 2a6X3-4a7X~ - 6asX~ + W2(t). (2) the eact stationary PDF solution is 2 2 p(3s,2,3, 4) = C ep{- al[~-( 2 + 4) + O'337~ + a4"~7~ + a5,;e6 + a6-~723 + a743 + as.z~]t, (22) where C is normalization constant. In the following analysis, the approimate PDF solu- tions obtained with the proposed local Galerkin method for different n values are compared with the above eact solution. It is noted that the equivalent stochastic lin- earization method or Gaussian closure method is a special case of the proposed local Galerkin method if F(X, t ) is ' selected to be the PDF from Gaussian closure. In this eample, the function F(X, t) is selected to be the stationary PDF from Gaussian closure. For a : a 3 : a 4 : a 6 =, a s = a 7 : a8:0.5 and arbitrary values of S~, S2, $22 and a2, the system is highly nonlinear and the approimate PDFs of X and X3 obtained with the presented method are compared with the eact PDF solutions in Figs. 5 and 6. It is ap- parent that the approimate solutions for n = 4 are very close to the eact solutions though the system is highly nonlinear. For n = 2, the results coincide with those from equivalent stochastic linearization or Gaussian clo- sure procedure. The PDF solutions for n = 4 are much improved comparing to those for n = 2. In order to show the tail behavior of the PDFs, the logarithmic PDFs are plotted in Figs. 7 and 8. It is obvious that the approimate PDFs for n = 4 are much close to eact PDF solutions even in the tails which are important for reliability analysis. Numerical results show that the PDFs of X2 and X4 are same as eact solutions in all cases. The above results validate the method for higherdimensional systems or higher-dimensional FPK equations. j_, 0y ~.8. /'.7 I/ 04 /./ 0.3 // Fig. 5 XI 2". Eact \... n= n= The PDFs of X, for eample "X \ t.'" 0.8 '/ l/o.,, /.; o.3, // 02 j," 0., -'~ i, n -I Fig. 6 Eaa ~\... n= n=2 '~, [, The PDFs of X3, for eample 2

5 Vol. 3 No. Mar. 999 Er G. K. : Local Galerkin Method for the Approimate g- Fig. 7 j!: Eact n=4... /2= l The logarithmic PDFs of X), for eample 2 plicative ecitations; (2) the approimate PDF model meets the probability theory; (3) the method is suitable for higher-dimensional problems; (4) for Ito's SDEs which are widely used for many problems in science and engineering, the resulted algebraic equations for stationary PDF are quadratic which are easy to be solved; and (5) the solution procedure is consistent and it is easy to write computer program and develop software with this method. Based upon the method, computer softwares were developed by the author. The numerical results presented in this papers were just obtained from the softwares. r~ Fig. 8 4 Conclusions Eact n'-4... n=2 "6 The logarithmic PDFs of X3, for eample 2 The idea proposed recently by the author for the approimate PDF solution of nonlinear random vibrations is etended for the solutions to more general FPK equations. Therefore, a new method has been proposed for the approimate solutions to more general FPK equations. With this method, the approimate solution to the FPK equation is assumed to be of the form of an eponential function of the polynomial in state variables. After the approimate PDF is substituted into the FPK equation, the FPK equation is not solved directly, but local residual error is separated. After that, a set of basic functions which span a finite-dimensional real space are formulated and the projection of the local residual error is made vanish on the finite-dimensional real space. Consequently, first order ordinary differential equations with respect to time or algebraic equations are formulated in terms of the unknown parameters. The approimate PDF solution to FPK equation can be determined by the solutions to the ordinary differential equations or algebraic equations. The derivation procedure and numerical results have shown that () the method is not limited by the degree of nonlinearity of SDE, the type of 8-correalted ecitations, and the eistence of multi- References Booton R. C., Nonlinear control systems with random inputs, IRE Transactions on Circuit Theory,954,CT-():9-9 2 Stratonovich R. L., Topics in the Theory of Random Noise Vol., Gordon and Breach, New York, Crandall S. H., Perturbation techniques for random vibration of nonlinear systems, J. Acoust. Soc. Am., 963,35 : Assaf S.A. and Zirkie L. D., Approimate analysis of nonlinear stochastic systems, Int. J. Control, 976,23: Lutes L. D., Approimate technique for treating random vibration of hysterestic systems, J. Acoust. Soc. Am., 970, 48 : Tagliani A., Principle of maimum entropy and probability distributions: definition and applicability field, Probab. Eng. Mech., 989,4: Er G. K., Crossing rate analysis with a non-gaussian closure method for nonlinear stochastic systems, Nonlinear Dynamics, 997,4 : Er G. K., Multi-Gaussian closure method for randomly ecited nonlinear systems, Int. J. Non-Linear Mech., 998,33: Er G. K., A new non-gaussian closure method for the PDF solution of non-linear random vibrations, Proceedings of 2th EM Division Conference, ASCE, La Jolla, USA, May, 998: Dimentberg M. F., An eact solution to a certain nonlinear random vibration problem, Int. J. Non-linear Mech., 982,7: Scheurkogel A. and Elishakoff I., Non-linear random vibra- tion of a two-degree-of-freedom system, Non-Linear Stochas- tic Engineering Systems, Eds. F. Ziegler and G. I. Schueller, Springer-Verlag, Berlin, 998 : Lin Y.K. and Cai G. Q., Probabilistic Structural Dynamics, McGrawHill, New York, 995 : 86-88

EPC procedure for PDF solution of nonlinear. oscillators excited by Poisson white noise

* Manuscript Click here to view linked References EPC procedure for PDF solution of nonlinear oscillators excited by Poisson white noise H.T. Zhu,G.K.Er,V.P.Iu,K.P.Kou Department of Civil and Environmental