DECOMPOSITION SOLUTION FOR DUFFING AND VAN DER POL OSCILLATORS

The decomposition method is applied to solve the Duffing and Van der Pol oscillators without customary restrictive assumptions (I-4) and without resort to perturbation methods.


I. INTRODUCTION
The Duffing equation is written + + By + yy3 x(t) (l.l) The Van   The deterministic problem is now solved.sinceall components of y are determined. n-i We use an n-term approximation n ..^Yi which, because of the rapid convergence, is generally sufficient with a very sVl n (say half a dozen or so terms) but easily carried as far as necessary since the integrals do not involve difficult Green's functions.Convergence has been previously established by Adomian [2,5] and has been shown [2] to be quite rapid.
For the stochastic case [2], none of the usual approximations of statistical inearization are necessary.The x(t) need not be stationary nor Gaussian nor delta-correlated.Further e,B,Y and the initial conditions can be stochastic.No "smallnesS' assumptions are necessary for the stochastic processes and the non- linearities.No linearization is used.We can allow <=> + {, B <B> + n, <y>noAn y (R)<y> + o and write Ly x <=>(d/dt)y <B>y {(d/dt)y y o Z A and proceed as before with Y Z^Y n" n u n=OTe result is a stochastic series fro which statistics are obtained without the problems of statistical separability of quantities such as <Ry> where R d/dt n which normally require closure approximations.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: der Pol equation can be written Y + J} + BY + y(d/dt)y 3 x(t) l, y /3, we have the form usually given.)Write L d2/dt2, R e(d/dt) + B, Ny yy3 in (l.l) and y(d/dt)y 3 in (I.2) Thus both are written Ly + Ry + Ny x(t) (1.3) in the standard form for the decomposition method [I-3] where definite integral from 0 to t.Then, Ly x(t) Ry-Ny.Assuming initial conditions y(O), y'(O) are specified, let y define YO by YO y(O) + ty'(O) + L-Ix(t).Then Yn+l -L-l(d/dt)Yn L-I BYn L-I [Ny] To get computable solutions, we need only substitute for Ny the sum y Z A n n=O for the Duffing case and y(d/dt) A n for the Van der Pol case where the A n are

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Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation