Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation

In this paper we study the use of the generalized polynomial chaos method to differential equations describing a model that depends on more than one random input. This random input can be in the form of parameters or of initial or boundary conditions. We investigate the effect of the choice of the probability density functions for the inputs on the output stochastic processes. The study is performed on the Airy's differential equation. This equation is a good test case since its solutions are highly oscillatory and errors can develop both in the amplitude and the phase. Several different situations are considered and, finally, conclusions are presented.

[1]  Juan Carlos Cortés,et al.  Mean square power series solution of random linear differential equations , 2010, Comput. Math. Appl..

[2]  N. Wiener.  The Homogeneous Chaos , 1938 .

[3]  I. Gihman,et al.  Stochastic Differential Equations , 1975 .

[4]  Juan Carlos Cortés,et al.  Random matrix difference models arising in long-term medical drug strategies , 2010, Appl. Math. Comput..

[5]  Sheldon M. Ross,et al.  A Course in Simulation , 1990 .

[6]  Michel Loève,et al.  Probability Theory I , 1977 .

[7]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[8]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[9]  Goodarz Ahmadi,et al.  Application of Wiener-Hermite Expansion to Nonstationary Random Vibration of a Duffing Oscillator , 1983 .

[10]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[11]  Nicola Bellomo,et al.  Nonlinear Stochastic Systems in Physics and Mechanics , 1987 .

[12]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

[13]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[14]  Juan Carlos Cortés,et al.  Numerical solution of random differential initial value problems: Multistep methods , 2011 .

[15]  G. Adomian,et al.  Solving frontier problems of physics : the decomposition methoc [i.e. method] , 1994 .

[16]  Magdy A. El-Tawil,et al.  Using Homotopy WHEP technique for solving a stochastic nonlinear diffusion equation , 2010, Math. Comput. Model..

[17]  Ali H. Nayfeh,et al.  Problems in Perturbation , 1985 .

[18]  Abraham J. Arenas,et al.  Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain , 2009, Biosyst..

[19]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[20]  Benito M. Chen-Charpentier,et al.  Epidemic models with random coefficients , 2010, Math. Comput. Model..

[21]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[22]  Magdy A. El-Tawil.  The Average Solution of a Stochastic Nonlinear Schrodinger Equation under Stochastic Complex Non-homogeneity and Complex Initial Conditions , 2009, Trans. Comput. Sci..

[23]  C. Braumann,et al.  Variable effort fishing models in random environments. , 1999, Mathematical biosciences.

[24]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[25]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[26]  Wilfred D. Iwan,et al.  On the existence and uniqueness of solutions generated by equivalent linearization , 1978 .