Numeric Solution of the Fokker-Planck-Kolmogorov Equation

Abstract

The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested; 2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.

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C. Floris, "Numeric Solution of the Fokker-Planck-Kolmogorov Equation," Engineering, Vol. 5 No. 12, 2013, pp. 975-988. doi: 10.4236/eng.2013.512119.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] A. T. Bharucha-Reid, “Elements of Theory of Markov Processes and Their Applications,” McGraw Hill, New York, 1960. [2] R. L. Stratonovich, “Topics in the Theory of Random Noise,” Gordon and Breach, New York, 1963. [3] Y. K. Lin and G. Q. Cai, “Probabilistic Structural Dynamics: Advanced Theory and Applications,” McGraw Hill, New York, 1995. [4] H. Risken, “Fokker-Planck Equation: Methods of Solution and Applications,” Springer, Berlin, 1996. [5] T. K. Caughey, S. H. Crandall and R. H. Lyon, “Derivation and Application of the Fokker-Planck Equation to Discrete Nonlinear Dynamic Systems,” Journal of Acoustical Society of America, Vol. 35, No. 11, 1963, pp. 16831692. http://dx.doi.org/10.1121/1.1918788 [6] T. K. Caughey and H. J. Payne, “On the Response of a Class of Self-Excited Oscillators to Stochastic Excitation,” International Journal of Non-Linear Mechanics, Vol. 2, No. 2, 1967, pp. 125-151. http://dx.doi.org/10.1016/0020-7462(67)90010-8 [7] A. T. Fuller, “Analysis of Nonlinear Stochastic Systems by Means of the Fokker-Planck Equation,” International Journal of Control, Vol. 9, No. 6, 1969, pp. 603-655. http://dx.doi.org/10.1080/00207176908905786 [8] S. C. Liu, “Solutions of Fokker-Planck Equation with Applications in Nonlinear Random Vibration,” Bell System Technical Journal, Vol. 48, No. 8, 1969, pp. 2031-2051. http://dx.doi.org/10.1002/j.1538-7305.1969.tb01163.x [9] M. F. Dimentberg, “An Exact Solution to a Certain NonLinear Random Vibration Problem,” International Journal of Non-Linear Mechanics, Vol. 17, No. 4, 1982, pp. 231-236. http://dx.doi.org/10.1016/0020-7462(82)90023-3 [10] T. K. Caughey and F. Ma, “The Steady-State Response of a Class of Dynamical Systems to Stochastic Excitation,” Journal of Applied Mechanics ASME, Vol. 49, No. 3, 1982, pp. 629-632. http://dx.doi.org/10.1115/1.3162538 [11] T. K. Caughey and F. Ma, “The Exact Steady-State Solution of a Class of Non-Linear Stochastic Systems,” International Journal of Non-Linear Mechanics, Vol. 17, No. 3, 1982, pp. 137-142. http://dx.doi.org/10.1016/0020-7462(82)90013-0 [12] T. K. Caughey, “On the Response of Non-Linear Oscillators to stochastic Excitation,” Probabilistic Engineering Mechanics, Vol. 1, No. 1, 1986, pp. 2-4. http://dx.doi.org/10.1016/0266-8920(86)90003-2 [13] Y. Yong and Y. K. Lin, “Exact Stationary Response Solution for Second Order Nonlinear Systems under Parametric and External White Noise Excitations,” Journal of Applied Mechanics ASME, Vol. 54, No. 2, 1987, pp. 414418. http://dx.doi.org/10.1115/1.3173029 [14] Y. K. Lin and G. Q. Cai, “Exact Stationary Response Solution for Second Order Nonlinear Systems under Parametric and External White Noise Excitations: Part II,” Journal of Applied Mechanics ASME, Vol. 55, No. 3, 1988, pp. 702-705. http://dx.doi.org/10.1115/1.3125852 [15] G. Q. Cai and Y. K. Lin, “On Exact Stationary Solutions of Equivalent Non-Linear Stochastic Systems,” International Journal of Non-Linear Mechanics, Vol. 23, No. 4, 1988, pp. 315-325. http://dx.doi.org/10.1016/0020-7462(88)90028-5 [16] C. Soize, “Steady-State Solution of Fokker-Planck Equation in Higher Dimension,” Probabilistic Engineering Mechanics, Vol. 3, No. 4, 1988, pp. 196-206. http://dx.doi.org/10.1016/0266-8920(88)90012-4 [17] W.-Q. Zhu, G. Q. Cai and Y. K. Lin, “On Exact Stationary Solutions of Stochastically Perturbed Hamiltonian Systems,” Probabilistic Engineering Mechanics, Vol. 5, No. 2, 1990, pp. 84-87. http://dx.doi.org/10.1016/0266-8920(90)90011-8 [18] W.-Q. Zhu, “Exact Solutions for Stationary Responses of Several Classes of Nonlinear Systems to Parametric and/ or External White Noise Excitations,” Applied Mathematics and Mechanics, Vol. 11, No. 2, 1990, pp. 165-175. http://dx.doi.org/10.1007/BF02014541 [19] C. Soize, “Exact Stationary Response of Multi-Dimensional Non-Linear Hamiltonian Dynamical Systems under Parametric and External Stochastic Excitations,” Journal of Sound and Vibration, Vol. 149, No. 1, 1991, pp. 1-24. http://dx.doi.org/10.1016/0022-460X(91)90908-3 [20] W.-Q. Zhu and Y. Q. Yang, “Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems,” Journal of Applied Mechanics ASME, Vol. 63, No. 2, 1996, pp. 493-500. http://dx.doi.org/10.1115/1.2788895 [21] R. Wang and K. Yasuda, “Exact Stationary Probability Density for Second Order Nonlinear Systems under External White Noise Excitation,” Journal of Sound and Vibration, Vol. 205, No. 5, 1997, pp. 647-665. http://dx.doi.org/10.1006/jsvi.1997.1052 [22] R. Wang and Z. Zhang, “Exact Stationary Response Solutions of Six Classes of Nonlinear Stochastic Systems under Stochastic Parametric and External Excitations,” Journal of Engineering Mechanics ASCE, Vol. 124, No. 1, 1998, pp. 18-23. http://dx.doi.org/10.1061/(ASCE)0733-9399(1998)124:1(18) [23] Z. Zhang, R. Wang and K. Yasuda, “On Joint Stationary Probability Density Function of Nonlinear Dynamic Systems,” Acta Mechanica, Vol. 130, No. 1, 1998, pp. 29-39. http://dx.doi.org/10.1007/BF01187041 [24] R. Wang, K. Yasuda and Z. Zhang, “A Generalized Analysis Technique of the Stationary FPK Equation in Nonlinear Systems under Gaussian White Noise Excitations,” International Journal of Engineering Science, Vol. 38, No. 12, 2000, pp. 1315-1330. http://dx.doi.org/10.1016/S0020-7225(99)00081-6 [25] R. Wang and Z. Zhang, “Exact Stationary Solutions of the Fokker-Planck Equation for Nonlinear Oscillators under Stochastic Parametric and External Exctations,” Nonlinearity, Vol. 13, No. 3, 2000, pp. 907-920. [26] Z. L. Huang and W.-Q. Zhu, “Exact Stationary Solutions of Stochastically and Harmonically Excited and Dissipated Integrable Hamiltonian Systems,” Journal of Sound and Vibration, Vol. 230, No. 3, 2000, pp. 709-720. http://dx.doi.org/10.1006/jsvi.1999.2634 [27] W.-Q. Zhu and Z. L. Huang, “Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems,” International Journal of Non-Linear Mechanics, Vol. 36, No. 1, 2001, pp. 3-48. http://dx.doi.org/10.1016/S0020-7462(99)00086-4 [28] R. G. Bhandari and R. E. Sherrer, “Random Vibration in Discrete Nonlinear Dynamic Systems,” Journal of Mechanical Engineering Science, Vol. 10, No. 2, 1968, pp. 168-174. http://dx.doi.org/10.1243/JMES_JOUR_1968_010_024_02 [29] G. Muscolino, G. Ricciardi and M. Vasta, “Stationary and Non-Stationary Probability Density Function for NonLinear Oscillators,” International Journal of Non-Linear Mechanics, Vol. 32, No. 6, 1997, pp. 1051-1064. http://dx.doi.org/10.1016/S0020-7462(96)00134-5 [30] G.-K. Er, “Multi-Gaussian Closure Method for Randomly Excited Non-Linear Systems,” International Journal of Non-Linear Mechanics, Vol. 33, No. 2, 1998, pp. 201-214. http://dx.doi.org/10.1016/S0020-7462(97)00018-8 [31] G.-K. Er, “A Consistent Method for the Solution to Reduced FPK Equation in Statistical Mechanics,” Physica A, Vol. 262, No. 1-2, 1999, pp. 118-128. http://dx.doi.org/10.1016/S0378-4371(98)00362-8 [32] M. Di Paola and A. Sofi, “Approximate Solution of the Fokker-Planck-Kolmogorov Equation,” Probabilistic Engineering Mechanics, Vol. 17, No. 4, 2002, pp. 369-384. http://dx.doi.org/10.1016/S0266-8920(02)00034-6 [33] W. Martens, U. von Wagner and V. Mehrmann, “Calculation of High-Dimensional Probability Density Functions of Stochastically Excited Nonlinear Mechanical Systems,” Nonlinear Dynamics, Vol. 67, No. 3, 2012, pp. 2089-2099. http://dx.doi.org/10.1007/s11071-011-0131-2 [34] J. D. Atkinson, “Eigenfunction Expansions for Randomly Excited Non-Linear Systems,” Journal of Sound and Vibration, Vol. 30, No. 2, 1973, pp. 153-172. http://dx.doi.org/10.1016/S0022-460X(73)80110-5 [35] Y.-K. Wen, “Approximate Method for Nonlinear Random Vibration,” Journal of Engineering Mechanics Division ASCE, Vol. 101, No. 4, 1975, pp. 389-401. [36] J. P. Johnson and R. A. Scott, “Extension of Eigenfunction Expansion of a Fokker-Planck Equation—I. First Order System,” International Journal of Non-Linear Mechanics, Vol. 14, No. 5-6, 1979, pp. 315-324.http://dx.doi.org/10.1016/0020-7462(79)90005-2 [37] J. P. Johnson and R. A. Scott, “Extension of Eigenfunction Expansion of a Fokker-Planck Equation—II. Second Order System,” International Journal of Non-Linear Mechanics, Vol. 15, No. 1, 1980, pp. 41-56.http://dx.doi.org/10.1016/0020-7462(80)90052-9 [38] U. von Wagner and W. V. Wedig, “On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions,” Nonlinear Dynamics, Vol. 21, No. 3, 2000, pp. 289-306.http://dx.doi.org/10.1023/A:1008389909132 [39] R. Courant and D. Hilbert, “Methods of Mathematical Physics,” John Wiley & Sons, New York, 1989. [40] J. B. Roberts, “First-Passage Time for Randomly Excited Non-Linear Oscillators,” Journal of Sound and Vibration, Vol. 109, No. 1, 1986, pp. 33-50.http://dx.doi.org/10.1016/S0022-460X(86)80020-7 [41] T. Blum and A. J. McKane, “Variational Schemes in the Fokker-Planck Equation,” Journal of Physics A: Mathematical and General, Vol. 29, No. 9, 1996, pp. 18591872. http://dx.doi.org/10.1088/0305-4470/29/9/003 [42] M. Dehghan and M. Tatari, “The Use of He’s Variational Iteration Method for Solving a Fokker-Planck Equation”, Physica Scripta, Vol. 74, No. 3, 2006, pp. 310-316.http://dx.doi.org/10.1088/0031-8949/74/3/003 [43] A. Quarteroni and A. Valli, “Numerical Approximation of Partial Differential Equations,” Springer, Berlin, 1994. [44] L. A. Bergman and J. C. Heinrich, “Solution of the Pontriagin-Vitt Equation for the Moments of Time to First Passage of the Randomly Accelerated Particle by the Finite Element Method,” International Journal for Numerical Methods in Engineering, Vol. 15, No. 9, 1980, pp. 1408-1412. http://dx.doi.org/10.1002/nme.1620150913 [45] L. A. Bergman and J. C. Heinrich, “On the Moments of Time to First Passage of Linear Oscillator,” Earthquake Engineering and Structural Dynamics, Vol. 9, No. 3, 1981, pp. 197-204.http://dx.doi.org/10.1002/eqe.4290090302 [46] B. F. Spencer Jr. and L. A. Bergman, “On the Reliability of a Simple Hysteretic System,” Journal of Engineering Mechanics, Vol. 111, No. 12, 1985, pp. 1502-1514.http://dx.doi.org/10.1061/(ASCE)0733-9399(1985)111:12(1502) [47] R. S. Langley, “A Finite Element Method for the Statistics of Non-Linear Random Vibration,” Journal of Sound and Vibration, Vol. 101, No. 1, 1985, pp. 41-54.http://dx.doi.org/10.1016/S0022-460X(85)80037-7 [48] H. P. Langtangen, “A General Numerical Solution Method for Fokker-Planck Equations with Applications to Structural Reliability,” Probabilistic Engineering Mechanics, Vol. 6, No. 1, 1991, pp. 33-48.http://dx.doi.org/10.1016/S0266-8920(05)80005-0 [49] B. F. Spencer Jr. and L. A. Bergman, “On the Numerical Solution of the Fokker-Planck Equation for Nonlinear Stochastic Systems,” Nonlinear Dynamics, Vol. 4, No. 4, 1993, pp. 357-372.http://dx.doi.org/10.1007/BF00120671 [50] H. U. K?ylüoglu, S. R. K. Nielsen and R. Iwankiewicz, “Reliability of Non-Linear Oscillators Subject to Poisson Driven Impulses,” Journal of Sound and Vibration, Vol. 176, No. 1, 1994, pp. 19-33.http://dx.doi.org/10.1006/jsvi.1994.1356 [51] L.-C. Shiau and T.-Y. Wu, “A Finite Element Method for Analysis of Non-Linear System under Stochastic ParaMetric and External Excitation,” International Journal of Non-Linear Mechanics, Vol. 31, No. 2, 1996, pp. 193201. http://dx.doi.org/10.1016/0020-7462(95)00049-6 [52] L. A. Bergman, S. F. Wojtkiewicz, E. A. Johnson and B. F. Spencer Jr., “Robust Numerical Solution of the Fokker-Planck Equation for Second Order Dynamical Systems under Parametric and External White Noise Excitation,” In: W. H. Kliemann, Ed., Nonlinear Dynamics and Stochastic Mechanics, American Mathematical Society, Providence, 1996, pp. 23-37. [53] A. Masud and L. A. Bergman, “Application of Multi-Scale Finite Element Methods to the Solution of the FokkerPlanck Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 12-16, 2005, pp. 15131526. http://dx.doi.org/10.1016/j.cma.2004.06.041 [54] M. Kumar, S. Chakravorty, P. Singla and J. L. Junkins, “The partition of Unity Finite Element Approach with Hp-Refinement for the Stationary Fokker-Planck Equation,” Journal of Sound and Vibration, Vol. 327, No. 1-2, 2009, pp. 144-162.http://dx.doi.org/10.1016/j.jsv.2009.05.033 [55] E. Wong and M. Zakai, “On the Relation between Ordinary and Stochastic Equations,” International Journal of Engineering Science, Vol. 3, No. 2, 1965, pp. 213-229.http://dx.doi.org/10.1016/0020-7225(65)90045-5 [56] R. L. Stratonovich, “Topics in Theory of Random Noise,” Taylor & Francis, New York, 1967. [57] K. It?, “On Stochastic Differential Equations,” Memoirs of American Mathematical Society, Vol. 4, 1951, pp. 1-51. [58] K. It?, “On a Formula Concerning Stochastic Differentials,” Nagoya Mathematical Journal, Vol. 3, No. 1, 1951, pp. 55-65. [59] M. Di Paola, “Stochastic Differential Calculus,” In: F. Casciati, Ed., Dynamic Motion: Chaotic and Stochastic Behaviour, Springer Verlag, Wien, 1993, pp. 29-92. [60] W. Q. Zhu, “Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation,” Applied Mechanics Review, Vol. 59, No. 4, 2006, pp. 230-248.http://dx.doi.org/10.1115/1.2193137 [61] J. B. Roberts and P. Spanos, “Random Vibration and Statistical Linearization,” John Wiley & Sons, New York, 1990.