# A Generalized Finite Difference Method for Solving Fokker-Planck-Kolmogorov Equations

• Zhao, Li (Department of Civil Engineering, The University of Akron) ;
• Yun, Gun Jin (Department of Mechanical and Aerospace Engineering, Seoul National University)
• Accepted : 2017.07.31
• Published : 2017.12.30

#### Abstract

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

#### Acknowledgement

Supported by : Seoul National University, University of Akron

#### References

1. Nigam, N. C., Introduction to Random Vibrations, MIT Press, Cambridge, 1983.
2. Caughey, T. K., "Derivation and Application of the Fokker-Planck Equation to Discrete Nonlinear Dynamic Systems Subjected to White Noise Excitation", Journal of the Acoustical Society of America, Vol. 35, No. 11, 1963, pp. 1683-1692. https://doi.org/10.1121/1.1918788
3. Chang, J. S. and Cooper, G., "A Practical Difference Scheme for Fokker-Planck Equations", Journal of Computational Physics, Vol. 6, No. 1, 1970, pp. 1-16. https://doi.org/10.1016/0021-9991(70)90001-X
4. Roberts, J. B., "First-Passage Time for Randomly Excited Non-linear Oscillators", Journal of Sound and Vibration, Vol. 109, No. 1, 1986, pp. 33-50. https://doi.org/10.1016/S0022-460X(86)80020-7
5. Zorzano, M. P., Mais, H. and Vazquez, L., "Numerical Solution of Two Dimensional Fokker-Planck Equations", Applied Mathematics and Computation, Vol. 98, No. 2-3, 1999, pp. 109-117. https://doi.org/10.1016/S0096-3003(97)10161-8
6. Langley, R. S., "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. https://doi.org/10.1016/S0022-460X(85)80037-7
7. Langtangen, H. P., "Numerical-Solution of 1st Passage Problems in Random Vibrations", Siam Journal on Scientific Computing, Vol. 15, No. 4, 1994, pp. 977-996. https://doi.org/10.1137/0915059
8. Spencer, B. F. and Bergman, L. A., "On the Numerical Solution of the Fokker-Planck Equation for Nonlinear Stochastic Systems", Nonlinear Dynamics, Vol. 4, No. 1993, pp. 357-362. https://doi.org/10.1007/BF00120671
9. Wehner, M. F. and Wolfer, W. G., "Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations. II. Restricted Stochastic-Processes", Physical Review A, Vol. 28, No. 5, 1983, pp. 3003-3011. https://doi.org/10.1103/PhysRevA.28.3003
10. Wehner, M. F. and Wolfer, W. G., "Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations", Physical Review A, Vol. 27, No. 5, 1983, pp. 2663-2670. https://doi.org/10.1103/PhysRevA.27.2663
11. Wehner, M. F. and Wolfer, W. G., "Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations. III. Time and Functionally Dependent Coefficients", Physical Review A, Vol. 35, No. 4, 1987, pp. 1795-1801. https://doi.org/10.1103/PhysRevA.35.1795
12. Kougioumtzoglou, I. A. and Spanos, P. D., "Nonstationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism", Journal of Engineering Mechanics, Vol. 140, No. 9, 2014, pp. 04014064. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000780
13. Cai, G. Q. and Lin, Y. K., "Reliability of Nonlinear Structural Frame Under Seismic Excitation", Journal of Engineering Mechanics, Vol. 124, No. 8, 1998, pp. 852-856. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:8(852)
14. Naess, A., Iourtchenko, D. and Batsevych, O., "Reliability of Systems with Randomly Varying Parameters by the Path Integration Method", Probabilistic Engineering Mechanics, Vol. 26, No. 1, 2011, pp. 5-9. https://doi.org/10.1016/j.probengmech.2010.05.005
15. Wojtkiewicz, S. F., Bergman, L. A. and Spencer, B. F., "High Fidelity Numerical Solutions of the Fokker-Planck Equation", Structural Safety and Reliability, Vols. 1-3, 1998, pp. 933-940.
16. Kumar, P. and Narayanan, S., "Solution of Fokker- Planck Equation by Finite Element and Finite Difference Methods for Nonlinear Systems", Sadhana, Vol. 31, No. 4, 2006, pp. 445-450. https://doi.org/10.1007/BF02716786
17. Ghaboussi, J., "Generalized Differences in Direct Integration Methods for Transient Analysis", International Conference on Numerical Methods in Engineering: Theory and Applications, Swansea, 1987.
18. Ballester, C. and Pereyra, V., "On the Construction of Discrete Approximation to Linear Differential Expressions", Mathematics of Computation, Vol. 21, No. 1967, pp. 297-302. https://doi.org/10.1090/S0025-5718-1967-0228167-8
19. Collatz, L., The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin, 1960.
20. Chandrasekhar, S., "Stochastic Problems in Physics and Astronomy", Reviews of Modern Physics, Vol. 43, No. 1, 1943, pp. 1-89. https://doi.org/10.1103/RevModPhys.43.1
21. Saad, Y., Iterative Methods for Sparse Linear Systems, PWS, 1996.
22. Naprstek, J. and Kral, R., "Finite Element Method Analysis of Fokker-Planck Equation in Stationary and Evolutionary Versions", Advances in Engineering Software, Vol. 72, No. 2014, pp. 28-38. https://doi.org/10.1016/j.advengsoft.2013.06.016