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Approximate analytical solutions for a class of nonlinear stochastic differential equations

Part of: Functional-differential and differential-difference equations Combinatorics, graph theory, probability theory, statistics Stochastic analysis

Published online by Cambridge University Press:  18 September 2018

A. T. MEIMARIS ,
I. A. KOUGIOUMTZOGLOU  and
A. A. PANTELOUS
A. T. MEIMARIS
Affiliation:
Department of Econometrics and Business Statistics, Monash Business School, Monash University, 20 Chancellors Walk, Wellington Road, Clayton, Victoria 3800, Australia emails: Antonios.Meimaris@monash.edu; Athanasios.Pantelous@monash.edu
I. A. KOUGIOUMTZOGLOU
Affiliation:
Department of Civil Engineering and Engineering Mechanics, The Fu Foundation School of Engineering and Applied Science, Columbia University, 500 West 120th Street, New York, NY 10027, USA email: ikougioum@columbia.edu
A. A. PANTELOUS
Affiliation:
Department of Econometrics and Business Statistics, Monash Business School, Monash University, 20 Chancellors Walk, Wellington Road, Clayton, Victoria 3800, Australia emails: Antonios.Meimaris@monash.edu; Athanasios.Pantelous@monash.edu
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Abstract

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

Keywords

Stochastic Differential EquationsStochastic DynamicsPath IntegralError quantificationCauchy-Schwarz inequality

MSC classification

Primary: 34K50: Stochastic functional-differential equations
Secondary: 60H10: Stochastic ordinary differential equations 60H35: Computational methods for stochastic equations 81P20: Stochastic mechanics (including stochastic electrodynamics) 97K60: Distributions and stochastic processes
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 928 - 944
Copyright
© Cambridge University Press 2018 

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References

Chaichian, M. & Demichev, A. (2001) Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics, CRC Press, Bath, UK.CrossRefGoogle Scholar
Di Matteo, A., Kougioumtzoglou, I. A., Pirrotta, A., Spanos, P. D. & Di Paola, M. (2014) Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral. Prob. Eng. Mech. 38, 127–135.CrossRefGoogle Scholar
Ewing, G. M. (1969) Calculus of Variations with Applications, Dover Publications, New York, USA.Google Scholar
Feynman, R. P. (1948) Space-time approach to non-relativistic quantum mechanics. Rev. Modern Phys. 20(2), 367.CrossRefGoogle Scholar
Forsgren, A., Gill, P. E. & Wright, M. H. (2002) Interior methods for nonlinear optimization. SIAM Rev. 44(4), 525–597.CrossRefGoogle Scholar
Grigoriu, M. (2002) Stochastic Calculus: Applications in Science and Engineering, Springer Science & Business Media, New York, USA.CrossRefGoogle Scholar
Kay, S. M. (1993) Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall PTR.Google Scholar
Kougioumtzoglou, I. A. (2017) A Wiener path integral solution treatment and effective material properties of a class of one-dimensional stochastic mechanics problems. ASCE J. Eng. Mech. 143(6), 04017014, 1–12. doi: 10.1061/(ASCE)EM.1943-7889.0001211.CrossRefGoogle Scholar
Kougioumtzoglou, I. A. & Spanos, P. D. (2009) An approximate approach for nonlinear system response determination under evolutionary stochastic excitation. Curr. Sci. 97(8), 1203–1211.Google Scholar
Kougioumtzoglou, I. A. & Spanos, P. D. (2012) An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators. Prob. Eng. Mech. 28, 125–131.CrossRefGoogle Scholar
Kougioumtzoglou, I. A. & Spanos, P. D. (2014) Nonstationary stochastic response determination of nonlinear systems: a Wiener path integral formalism. ASCE J. Eng. Mech. 140(9), 04014064, 1–14. doi: 10.1061/(ASCE)EM.1943-7889.0000780.CrossRefGoogle Scholar
Kougioumtzoglou, I. A., Di Matteo, A., Spanos, P. D., Pirrotta, A. & Di Paola, M. (2015) An efficient Wiener path integral technique formulation for stochastic response determination of nonlinear MDOF systems. ASME J. Appl. Mech. 82(10), 101005.CrossRefGoogle Scholar
Li, J. & Chen, J. (2009) Stochastic Dynamics of Structures, John Wiley & Sons, (Asia) Pte Ltd.CrossRefGoogle Scholar
Meimaris, A. T., Kougioumtzoglou, I. A. & Pantelous, A. A. (2018) A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations. Prob. Eng. Mech., 54, 87–94. doi: 10.1016/j.probengmech.2017.07.005.CrossRefGoogle Scholar
Mukhopadhyay, N. (2000) Probability and Statistical Inference, Marcel Dekker Inc, New York, USA.Google Scholar
Naess, A. & Johnsen, J. M. (1993) Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Prob. Eng. Mech. 8(2), 91–106.CrossRefGoogle Scholar
Nocedal, J. & Wright, S. J. (1999) Numerical Optimization, Springer, New York, USA.CrossRefGoogle Scholar
Roberts, J. B. & Spanos, P. D. (2003) Random Vibration and Statistical Linearization, Courier Corporation, New York, USA.Google Scholar
Sonnad, J. R. & Goudar, C. T. (2004) Solution of the Haldane equation for substrate inhibition enzyme kinetics using the decomposition method. Math. Comput. Model. 40(5–6), 573–582.CrossRefGoogle Scholar
Spanos, P. D., Kougioumtzoglou, I. A., dos Santos, K. R. M. & Beck, A. T. (2018) Stochastic averaging of nonlinear oscillators: Hilbert transform perspective. ASCE J. Eng. Mech. 144(2), 04017173, 1–9.CrossRefGoogle Scholar
Steele, J. M. (2004) The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, Cambridge University Press, New York, USA.CrossRefGoogle Scholar
Taniguchi, T. & Cohen, E. G. D. (2008) Inertial effects in nonequilibrium work fluctuations by a path integral approach. J. Stat. Phys. 130(1), 1–26.CrossRefGoogle Scholar
Touchette, H. (2009) The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69.CrossRefGoogle Scholar
Van Kampen, N. G. (1992) Stochastic Processes in Physics and Chemistry, Elsevier, New York, USA.Google Scholar
Wehner, M. F. & Wolfer, W. G. (1983) Numerical evaluation of path-integral solutions to Fokker–Planck equations. II. Restricted stochastic processes. Phys. Rev. A 28, 3003.CrossRefGoogle Scholar
Wiener, N. (1921) The average of an analytic functional and the Brownian movement. Proc. Natl Acad. Sci. 7(10), 294–298.CrossRefGoogle ScholarPubMed