Local Solutions of Laplace, Heat, and Other Equations by Itô Processes
Publication: Journal of Engineering Mechanics
Volume 123, Issue 8
Abstract
A method is presented for finding the solution of deterministic partial differential equations at an arbitrary point of the domain of definition of these equations referred to as the local solution. The method is based on the Itô calculus, properties of diffusion processes, and Monte Carlo simulation. The theoretical background of the proposed method is relatively difficult. However, the method has attractive features for applications. For example, the numerical algorithms based on the proposed method are simple, stable, accurate, local, and ideal for parallel computation. Numerical examples from mechanics are presented to demonstrate the use and the accuracy of the proposed method.
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References
1.
Boresi, A. P., and Chong, K. P. (1987). Elasticity in engineering mechanics. Elsevier Applied Science, New York.
2.
Chung, K. L., and Williams, R. J. (1990). Introduction to stochastic integration, 2nd Ed., Birkhäuser, Boston, Mass.
3.
Courant, R., and Hilbert, D. (1966). Methods of mathematical physics, Vol. 2, Wiley Interscience Publishers, New York.
4.
Durrett, R. (1984). Brownian motion and martingales in analysis. Wadsworth Advanced Books & Software, Belmont, Calif.
5.
Gihman, I. I., and Skorohod, A. V. (1972). Stochastic differential equations. Springer-Verlag New York, Inc., New York.
6.
Greenberg, M. D. (1978). Foundations of applied mathematics. Prentice-Hall, Inc., Englewood Cliffs, N.J.
7.
Grigoriu, M. (1995). Applied non-Gaussian processes. Examples, theory, simulation, linear random vibration, and MATLAB solutions. Prentice-Hall, Inc., Englewood Cliffs, N.J.
8.
Kloeden, P. E., and Platen, E. (1992). Numerical solution of stochastic differential equations. Springer-Verlag New York Inc., New York.
9.
Muller, M. E.(1956). “Some continuous Monte Carlo methods for the Dirichlet problem.”The Annals of Math. Statistics, 27(3), 569–589.
10.
∅ksendal, B. (1992). Stochastic differential equations, 3rd Ed., Springer-Verlag New York Inc., New York.
11.
Suresh, K. (1966). “Fast point solution techniques in engineering analysis.”Advances in mechanical engineering, T. S. Mruthyunjaya, ed., Narosa Publishing House, New Delhi, India, 385–400.
12.
Zagajac, J. (1995). “A fast method for estimating discrete field values in early engineering design.”Proc., 3rd Symp. on Solid Modeling and Applications, C. Hoffmann and J. Rossignac, eds., 420–430.
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Information
Published In
Journal of Engineering Mechanics
Volume 123 • Issue 8 • August 1997
Pages: 823 - 829
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Aug 1, 1997
Published in print: Aug 1997
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