Solutions of Random Initial Value Problems

  • D. G. Harlow
  • T. J. Delph
Conference paper
Part of the IUTAM Symposia book series (IUTAM)


We consider the random nonlinear initial value problem
$$ \frac{{dX}}{{dt}} = g\left( {t,X,A} \right);X\left( {{t_0}} \right) = {X_0} $$
where X = (X 1 , X 2 ,…, X n ) T , g = (g 1 , g 2 ,…, g n ) T , and T denotes transpose. The randomness in the problem arises from the vector of initial conditions X 0 = (X 1 , X 2 ,…, X n ) T , whose components are random variables with joint probability density function (jpdf) \( {f_{{{x_0}}}} \) (x 0 ), and the m random parameters A = (A 1 , A 2 ,…, A m ), which appear on the right-hand side of eqn. (1) and which have the jpdf fA (a). In general, A and X 0 may be jointly distributed; however they are often independent in many applications.


Crack Length Random Parameter Joint Probability Density Function Jacobian Determinant Finite Element Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T.T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York (1973).MATHGoogle Scholar
  2. 2.
    G. Adomian, Stochastic Systems, Academic Press, New York (1983).MATHGoogle Scholar
  3. 3.
    N. Bellomo and G. Pistone, J. Math. Anal. and Appl. 77, 215 (1980).CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    N. Bellomo and G. Pistone, Mech. Res. Comm. 6, 75 (1979).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    D.G. Harlow and T.J. Delph, Math. & Computers in Simulation, to appear.Google Scholar
  6. 6.
    P. Billingsley, Probability and Measure, John Wiley & Sons, New York (1986).MATHGoogle Scholar
  7. 7.
    D. Xiao, D.G. Harlow, and T.J. Delph, Eng. Fract. Mech., to appear.Google Scholar
  8. 8.
    D.A. Virkler, B.M. Hillberry, and P.K. Goel, J. Eng. Mat. Tech. 101, 148 (1979).CrossRefGoogle Scholar
  9. 9.
    O. Ditlevsen and R. Olesen, Eng. Frac. Mech. 25, 177 (1986).CrossRefGoogle Scholar
  10. 10.
    J.E. Yukich and T.J. Delph, submitted for publication (1991).Google Scholar
  11. 11.
    I. Sprung and V.A. Zilberstein, in Understanding Variability in Creep and Rupture Behavior, ed. M Prager and J.D. Parker, Amer. Soc. Mech. Eng., New York (1988).Google Scholar
  12. 12.
    T.J. Delph, submitted for publication (1991).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • D. G. Harlow
    • 1
  • T. J. Delph
    • 1
  1. 1.Lehigh UniversityBethlehemUSA

Personalised recommendations