Computation of the moments of solutions of certain random two point boundary value problems

  • R. Huss
  • R. Kalaba
Invited Papers
Part of the Lecture Notes in Mathematics book series (LNM, volume 228)


Assume that the linear two-point boundary value problem
$$\begin{gathered}\mathop x\limits^\infty + [p(t) + \lambda q(t)]x = - g(t),0 \leqslant t \leqslant 1, \hfill \\x(0) = 0,x(1) = c \hfill \\\end{gathered}$$
possesses a unique solution for all λ in the interval 0 ≤ λ ≤ Λ. Consider λ to be a random variable with probability density function f(λ), 0 ≤ λ ≤ Λ. A method for determining the moments
$$\begin{gathered}E[x^n (t,\lambda )] = \int_0^\Lambda {x^n (t,\lambda )f(\lambda )d\lambda } , \hfill \\n = 1,2, \cdots , \hfill \\\end{gathered}$$
is presented. Numerical experiments show the computational feasibility of the new approach.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adomian, G., J. Math. Phys. 11, 1069–1084 (1970).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bekey, G. A. and W. Karplus, Hybrid Computation (John Wiley & Sons, New York, 1968).MATHGoogle Scholar
  3. 3.
    Bellman, R. and R. Kalaba, Proc. Nat. Acad. Sci. USA 47, 336–338 (1961).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berezin, I. S. and N. P. Zhidkhov, Computing Methods (Addison-Wesley Publishing Company, Palo Alto, 1965).Google Scholar
  5. 5.
    Buell, J., R. Kalaba and E. Ruspini, Int. J. Engin. Sci. 7, 1167–1172 (1969).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Casti, J. and R. Kalaba, Information Sciences 2, 51–67 (1970).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Casti, J., R. Kalaba and B. Vereeke, J. Opt. Th. Applic. 3, 81–88 (1969).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Courant, R. and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York, 1953).MATHGoogle Scholar
  9. 9.
    Huss, R. and R. Kalaba, "Invariant Imbedding and the Numerical Determination of Green's Functions", J. Opt. Th. Applic., in press.Google Scholar
  10. 10.
    Huss, R., H. Kagiwada and R. Kalaba, "A Cauchy System for the Green's Function and the Solution of Two-Point Boundary Value Problems", J. Franklin Institute, in press.Google Scholar
  11. 11.
    Kagiwada, H. and R. Kalaba, J. Opt. Th. Applic. 1, 33–39 (1967).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kagiwada, H. and R. Kalaba, J. Math. Anal. Applic. 23, 540–550 (1968).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kagiwada, H. and R. Kalaba, J. Opt. Th. Applic. 2, 378–385 (1968).CrossRefGoogle Scholar
  14. 14.
    Kagiwada, H., R. Kalaba and Y. Thomas, J. Opt. Th. Applic. 5, 11–21 (1970).MathSciNetGoogle Scholar
  15. 15.
    Kalaba, R. and E. Ruspini, Int. J. Engin. Sci. 7, 1091–1101 (1969).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • R. Huss
  • R. Kalaba

There are no affiliations available

Personalised recommendations