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Method of particular solutions for linear, two-point boundary-value problems

  • Angelo Miele
Contributed Papers

Abstract

The methods commonly employed for solving linear, two-point boundary-value problems require the use of two sets of differential equations: the original set and the derived set. This derived set is the adjoint set if the method of adjoint equations is used, the Green's functions set if the method of Green's functions is used, and the homogeneous set if the method of complementary functions is used.

With particular regard to high-speed digital computing operations, this paper explores an alternate method, the method of particular solutions, in which only the original, nonhomogeneous set is used. A general theory is presented for a linear differential system ofnth order. The boundary-value problem is solved by combining linearly several particular solutions of the original, nonhomogeneous set. Both the case of an uncontrolled system and the case of a controlled system are considered.

Keywords

Differential Equation Control System General Theory Differential System Adjoint 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.

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References

  1. 1.
    Miele, A.,Method of Particular Solutions for Linear, Two-Point Boundary-Value Problems, Part 1, Preliminary Examples, Rice University, Aero-Astronautics Report No. 48, 1968.Google Scholar
  2. 2.
    Miele, A.,Method of Particular Solutions for Linear, Two-Point Boundary-Value Problems, Part 2, General Theory, Rice University, Aero-Astronautics Report No. 49, 1968.Google Scholar
  3. 3.
    Bliss, G. A.,Mathematics for Exterior Ballistics, John Wiley and Sons, New York, 1944.Google Scholar
  4. 4.
    Goodman, T. R., andLance, C. N.,The Numerical Integration of Two-Point Boundary-Value Problems, Mathematical Tables and Other Aids to Computation, Vol. 10, No. 54, 1956.Google Scholar
  5. 5.
    Bellman, R.,Introduction to the Mathematical Theory of Control Processes, Vol. 1: Linear Equations and Quadratic Criteria, Academic Press, New York, 1967.Google Scholar
  6. 6.
    Miller, K. S.,Linear Differential Equations in the Real Domain, W. W. Norton and Company, New York, 1963.Google Scholar
  7. 7.
    Gura, I. A.,State Variable Approach to Linear Systems, Instruments and Control Systems, Vol. 40, No. 10, 1967.Google Scholar
  8. 8.
    Ince, E. L.,Ordinary Differential Equations, Dover Publications, New York, 1956.Google Scholar
  9. 9.
    Tifford, A. N.,On the Solution of Total Differential, Boundary-Value Problems, Journal of the Aeronautical Sciences, Vol. 18, No. 1, 1951.Google Scholar
  10. 10.
    Fox, L.,The Numerical Solution of Two-Point Boundary Problems in Ordinary Differential Equations, The Clarendon Press, Oxford, England, 1957.Google Scholar
  11. 11.
    Fox, L., Editor,Numerical Solution of Ordinary and Partial Differential Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1962.Google Scholar
  12. 12.
    Boyce, W. E., andDiPrima, R. C.,Elementary Differential Equations and Boundary Value Problems, John Wiley and Sons, New York, 1965.Google Scholar
  13. 13.
    Dennis, S. C. R., andPoots, G.,The Solution of Linear Differential Equations, Proceedings of the Cambridge Philosophical Society, Vol. 51, No. 3, 1955.Google Scholar
  14. 14.
    Clenshaw, C. W.,The Numerical Solution of Linear Differential Equations in Chebyshev Series, Proceedings of the Cambridge Philosophical Society, Vol. 53, No. 1, 1957.Google Scholar
  15. 15.
    Kagiwada, H. H., andKalaba, R. E.,A Practical Method for Determining Green's Functions Using Hadamard's Variational Formula, Journal of Optimization Theory and Applications, Vol. 1, No. 1, 1967.Google Scholar
  16. 16.
    Kagiwada, H. H., Kalaba, R. E., Schumitzky, A., andSridhar, R.,Cauchy and Fredholm Methods for Euler Equations, Journal of Optimization Theory and Applications, Vol. 2, No. 4, 1968.Google Scholar
  17. 17.
    Kagiwada, H. H., andKalaba, R. E.,Derivation and Validation of an Initial-Value Method for Certain Nonlinear Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 2, No. 6, 1968.Google Scholar
  18. 18.
    Heideman, J. C.,Use of the Method of Particular Solutions in Nonlinear Two-Point Boundary-Value Problems, Part 1, Uncontrolled Systems, Rice Univetsity, Aero-Astronautics Report No. 50, 1968.Google Scholar
  19. 19.
    Heideman, J. C.,Use of the Method of Particular Solutions in Nonlinear, Two-Point Boundary-Value Problems, Part 2, Controlled Systems, Rice University, Aero-Astronautics Report No. 51, 1968.Google Scholar

Copyright information

© Plenum Publishing Corporation 1968

Authors and Affiliations

  • Angelo Miele
    • 1
  1. 1.Department of Mechanical and Aerospace Engineering and Materials ScienceRice UniversityHouston

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