Advertisement

Numerical solution of singular perturbed boundary value problems using a collocation method with tension splines

  • Maximilian R. Maier
Part of the Progress in Scientific Computing book series (PSC, volume 5)

Abstract

A collocation method is given for the numerical solution of a singular perturbed boundary value problem of the form:
$$ \begin{gathered} y' = f\left( {x,y,\varepsilon } \right){\kern 1pt} a \leqslant x \leqslant b \hfill \\ r\left( {y\left( a \right),y\left( b \right)} \right) = 0 \hfill \\ y = {\left( {{y_1}, \ldots ,{y_n}} \right)^t}{\kern 1pt} {y_i}:\left[ {a,b} \right] \to R \hfill \\ f = {\left( {{f_1}, \ldots ,{f_n}} \right)^t}{\kern 1pt} {f_i}:\left[ {a,b} \right] \to R \hfill \\ r = {\left( {{r_1}, \ldots ,{r_n}} \right)^t}{\kern 1pt} {r_i}:R \times R \to R \hfill \\ \end{gathered} $$
(1.1)

Keywords

Boundary Layer Collocation Method Space Charge Region Tension Parameter Boundary Layer Type 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrahamsson, L.: Numerical solution of a boundary value problem in semiconductor physics, Uppsala University Report No. 81 (1979).Google Scholar
  2. 2.
    Abrahamsson, L., Keller, H.B., Kreiss, H.O.: Difference approximations for singular perturbations of systems of ordinary differential equations, Numer. Math. 22, 367–391 (1974).MathSciNetMATHGoogle Scholar
  3. 3.
    Ascher, U.: Solving boundary value problems with a spline-collocation code, J. Comput. Phys. 34, 401–413 (1980).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ascher, U., Weiss, R.: Collocation for singular perturbation problems. First-order systems with constant coefficients, SIAM J. Num. 20 (3), 537–557 (1983).MathSciNetMATHGoogle Scholar
  5. 5.
    de Boor, C.: Good approximation by splines with variable knots II. Lecture notes in Math., vol.363, Springer 1973Google Scholar
  6. 6.
    Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19, 557–593 (1965).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application in multiple shooting, Numer. Math. 22, 289–315 (1974).MathSciNetMATHGoogle Scholar
  8. 8.
    Deuflhard, P.: A relaxation strategy for the modified Newton method, Springer Lecture Notes, Optimization and Optimal Control (ed. Bulirsch, Oettli, Stoer) 477, 59–73 (1975).CrossRefGoogle Scholar
  9. 9.
    Dickmanns, E.D., Well, K.H.: Approximate solution of optimal control problems using third order Hermite polynomial functions, Lecture Notes in Comp. Science 27, 158–166 (1975).Google Scholar
  10. 10.
    Flaherty, J.E., Mathon, W.: Collocation with polynomial and tension splines for singularly-perturbed boundary value problems, SIAM J. Sci. Stat. Comput. 1, 260–289 (1980).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Flaherty, J.E., O’Malley, R.E., Jr.: The numerical solution of boundary value problems for stiff differential equations, Math. Comp. 31, 66–93 (1977).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Howes, F.A.: Some old and new results on singularly perturbed boundary value problems, Singular Perturbations and Asymptotics (ed. Meyer, R.E., Parter, S.V.) Academic Press (1980).Google Scholar
  13. 13.
    Kreiss, H.O.: Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 15, 21–58 (1978).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Maier, M.R., Smith, D.R.: Numerical solution of a symmetric diode model, J. Comput. Phys. 42, 309–326 (1981).MATHCrossRefGoogle Scholar
  15. 15.
    Miranker, W.L.: Numerical methods of boundary layer type for stiff systems of differential equations, Computing 11, 221–234 (1973).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    O’Malley, R.E., Jr.: Introduction to Singular Perturbations, Academic Press (1974).Google Scholar
  17. 17.
    Pearson, C.E.: On a differential equation of boundary layer type, J. Math. Phys. 47, 134–154 (1968).MATHGoogle Scholar
  18. 18.
    Pearson, C.E.: On nonlinear differential equations of boundary layer type, J. Math. Phys. 47, 351–358 (1968).MATHGoogle Scholar
  19. 19.
    Rentrop, P.: An algorithm for the computation of the exponential spline, Numer. Math. 35, 81–93 (1980).MathSciNetMATHGoogle Scholar
  20. 20.
    Rieger, J.: Eindimensionale numerische Berechnung von physikalischen Grundgrößen einer pn-Diode mit beliebigem Dotierungsprofil. Technische Universität München, Institut für Technische Elektronik, Diplomarbeit 1981.Google Scholar
  21. 21.
    Russell, R.D.: Collocation for systems of boundary value problems, Numer. Math. 23, 119–133 (1974).MATHGoogle Scholar
  22. 22.
    Russell, R.D., Christiansen, J.: Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15 (1), 59–80 (1978).MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sigfridsson, B., Lindström, J.L.: Electrical properties of electron-irradiated n-type silicon, J. Appl. Phys. 47, 4611–4620 (1976).CrossRefGoogle Scholar
  24. 24.
    Smith, D.R.: On a singularly perturbed boundary value problem arising in the physical theory of semiconductors, Technische Universität München, Institut für Mathematik, TUM-M8021 (1980).Google Scholar
  25. 25.
    Stoer,J., Bulirsch, R.: Introduction to Numerical Analysis, Springer (1980).Google Scholar
  26. 26.
    Vasil’eva, A.B., Stel’makh, V.G.: Singularly disturbed systems of the theory of semiconductor devices, USSR Comp. Math. and Phys. 17, 48–58 (1977).Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1985

Authors and Affiliations

  • Maximilian R. Maier

There are no affiliations available

Personalised recommendations