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)


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


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

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© Birkhäuser Boston, Inc. 1985

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  • Maximilian R. Maier

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