Numerical solution of boundary value problems for selfadjoint differential equations of 2 n th order

Jiří Taufer

Applications of Mathematics (2004)

  • Volume: 49, Issue: 2, page 141-164
  • ISSN: 0862-7940

Abstract

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The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2 n th order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.

How to cite

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Taufer, Jiří. "Numerical solution of boundary value problems for selfadjoint differential equations of $2n$th order." Applications of Mathematics 49.2 (2004): 141-164. <http://eudml.org/doc/33180>.

@article{Taufer2004,
abstract = {The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of $2n$th order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.},
author = {Taufer, Jiří},
journal = {Applications of Mathematics},
keywords = {ODE; two-point boundary value problem; transfer of boundary conditions; self-adjoint differential equation; numerical solution; Riccati differential equation; ODE; two-point boundary value problem; transfer of boundary conditions; self-adjoint differential equation; numerical solution; Riccati differential equation},
language = {eng},
number = {2},
pages = {141-164},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Numerical solution of boundary value problems for selfadjoint differential equations of $2n$th order},
url = {http://eudml.org/doc/33180},
volume = {49},
year = {2004},
}

TY - JOUR
AU - Taufer, Jiří
TI - Numerical solution of boundary value problems for selfadjoint differential equations of $2n$th order
JO - Applications of Mathematics
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 2
SP - 141
EP - 164
AB - The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of $2n$th order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.
LA - eng
KW - ODE; two-point boundary value problem; transfer of boundary conditions; self-adjoint differential equation; numerical solution; Riccati differential equation; ODE; two-point boundary value problem; transfer of boundary conditions; self-adjoint differential equation; numerical solution; Riccati differential equation
UR - http://eudml.org/doc/33180
ER -

References

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  1. Theory of Linear Operators in Hilbert Space, Pitman, Boston, 1981. (1981) 
  2. Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996. (1996) Zbl0847.65010MR1368629
  3. Initial Value Methods for Boundary Value Problems. Theory and Applications of Invariant Imbedding, Academic Press, New York, 1973. (1973) MR0488791
  4. Riccati Differential Equations, Academic Press, New York, 1972. (1972) Zbl0254.34003MR0357936
  5. Invariant Imbedding and its Applications to Ordinary Differential Equations. An Introduction, Addison-Wesley Publishing Company, , 1973. (1973) Zbl0271.34001MR0351102
  6. Lösung der Randwertprobleme für Systeme von linearen Differentialgleichungen, Rozpravy Československé akademie věd, Řada Mat. přírod. věd 83 (1973). (1973) Zbl0276.34009
  7. Solution of Boundary Value Problems for Systems of Linear Differential Equations, Nauka, Moscow, 1981. (Russian) (1981) Zbl0516.34002MR0635932

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