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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
Summary:
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.
References:
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[2] N. J. Higham: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 1996. MR 1368629 | Zbl 0847.65010
[3] G. H. Meyer: Initial Value Methods for Boundary Value Problems. Theory and Applications of Invariant Imbedding. Academic Press, New York, 1973. MR 0488791
[4] W. T. Reid: Riccati Differential Equations. Academic Press, New York, 1972. MR 0357936 | Zbl 0254.34003
[5] M. R. Scott: Invariant Imbedding and its Applications to Ordinary Differential Equations. An Introduction. Addison-Wesley Publishing Company, , 1973. MR 0351102 | Zbl 0271.34001
[6] J. Taufer: Lösung der Randwertprobleme für Systeme von linearen Differentialgleichungen. Rozpravy Československé akademie věd, Řada Mat. přírod. věd 83 (1973). Zbl 0276.34009
[7] J. Taufer: Solution of Boundary Value Problems for Systems of Linear Differential Equations. Nauka, Moscow, 1981. (Russian) MR 0635932 | Zbl 0516.34002
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