An Extended Precise Integration Method for Solving Inhomogeneous Two-Point Boundary Value Problems of Linear Time-Invariant Systems

TAN Shu-jun; ZHOU Wen-ya; WU Zhi-gang

doi:10.3879/j.issn.1000-0887.2015.11.003

Volume 36 Issue 11

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TAN Shu-jun, ZHOU Wen-ya, WU Zhi-gang. An Extended Precise Integration Method for Solving Inhomogeneous Two-Point Boundary Value Problems of Linear Time-Invariant Systems[J]. Applied Mathematics and Mechanics, 2015, 36(11): 1145-1157. doi: 10.3879/j.issn.1000-0887.2015.11.003

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An Extended Precise Integration Method for Solving Inhomogeneous Two-Point Boundary Value Problems of Linear Time-Invariant Systems

doi: 10.3879/j.issn.1000-0887.2015.11.003

¹School of Aeronautics and Astronautics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China;²State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology), Dalian, Liaoning 116024, P.R.China

Funds: The National Natural Science Foundation of China（11002032;11372056;11432010）

Received Date: 2015-06-16
Rev Recd Date: 2015-10-08
Publish Date: 2015-11-15

Abstract

Abstract

An extended precise integration method (EPIM) for solving inhomogeneous two-point boundary value problems (TPBVPs) of linear time-invariant systems was proposed. Firstly, the interval quantities of the interval matrices and vectors were introduced to describe the discretization of the differential equations for the TPBVPs. Thus a general framework for solving the TPBVPs was established, where the interval quantities for different intervals were computed in parallel, and the assembled algebraic equations for global analysis were independent of the boundary conditions. Secondly the interval response matrices corresponding to the interval vectors were used to deal with the inhomogeneous terms. The addition theorems for the interval response matrices were derived with the inhomogeneous terms in the forms of polynomial function, sine/cosine function, exponential function and their combinations. Then the extended precise integration method was proposed in combination with the incremental storage technique, of which the accuracy approached the machine precision for the inhomogeneous terms in the above forms. The general forms of the inhomogeneous terms can be approximated with the mentioned forms. In comparison with the analytical methods, two numerical examples of stiff problems give results showing the high accuracy and stability of the proposed method.
- two-point boundary problem,
- inhomogeneous term,
- response matrix,
- precise integration method,
- addition theorem

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References

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