Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
In this thesis, an efficient algorithm and a code BVPDI is developed for solving Boundary Value Problems (BVPs) for Ordinary Differential Equations (ODEs). A generalised variable order variable stepsize Direct Integration (01) method, a generalised Backward Differentiation method (BDF) and shooti...
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| Format: | Thesis |
| Language: | English English |
| Published: |
1999
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/9505/1/FSAS_1999_44_A.pdf |
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| Summary: | In this thesis, an efficient algorithm and a code BVPDI is developed for
solving Boundary Value Problems (BVPs) for Ordinary Differential Equations
(ODEs). A generalised variable order variable stepsize Direct Integration (01)
method, a generalised Backward Differentiation method (BDF) and shooting
techniques are used to solve the given BVP. When using simple shooting
technique, sometimes stability difficulties arise when the differential operator of
the given ODE contains rapidly growing and decaying fundamental solution
modes. Then the initial value solution is very sensitive to small changes in the
initial condition. In order to decrease the bound of this error, the size of domains
over which the Initial Value Problems (IVPs) are integrated has to be restricted.
This leads to the multiple shooting technique, which is generalisation of the
simple shooting technique. Multiple shooting technique for higher order ODEs
with automatic partitioning is designed and successfully implemented in the
code BVPDI, to solve the underlying IVP. The well conditioning of a higher order BVP is shown to be related to
bounding quantities, one involving the boundary conditions and the other
involving the Green's function. It is also shown that the conditioning of the
multiple shooting matrix is related to the given BVP. The numerical results are
then compared with the only existing direct method code COLNEW. The
advantages in computational time and the accuracy of the computed solution,
especially, when the range of interval is large, are pointed out. Also the
advantages of BVPDI are clearer when the results are compared with the NAG
subroutine D02SAF (reduction method).
Stiffness tests for the system of first order ODEs and the techniques of
identifying the equations causing stiffness in a system a rediscussed. The
analysis is extended for the higher order ODEs. Numerical results are discussed
indicating the advantages of BVPDI code over COLNEW.
The success of the BVP DI code applied to the general class of BVPs is
the motivation to con sider the same code for a special class of second order
BVPs called Sturm-Liouville (SL) problems. By the application of Floquet theory
and shooting algorithm, eigenvalues of SL problems with periodic boundary
conditions are determined without reducing to the first order system of
equations. Some numerical examples are given to illustrate the success of the
method. The results are then compared, when the same problem is reduced to
the first order system of equations and the advantages are indicated. The code
BVPDI developed in this thesis clearly demonstrates the efficiency of using DI
Method and shooting techniques for solving higher order BVP for ODEs. |
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