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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Main Author:	V., Malathi
Format:	Thesis
Language:	English English
Published:	1999
Subjects:	Differential equations Boundary value problems - Numerical solutions
Online Access:	http://psasir.upm.edu.my/id/eprint/9505/1/FSAS_1999_44_A.pdf
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spelling	my-upm-ir.95052013-09-26T01:00:33Z Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques 1999-10 V., Malathi 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. Differential equations Boundary value problems - Numerical solutions 1999-10 Thesis http://psasir.upm.edu.my/id/eprint/9505/ http://psasir.upm.edu.my/id/eprint/9505/1/FSAS_1999_44_A.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Differential equations Boundary value problems - Numerical solutions Faculty of Science and Environmental Studies English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Differential equations Boundary value problems - Numerical solutions
spellingShingle	Differential equations Boundary value problems - Numerical solutions V., Malathi Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
description	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.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	V., Malathi
author_facet	V., Malathi
author_sort	V., Malathi
title	Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
title_short	Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
title_full	Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
title_fullStr	Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
title_full_unstemmed	Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques
title_sort	solving boundary value problems for ordinary differential equations using direct integration and shooting techniques
granting_institution	Universiti Putra Malaysia
granting_department	Faculty of Science and Environmental Studies
publishDate	1999
url	http://psasir.upm.edu.my/id/eprint/9505/1/FSAS_1999_44_A.pdf
_version_	1747810968571740160

Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques

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