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

Citation

V., Malathi (1999) Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques. PhD thesis, Universiti Putra Malaysia.

Abstract

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.