Malathi, V.

Suleiman, M. B.

Universiti Putra Malaysia Institutional Repository

UPM Research Report 1998: Volume 2DEVELOPING A CODE FOR AUTOMATICSOLUTION OF HIGHER ORDER BOUNDARY VALUE'PROBLEMS USING THE SHOOTING TECHNIQUEM.B. Suleiman and V; Malathi'Faculty of Science and Environmental StudiesUniversiti Putra Malaysia, 43400 UPM, Serdang, Selangor,MalaysiaKeywords: boundary value problems, ordinary differ-ential equations, shooting technique, direct integrationmethod, sturm-liouvillie eigen value problems.IntroductionSince the advent of computers, the numerical solution ofboundary value problems for ordinary differential equationshas been the subject of research. Boundary value problemsmanifest themselves in almost all branches of science, engi-neering and technology. In all the methods and codes devel-oped using initial value techniques, the higher order equa-tions are reduced to first order equations making it incompe-tent with other methods, which solve the equations directly.As there is a close relationship between boundary valueproblems and initial value problems, it is appropriate then, toconstruct a numerical method for a given boundary valueproblem by relating the problem to the corresponding initialvalue problems and solving the latter numerically. The codeBVPDI is designed to solve mixed order systems of linearand nonlinear boundary value problems using direct integra-tion method. This is in contrast to the other codes developedso far, which require conversion of the given problem to firstorder system, thereby increasing the number of equations andchanging the algebraic structure of the discretized problem.Numerical results are compared with the existing codeCOLNEW (Bader and Ascher, 1987), which solves thehigher order equations directly.Materials and MethodsA generalised variable order variable stepsize direct integra-tion method (Suleiman, 1989), a generalised backward dif-ferentiation method and shooting technique are used to solvethe given boundary value problem. When using simpleshooting technique, sometimes stability difficulties arisewhen the differential operator of the given ordinary differen-tial equation contains rapidly growing and decaying funda-mental solution modes. In order to decrease the bound of thiserror, the size of domains over which the initial value prob-lems are integrated has to be restricted. This leads to themultiple shooting technique, which is a generalisation of thesimple shooting technique. Multiple shooting technique forhigher order ordinary differential equations with automaticpartitioning is designed and successfully implemented in thecode BVPDI, to solve the underlying initial value problem.Stiffness tests for the system of first order ordinary differen-tial equations and the techniques of identifying the equationscausing stiffness in a system are discussed. The analysis isextended for the higher order ordinary differential equations.By the application of Ploquet theory (Eastham, 1973) andshooting algorithm, eigenvalues of Sturrn-Liouvillie prob-306lems with periodic boundary conditions are determined with-out reducing to the first order system of equations.Results and DiscussionThe well-conditioning .of a higher order boundary valueproblem is shown to be related to the bounds involving theboundary conditions and the green's function. It is shownhow the conditioning of the multiple shooting matrix is re-lated to the given boundary value problem. The conditioningconstants for the given boundary value problem are derivedfor higher order boundary value problem. It is also. shownthat small perturbations in the boundary conditions inducesmall perturbations in the solution.The multiple shooting technique is designed for higher orderordinary differential equations. A new strategy is developedfor the automatic partitioning of the interval and has beensuccessfully implemented in the code BVPDI. The functionalvalues at the shooting nodes are determined by using a poly-nomial interpolation. New integration coefficients are de-rived for this purpose. For stiff boundary value problems,separate tests are undertaken to isolate the stiff equations andthey are solved by using generalised BDF method. The stiff-ness test developed has been successful on the problemstested making it very efficient to solve higher order systemsof equations. The numerical results compared with the onlyexisting direct method code COLNEW, show that BVPDIperforms better in all the aspects than COLNEW. The ad-vantages in computational time and the accuracy of the com-puted solution when the range of interval is large is very sig-nificant. Also that BVPDI has no difficulty at all when theproblem is higher than the fourth order, whereas COLNEWhas to reduce to system of smaller order equations when thegiven problem is of order more than four.The success of the BVPDI code applied to the general classof boundary value problems is the motivation to consider thesame code for a special class of second order eigenvalueboundary value problems called Sturm-Liouville eigenvalueproblems. The shooting algorithm developed for this purposesolves the eigenvalue of the given problem very efficiently.Some numerical examples are given to illustrate the successof the method. The results are compared, when the sameproblem is reduced to the first order system of equations andthe advantages in storage and speed are more pronounced.ConclusionsThe results suggest that initial value methods when comparedwith other fmite difference methods, can perform better byusing the combination of multiple shooting and direct inte-gration procedures. Our new approach not previously re-ported is flexible and efficient to solve the boundary valueproblems for ordinary differential equations directly.ReferencesBader, G. and Ascher, U. 1987. A New Basis Implementation for aMixed Order Boundary ODE Solver. SlAM. J. Sci. Stat. Compo 8:483-500.Eastham, M.S.P. 1973. The Separated Theory of Periodic Differen-tial Equations. Scottish Academic Press, Edinburgh.Suleiman, M.B. 1989. Solving Nonstiff Higher Order ODEs Directlyby Direct Integration Method. Applied Mathematics and Com-putation. 33: 197-219.Supported by IRPA Grant 09-02-04-0055

Developing a code for automatic solution of higher order boundary value problems using the shooting technique

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Developing a code for automatic solution of higher order boundary value problems using the shooting technique

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