APPROXIMATION OF PARAMETER-DEPENDENT TWO-POINT BOUNDARY VALUE PROBLEMS BY COLLOCATION

Marwan  A.  TANNIR

In this thesis we consider the approximation of parameter‑dependent two‑point Boundary Value Problems (BVP's) by collocation. The collocation method is a very powerful numerical scheme for solving two‑point BVP's, known for its superconvergence at the discretization points.

In Part I of the thesis we study the convergence of approximate solution branches obtained by collocation to exact solution branches of the parameter‑dependent two‑point BVP. For a non‑singular solution branch we obtain the convergence results by a simple extension to standard theory. However, we concentrate on solution branches through certain singular points, namely turning point, bifurcation point and symmetry‑breaking bifurcation point, proving that we still have superconvergence for such singular solutions.

In Part II of the thesis we consider the computation of periodic solution branches. A periodic solution branch is in fact a solution of a special type of parameter‑dependent two‑point BVP and hence the convergence results of collocation approximation to such solution branches are the same as in Part I. However, in this part a new method which uses arclength parameterization for periodic solutions is considered. We show how this new method can be implemented in the existing continuation and bifurcation packages.

Finally, we consider the effects of using this new method in computing periodic solution branches which approach homoclinic orbits or heteroclinic cycles.