Numerical solution of nonlinear boundary value problems for ordinary differential equations in the continuous framework

Thesis

Ordinary differential equations (ODEs) play an important role in mathematics. Although intrinsically, the setting for describing ODEs is the continuous framework, where differential operators are considered as maps from one function space to another, common numerical algorithms for ODEs discretise problems early on in the solution process. This thesis is about continuous analogues of such discrete algorithms for the numerical solution of ODEs.

This thesis shows how Newton's method for finite dimensional system can be generalised to function spaces, where it is known as Newton-Kantorovich iteration. It presents affine invariant damping strategies for increasing the chance of convergence for the Newton-Kantorovich iteration.

The derivatives required in this continuous setting are Fréchet derivatives, the continuous analogue of Jacobian matrices. In this work, we present how automatic differentiation techniques can be applied to compute Fréchet derivatives. We introduce chebop, a Matlab solver for nonlinear boundary-value problems, which combines damped Newton iteration in function space and automatic Fréchet differentiation.

By proving that affine operators have constant Fréchet derivatives, it is demonstrated how automatic linearity detection of computed quantities can be implemented. This is valuable for black-box solvers, which can use the information to determine whether an iteration scheme has to be employed for solving a problem.

Like nonlinear systems of equations, nonlinear boundary-value problems can have multiple solutions. This thesis present two techniques for obtaining multiple solutions of operator equations: deflation and path-following. An algorithm combining the two techniques is proposed.

Authors: Birkisson, A

• Publication date: 2013
• DOI: 10.5287/ora-rv2qmjkyq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: Chebfun; automatic linearity detection; deflation; Frechet derivatives; automatic differentiation; nonlinear boundary value problems
• Subjects: Ordinary differential equations; Mathematics; Numerical analysis