The purpose of this course is to learn a variety of mathematical methods for deriving useful approximate solutions of the differential equations and integrals found in the Mathematical Sciences. The course will be structured as follows:

1. Existence and uniqueness results for ordinary differential equations: The Lipschitz condition and Picard’s theorem. Comparison theorems.
2. Integral Equations: The Volterra integral equation and initial value problems, the Fredholm integral equation and boundary value problems.
3. Sturm-Liouville Theory: The adjoint differential operator, the Sturm-Liouville problem, basic properties of a Sturm-Liouville eigenvalue problem, unboundedness of the eigenvalues, completeness in the appropriate sense of the set of eigenfunctions
4. Theory of Infinite-dimensional vector spaces: Inner product spaces, complete metric spaces, Hilbert spaces, square summable series and square integrable functions, Least squares approximation, projection theorem, generalized Fourier coefficients, Bessel’s inequality, Parseval’s equality and completeness

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to:

1. Understand conditions guaranteeing existence and uniqueness results for ordinary differential equations and recognize examples where those conditions do not hold;
2. State and prove Picard’s theorem;
3. Transform between an initial value problem and the corresponding Volterra integral equation;
4. Transform between a boundary value problem and the corresponding Fredholm integral equation;
5. State the axiomatic properties of the Green function for a second order initial value problem and boundary value problem;
6. Understand the concept of the adjoint differential operator;
7. Recognise a Sturm-Liouville eigenvalue problem and prove the basic properties of eigenvalues and eigenfunctions;
8. Understand the fundamental properties of infinite dimensional vectors spaces.
9. Prove key results such as Bessel’s inequality, Parseval’s equality and its relationship to completeness

Student Effort Hours:
Student Effort Type Hours
Lectures

36

Specified Learning Activities

24

Autonomous Student Learning

40

Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Assignment: Take-home assignments Throughout the Trimester n/a Standard conversion grade scale 40% No

20

No
Examination: 2 hour end of trimester exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

No

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour