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.

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 | Component Scale | % of Final Grade | |||
---|---|---|---|---|---|---|

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

No

Resit In | Terminal Exam |
---|---|

Autumn | Yes - 2 Hour |

Feedback Strategy/Strategies

• Feedback individually to students, post-assessment

• Group/class feedback, post-assessment

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