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## MTH3042 - Integral Equations (2019)

MODULE TITLE | Integral Equations | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | MTH3042 | MODULE CONVENER | Dr Layal Hakim (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 | 0 |

Number of Students Taking Module (anticipated) | 150 |
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Similarly to differential equations, integral equations provide an effective way to model real life situations, particularly those that arise in physics and engineering. Using certain techniques, many initial and boundary value problems can be converted to integral equations where the unknown function lies in the integrand. This module will introduce students to the mathematics of integral equations, techniques of analysing such equations, and methods of solving them, analytically or numerically.

Differential Equations (MTH2003) and Analysis (MTH2001) are essential prerequisite for this.

Following the introduction of what integral equations, students will be introduced to a large class of integral equation. Volterra integral equations and Fredholm integral equations will be explained and methods on how to solve these equations will be described for cases of having integral equations of the first kind and the second kind, as well as looking at homogeneous and nonhomogeneous equations. Examples, that model real life problems, will be given and solutions will be interpreted.

On successful completion of this module ** you should be able to**:

**Module Specific Skills and Knowledge:**

1. Classify integral equations;

2. Define the Laplace transform and implement its use to solve integral equations;

3. Explicitly solve several classes of integral equations, both analytically and numerically;

4. Deploy the analysis of integral equations;

5. Illustrate the use of integral equations to model real life problems.

**Discipline Specific Skills and Knowledge:**

6. Analyse qualitative information about the solution;

7. Develop further the ability of problem structuring, problem solving, and logical thinking;

8. Assemble the necessary parts of a proof that form a final result by a chain of reasoning.

**Personal and Key Transferable / Employment Skills and Knowledge:**

9. Describe real life applications using integral equations through examples and exercises;

10. Build the ability to identify which techniques are suitable for which problems;

11. Communicate ideas effectively by learning the analysis of integral equations.

Classification of integral equations:

- Linear/nonlinear, Fredholm/Volterra, homogeneous/inhomogeneous, first/second kind.

Structure of kernels:

- convolution/non-convolution type, separable kernels, finite rank kernels, and weakly singular kernels.

Laplace transforms:

- An introduction to Laplace transforms with a focus on integral equations with a convolution type kernel;

- When to use, and how to obtain, the inverse Laplace transform.

iii. Linear integral equations:

- Conditions under which the solutions to first and second kind integral equations exist;

- Linear operators and The Fredholm Alternative;

- Methods of obtain the exact and numerical solutions to Fredholm and Volterra integral equations;

- Bounded linear integral operators and how to find the bound using norms of integral operators;

- Iterative techniques, particularly the Neumann iteration method, for second kind equations;

- Approximation techniques using Taylor series;

- Analysis of integral equations: criteria for convergence; existence and uniqueness of solutions; continuity of integral operators;

- The Fredholm Theorem and its proof, the Herbert-Shmidt Theorem for self-adjoint kernels and its proof;

- Error estimates in the numerical solution;

- Nonlinear integral equations: Methods of obtaining solutions to simple cases of the Fredholm integral equation of the Hammerstein type;

- Applications to where integral equations are used to model the behaviour of real life problems.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad |
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Category | Hours of study time | Description |

Lectures | 28 |
Definitions are stated and explained, theorems are stated and proved, examples are thoroughly carried out interactively, and results and techniques are discussed. |

Tutorials | 5 | Students can practice exercises, examples and ask questions in a supportive environment. |

Independent Study | 117 |
Independent reading and problem solving. |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Exercise sheets | 5 x 10 hours | All |
The lecturer will discuss problems during tutorials. Solutions to the sheets will be uploaded onto the VLE at some point after the tutorial. |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Coursework – based on questions submitted for assessment | 20 | 2 assignments, 30 hours total | All | Annotated script and written/verbal feedback |

Written Exam – closed book | 80 | 2 hours | All | Written/verbal on request, SRS |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
---|---|---|---|

Written May/June Exam | Written Exam (100%) | All | During the August Ref/Def Period |

For students taking Ref/Def assessments, only that exam will be used to calculate their overall mark for this module.

information that you are expected to consult. Further guidance will be provided by the Module Convener

**Basic reading:**

ELE – http://vle.exeter.ac.uk/

**Web based and electronic resources:**

**Other resources:**

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | RP Kanwal | Linear Integral Equations | 2nd ed. | Birkhauser, Boston | 1997 | [Library] | |

Set | AC Pipkin | A Course on Integral Equations | Springer-Verlag, New York | 1991 | [Library] | ||

Set | BJ Moiseiwitsch | Integral Equations | Longman, London | 1977 | [Library] | ||

Set | D Porter | Integral Equations. A practical treatment, from Spectral Theory to Applications | DSG, Stirling, CUP | 1990 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | MTH2001, MTH2003, MTH2005 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Wednesday 03 April 2019 | LAST REVISION DATE | Friday 30 August 2019 |

KEY WORDS SEARCH | Integral equations; Volterra integral equations; Fredholm integral equations; Laplace transforms. |
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