Mathematics

MTH3042 - Integral Equations (2019)

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MODULE TITLEIntegral Equations CREDIT VALUE15
MODULE CODEMTH3042 MODULE CONVENERDr Layal Hakim (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
Number of Students Taking Module (anticipated) 150
DESCRIPTION - summary of the module content

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.

AIMS - intentions of the module

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.

 

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.

SYLLABUS PLAN - summary of the structure and academic content of the module

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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33.00 Guided Independent Study 117.00 Placement / Study Abroad
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
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.

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
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.

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
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

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
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

 

RE-ASSESSMENT NOTES

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

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
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
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 6 AVAILABLE AS DISTANCE LEARNING No
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.