Integral Equations - 2024 entry

Following the introduction integral equations, you will be introduced to a large class of integral equations. 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.

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.

Web based and electronic resources:

ELE

Reading list for this module: