This course focuses on advanced mathematical techniques useful in addressing problems in various fields of application. For an efficient use of these techniques, the underlying theoretical principles for analytical and numerical analyses and problem solving need to be formulated and understood. The course is accessible to students interested in solving differential equations and manipulating integrals. Illustrative examples are presented in a self-contained manner and a specific background in a particular area of application is not mandatory.

Topic 1. Asymptotic methods

This topic addresses asymptotic and perturbation methods together with applications. These methods are important when a small or large parameter can be identified and utilised to approximate a problem and its solution.

Topic 2. Integral equations

Many physical problems are cast as integral equations where boundary conditions are incorporated within the mathematical formulation. This confers certain advantages to their solution approach in terms of infinite series. In this module, different types of integral equations and methods for their solution derivation are discussed.

Topic 3. Calculus of variations

Calculus of variations provides valuable techniques for investigating maxima and minima of integral functionals representing some physically or geometrically meaningful quantity. This module considers the classical approach to these problems.

Topic 4. Numerical methods

Within this topic, numerical integration methods for solving ordinary differential equations (ODEs) are presented and applied.

Topic 5. Elements of learning

Neural Networks (NN) is the newest most important tool of applied mathematics which can be used to approximate a function. In this course, the mathematical foundation of this methodology is introduced.

Pre-requisite modules: MA1001, MA1005, MA1006, MA2008

• Demonstrate foundational knowledge and understanding of fundamental analytical and computational techniques, and their applications and limitations.
• Construct perturbation and asymptotic expansions to solutions of linear and nonlinear boundary value problems.
• Analyse differential equations formulated as equivalent integral equations.
• Derive and solve equations for minimisation problems using calculus of variations.
• Solve numerically a range of boundary value problems formulated as ODEs.
• Formulate the optimisation problem for the approximation of a function with NN.

Students will be guided through learning activities appropriate to the module, including:

• Weekly face to face classes (lectures, exercise classes)
• Electronic resources to support the learning (e.g. exercise sheets, lecture notes)

Every student is expected to undertake at least 100 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

Skills that will be practised and developed:

• Analytical and numerical skills associated with the analysis and approximation of functions, differential and integral equations and variational problems.

Transferable skills:

• Ability to relate mathematical theory to explicit computations.
• Clear and precise formulation of mathematical reasoning.
• Logical thinking.

• Formative assessment is carried out by means of optional coursework exercises. Feedback to students on their solutions and their progress towards the learning outcomes is provided in lectures.
• The summative assessment is a 3-hour written examination (weighing 100%) at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above-average marks.

Syllabus content:

1. Asymptotic methods
   1. Introduction to perturbation theory
   2. Asymptotic series and approximations
   3. Boundary layers and matched asymptotic expansions
   4. Exercises
2. Integral equations
   1. Relations between differential and integral equations
   2. Neumann series
   3. Singular integral equations
   4. Exercises
3. Calculus of variations
   1. Maxima and minima
   2. Constraints and Lagrange multipliers
   3. The Euler-Lagrange equation
   4. Exercises
4. Numerical integration
   1. The Euler method
   2. Leapfrog methods
   3. Runge-Kutta methods
   4. Exercises
5. Elements of deep learning
   1. Function approximation with neural networks
   2. Automatic differentiation
   3. Exercises