Module MATH4221: Approximation Theory and Solutions to ODEs IV

Department: Mathematical Sciences

MATH4221: Approximation Theory and Solutions to ODEs IV

Prerequisites

• Mathematics modules to the value of 100 credits at Years 2 and 3, with at least 40 credits at Level 3, and including Numerical Analysis II (MATH2051).

Corequisites

• None.

Excluded Combination of Modules

• Approximation Theory & Solutions to ODEs III (MATH3081)

Aims

• To build on the foundations laid in the level II Numerical Analysis module and to enable students to gain a deeper knowledge and understanding of two particular areas of numerical analysis.

Content

• Approximation theory: Piecewise polynomial approximation and spline functions.
• Approximation by rational functions.
• Trigonometric polynomials and fast Fourier transforms.
• Numerical solution of ordinary differential equations: Introduction to numerical methods for initial-value problems.
• Local and global truncation errors, convergence.
• One-step methods, with emphasis on explicit Runge-Kutta methods.
• Practical algorithms.
• Linear multistep methods.
• Predictor - corrector methods.
• Shooting methods.
• Specialist software will be used for computational work throughout the module.
• Reading material on one or more aspects of implicit Runge-Kutta methods.

Learning Outcomes

• By the end of the module students will: be able to solve novel and/or complex problems in Approximation Theory and Solution of Ordinary Differential Equations.
• have a systematic and coherent understanding of theoretical mathematics in the field of Approximation Theory and Solution of Ordinary Differential Equations.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Consistency, convergence and linear stability of numerical methods.
• One-step methods, particularly Runge-Kutta methods.
• Linear multi-step methods.
• Approximation theory.
• Piecewise polynomial approximation and spline functions.
• Approximation by rational functions.
• Trigonometric polynomials and fast Fourier transforms.
• Knowledge and understanding of a topic in the areas listed under content, for implicit Runge-Kutta methods.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, Computation.
• Ability to read independently to acquire knowledge and understanding in the area of implicit Runge-Kutta methods.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University