Module MATH4041: PARTIAL DIFFERENTIAL EQUATIONS IV

Department: Mathematical Sciences

MATH4041: PARTIAL DIFFERENTIAL EQUATIONS IV

Prerequisites

• Analysis in Many Variables II (MATH2031) AND other Mathematics modules to the value of 80 credits in Years 2 and 3, with at least 40 credits at Level 3

Corequisites

• None.

Excluded Combination of Modules

• Partial Differential Equations III (MATH3291)

Aims

• To develop a basic understanding of the theory and methods of solution for Partial Differential Equations.
• To develop a basic understanding of the ideas of approximate (numerical) solution to certain Partial Differential Equations.

Content

• First order equations and characteristics. Conservation laws.
• Systems of first-order equations, conservation laws and Riemann invariants.
• Hyperbolic systems and discontinuous derivatives. Acceleration waves.
• Classification of general second order quasi-linear equations and reduction to standard form for each type (elliptic, parabolic and hyperbolic).
• Energy methods for parabolic equations. Well-posed problems.
• Maximum principles for parabolic equations.
• Finite difference solution to parabolic and elliptic equations.
• Stability and convergence for solution to finite difference equations.
• Iterative methods of solving Ax=b.
• Reading material on one of the following topics: Strict form of the maximum principle for parabolic equations; Error estimates for numerical solutions of partial differential equations; Further aspects of non-linear partial differential equations.

Learning Outcomes

• By the end of the module students will: be able to solve problems in Partial Differential Equations.
• have an understanding of theoretical mathematics in the field of Partial Differential Equations.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Solution of hyperbolic equations and systems.
• Classification of second order PDEs, and their solution.
• Maximum principles for a parabolic equation.
• Energy estimates for a parabolic equation.
• Finite difference methods for PDEs.
• have an appreciation of the techniques used in one of the following areas: maximum principle for parabolic equations, error estimates for numerical solutions of partial differential equations.
• have an advanced understanding in one of the following areas: Maximum principles for parabolic equations; Error estimates; Non-linear partial differential equations.

• Students will have highly specialised and advanced mathematical skills in the following areas: Modelling, Numerical Mathematics.

• Students will have an appreciation of Partial Differential Equations in the real world and how to solve them.
• Students will be able to study independently to further their knowledge of an advanced topic.

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