Student Records
Programme & Unit Catalogues

## MA40203: Theory of partial differential equations

Academic Year: 2012/3
Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: MA40203 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA20218 and take MA20219 and take MA30041 and take MA40043
Description: Aims:
To introduce students to modern PDE theory through the study of qualitative properties of solutions, principally those that derive from Maximum Principles. The treatment will be rigorous and will lead to specific nonlinear examples.

Learning Outcomes:
Students should be able to state definitions, and state and prove theorems, in the analysis of partial differential equations. They should be able to apply maximum principles to questions of existence, uniqueness, symmetry and boundedness/blow-up for solutions of PDE.

Skills:
Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).

Content:
Weak maximum principles for twice continuously differentiable solutions of linear elliptic PDE; interior ball property, Hopf boundary point lemma, strong maximum principles. Applications might include; uniqueness of solutions of linear Poisson equations; symmetry of non-negative solutions of nonlinear Poisson equation in a ball; Perron approach to existence. Maximum principles for parabolic equations; comparison theorem for nonlinear case. Applications might include: bounds for diffusive Burgers' equation via upper solutions; proof of blow-up for nonlinear diffusion equations via lower solutions; connection with similarity solutions; systems of reaction-diffusion equations.
Programme availability:

#### MA40203 is Optional on the following programmes:

Department of Computer Science
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 4
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 5
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 5
Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time) - Year 1
• TSMA-APM09 : MSc Mathematical Sciences (Part-time) - Year 1
• TSMA-APM09 : MSc Mathematical Sciences (Part-time) - Year 2
Department of Physics
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 4

Notes:
* This unit catalogue is applicable for the 2012/13 academic year only. Students continuing their studies into 2013/14 and beyond should not assume that this unit will be available in future years in the format displayed here for 2012/13.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.