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Programme & Unit Catalogues

MA40203: Theory of partial differential equations

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment Summary: EX 100%
Assessment Detail:
• Examination (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 (Year 4)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
• TSMA-AFM09 : MSc Mathematical Sciences
• TSMA-APM09 : MSc Mathematical Sciences
• USMA-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
Department of Physics

Notes:
* This unit catalogue is applicable for the 2014/15 academic year only. Students continuing their studies into 2015/16 and beyond should not assume that this unit will be available in future years in the format displayed here for 2014/15.
* 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.