Academic Year:
 2013/4 
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/blowup 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 nonnegative 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 blowup for nonlinear diffusion equations via lower solutions; connection with similarity solutions; systems of reactiondiffusion equations.

Programme availability: 
MA40203 is Optional on the following programmes:
Department of Computer Science
 USCMAFM14 : MComp (hons) Computer Science and Mathematics (Fulltime)  Year 4
 USCMAKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Fulltime with Thick Sandwich Placement)  Year 5
 USCMAAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 5
Department of Mathematical Sciences
 USMAAFB15 : BSc (hons) Mathematical Sciences (Fulltime)  Year 3
 USMAAKB16 : BSc (hons) Mathematical Sciences (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFB13 : BSc (hons) Mathematics (Fulltime)  Year 3
 USMAAKB14 : BSc (hons) Mathematics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAFB01 : BSc (hons) Mathematics and Statistics (Fulltime)  Year 3
 USMAAKB02 : BSc (hons) Mathematics and Statistics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAAB14 : BSc (hons) Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFB05 : BSc (hons) Statistics (Fulltime)  Year 3
 USMAAKB06 : BSc (hons) Statistics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB06 : BSc (hons) Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFM14 : MMath Mathematics (Fulltime)  Year 3
 USMAAFM14 : MMath Mathematics (Fulltime)  Year 4
 USMAAAM15 : MMath Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 TSMAAFM09 : MSc Mathematical Sciences (Fulltime)
 TSMAAPM09 : MSc Mathematical Sciences (Parttime)
Department of Physics
 USXXAFM01 : MSci (hons) Mathematics and Physics (Fulltime)  Year 4
