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## MA40050: Nonlinear equations & bifurcations

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment: CW 25%, EX 75%
Supplementary Assessment: MA40050 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA20218 and take MA20222
Description: Aims & Learning Objectives:
Aims:
To extend the real analysis of implicitly defined functions into the numerical analysis of iterative methods for computing such functions and to teach an awareness of practical issues involved in applying such methods.
Objectives: The students should be able to solve a variety of nonlinear equations in many variables and should be able to assess the performance of their solution methods using appropriate mathematical analysis.

Content:
Topics will be chosen from the following: Solution methods for nonlinear equations: Newtons method for systems. Quasi-Newton Methods. Eigenvalue problems. Theoretical Tools: Local Convergence of Newton's Method. Implicit Function Theorem. Bifurcation from the trivial solution. Applications: Exothermic reaction and buckling problems. Continuous and discrete models. Analysis of parameter-dependent two-point boundary value problems using the shooting method. Practical use of the shooting method. The Lyapunov-Schmidt Reduction. Application to analysis of discretised boundary value problems. Computation of solution paths for systems of nonlinear algebraic equations. Pseudo-arclength continuation. Homotopy methods. Computation of turning points. Bordered systems and their solution. Exploitation of symmetry. Hopf bifurcation. Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods.
Programme availability:

#### MA40050 is Optional on the following programmes:

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
• TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time) - Year 1
• TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time) - Year 1

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.