NOT OFFERED IN 2020

## Paper Description

Solution methods for boundary-value problems that frequently arise in mathematical physics. Analytic solutions use Green's functions and distribution theory. Computed solutions use the boundary element method.

Students gain a familiarity with the common boundary value problems that arise as mathematical descriptions of physical systems, including heat, wave, and potential models. The paper introduces generic analytic and computational tools for understanding and solving these problems, based on Green's function methods and the boundary-element method. By understanding the theoretical basis of these methods, students are able to understand the properties of solutions, and to apply these methods to new problems.

**Prerequisites:**

None

This paper consists of 15 lectures and 6 tutorials. There are 3 assignments.

**Assesment:**

Final Exam 70%, Assignments 30%

Important information about assessment for ELEC443

**Course Coordinator:**

Dr Daniel Schumayer

After completing this paper students are expected to have achieved the following major learning objectives:

- Perform simple manipulations using distributional calculus, in particular find distributional solutions to some simple differential equations
- Know the defining properties of classical, weak, and distributional solutions to differential equations
- Know the defining properties of fundamental solutions and Green's functions
- Find Green's functions for linear ordinary differential boundary value problems, including initial-value problems
- Find the adjoint operator and adjoint boundary value problem associated with a linear partial differential boundary value problem
- Derive Green's theorem for arbitrary second order linear partial differential equations with (classical) boundary conditions
- Use fundamental solutions or Green's functions within Green's theorem to write an integral solution to a linear boundary value problem
- Derive the boundary integral equation for second-order elliptic boundary value problems
- State the steps required for a rudimentary boundary element method
- Code up a boundary element method in MatLab or python that solves an elliptic problem

**Additional outcomes:**

An overall goal is to provide each student with confidence in their ability to recognize and solve boundary value problems in partial differential equations that they are likely to meet in their future studies.

**Topics:**

- Transversely loaded string
- Concept of weak solution
- Distribution theory: definition
- Distribution theory: algebraic operations
- Analytic operations on distributions
- Fundamental solution, weak solutions, classification of PDEs
- n-th order ODE ,Matrix inversion analogy
- Boundary value problems: notation
- Adjoint BC, bilinear concommitant
- Remarkable result, Green's theorem, integral solutions
- Boundary integral equation for negative Laplacian
- Boundary element method
- Green's functions in canonical examples
- Solutions in canonical examples

# Formal University Information

The following information is from the University’s corporate web site.

## Details

Solution methods for boundary-value problems that frequently arise in mathematical physics. Analytic solutions using Green's functions and distribution theory. Computed solutions using the boundary element method.

Paper title | Boundary Value Problems of Mathematical Physics |
---|---|

Paper code | ELEC443 |

Subject | Electronics |

EFTS | 0.0833 |

Points | 10 points |

Teaching period | Not offered in 2021 (On campus) |

Domestic Tuition Fees (NZD) | $673.90 |

International Tuition Fees (NZD) | $2,981.97 |

- Limited to
- BSc(Hons), PGDipSci, MSc, MAppSc
- Contact
Associate Professor Colin Fox (colin.fox@otago.ac.nz)

- Teaching staff
To be advised when next offered

- Textbooks
- Textbooks are not required for this paper.
- Graduate Attributes Emphasised
- Global perspective, Interdisciplinary perspective, Lifelong learning, Scholarship,
Communication, Critical thinking, Information literacy, Self-motivation, Teamwork.

View more information about Otago's graduate attributes. - Learning Outcomes
- By the end of the module students are expected to be able to:
- Perform simple manipulations using distributional calculus, in particular find distributional solutions to some simple differential equations
- Know the defining properties of classical, weak and distributional solutions to differential equations
- Know the defining properties of fundamental solutions and Green's functions
- Find Green's functions for linear ordinary differential boundary value problems, including initial-value problems
- Find the adjoint operator and adjoint boundary value problem associated with a linear partial differential boundary value problem
- Derive Green's theorem for arbitrary second-order linear partial differential equations with (classical) boundary conditions
- Use fundamental solutions or Green's functions within Green's theorem to write an integral solution to a linear boundary value problem
- Derive the boundary integral equation for second-order elliptic boundary value problems
- State the steps required for a rudimentary boundary element method
- Code up a boundary element method in MatLab or Python that solves an elliptic problem