ENGINEERING TRIPOS PART IIB – 2012/2013

Module 4M12 - Partial Differential Equations and Variational Methods

AIMS

Partial differential equations (PDEs) occur widely in all branches of engineering science, and this course provides an introduction to the various classes of PDE and the physical nature of their solution. The second part of the course demonstrates how variational calculus can be used to derive both ordinary and partial differential equations, and also how the technique can be used to obtain approximate solutions to these equations. The final section on the summation convention provides a powerful mathematical tool for the manipulation of equations that arise in engineering analysis.

LECTURE SYLLABUS

Suffix notation and the summation convention (2L Dr G Wells)

• Index notation for scalar, vector, and matrix products, and for grad, div and curl. Applications including Stokes’ theorem and the divergence theorem.

Variational methods in engineering analysis (6L Dr G Wells)

• Introduction to variational calculus. Functionals and their first variation. Derivation of differential equations and boundary conditions from variational principles. The Euler-Lagrange equations. The effect of constraints. Applications in mechanics, optics, stress analysis, and optimal control.

Partial Differential Equations (8L Prof. P. A. Davidson)

• What is a PDE? Classification of PDEs: elliptic/parabolic/hyperbolic types. Canonical examples of each type: Laplace/diffusion/wave equations. solving the diffusion equation. Solving the wave equation. Solving the Laplace equation.

OBJECTIVES

On completion of the module students should:

• Understand the various types of PDE and the physical nature of their solutions;
• Understand various solution methods for PDEs and be able to apply these to a range of problems;
• Understand the formulation of various physical problems in terms of variational statements;
• Be able to estimate solutions using trial functions and direct minimisation;
• Be able to calculate an Euler-Lagrange differential equation from a variational statement, and to find the corresponding natural boundary conditions;
• Be able to perform vector manipulations using suffix notation.

REFERENCES

Please see the Booklist for Group R Courses for references for this module.

Last updated: September2012