Synopsis for Differential Equations

Overview

The aim of this course is to introduce all students reading mathematics to the basic theory of ordinary and partial differential equations.

The course will be example-led and will concentrate on equations that arise in practice rather than those constructed to illustrate a mathematical theory. The emphasis will be on solving equations and understanding the possible behaviours of solutions, and the analysis will be developed as a means to this end. The course will furnish undergraduates with the necessary skills to pursue any of the applied options in the third year and will also form the foundation for a deeper and more rigorous course in partial differential equations.

Learning Outcomes

On completion of the course, students will have acquired a sound knowledge of a range of techniques for solving linear ordinary and partial differential equations. They will have gained an appreciation of the importance of existence and uniqueness of solution and will be aware that explicit analytic solutions are the exception rather than the rule.

Synopsis

(1–4) Picard's Theorem for with proof. Extension to systems stated but not proved. Examples with blow-up and non-uniqueness.

(Collins section 2.1. Boyce & DiPrima section 2.12. Kreyszig section 1.9.)
(5–7) Second-order ODEs: variation of parameters, Wronskian and Green's function.
(Collins chapters 3, 4. Boyce & DiPrima sections 3.1–3.5, 3.6, 3.6.2. Hildebrand chapter 3. Kreyszig sections 2.1, 2.7–2.10.)

(8–11) Phase planes, critical points, Poincaré-Bendixson criterion. Examples including conservative nonlinear oscillators, Van der Pol's equation and Lotka-Volterra equations. Stability of periodic solutions.
(Collins chapters 3, 4. Boyce & DiPrima sections 9.1–9.4. Kreyszig sections 3.3–3.5.)

(12–14) Characteristic methods for first-order quasilinear PDEs (using parameterisation). Examples from conservation laws. Multivalued solutions and shocks. (Charpit's method and artificial examples excluded.)

(Collins Chapter 5. Carrier & Pearson Chapters 6, 13. Ockendon et al. Chapter 1.)

(15–18) Classification of second-order linear PDEs. Ideas of uniqueness and well-posedness for Laplace, Wave and Heat equations. Revision of separation of variables from Mods and illustration of suitable boundary conditions by example. Multi-dimensional Laplacian operator giving rise to Bessel's and Legendre's equations.

(Collins chapters 6, 7. Carrier & Pearson Chapters 1, 3, 4, 5, 7. Strauss Chapter 1. Kreyszig sections 11.7–11.11.)

(19–24) Theory of Fourier and Laplace transforms, inversion, convolution. Inversion of some standard Fourier and Laplace transforms via contour integration. Use of Fourier transform in solving Laplace's equation and the Heat equation. Use of Laplace transform in solving the Heat equation.

(Collins chapter 14. Carrier & Pearson chapters 2, 15. Kreyszig chapter 5, sections 10.8–10.11, 11.6. Priestley chapter 9.)

The best single text is:

1. P. J. Collins, Differential and Integral Equations (O.U.P., 2006), Chapters 1-7, 14,15.

Alternatives

1. W. E. Boyce & R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (7th edition, Wiley, 2000).
2. Erwin Kreyszig, Advanced Engineering Mathematics (8th Edition, Wiley, 1999).
3. F. B. Hildebrand, Methods of Applied Mathematics (Dover, 1992).
4. W. A. Strauss, Partial Differential Equations: an Introduction (Wiley, 1992).
5. G. F. Carrier & C E Pearson, Partial Differential Equations — Theory and Technique (Academic, 1988).
6. H. A. Priestley, Introduction to Complex Analysis (Second edition, Oxford, 2003).
7. J. Ockendon, S. Howison, A. Lacey & A. Movchan, Applied Partial Differential Equations (Oxford, 1999). [More advanced.]