- Lecturer: Ian Griffiths
Course Term: Hilary
Course Lecture Information: 8 lectures
Course Level: M
Course Overview:
The principal aim of this course will be to introduce analytical techniques for solving partial differential equations (PDEs) that may be useful in the MMSC research project and for subsequent research. An emphasis will be placed on studying PDEs that emerge from real world situations and on acquiring skills to identify how such PDEs may be solved.
Course Synopsis:
1. Similarity solutions (~4 lectures)
Identifying similarity solutions. Invariants.
Similarity solution for the heat equation and for the Rayleigh problem.
Similarity solutions of the first kind.
Similarity solutions of the second kind.
2. Free-boundary problems (~4 lectures)
Stefan problems. One-phase Stefan problem. Connection back to similarity solutions section. Two-phase Stefan problem. Two-dimensional Stefan problem. Stability of interface. Connection to Hele–Shaw problem. Regularization and mushy regions. Examples, including electric welding.
Co-dimension 2 free boundary problems. Examples, including welding and electropainting.
Useful texts
[1] Hydon, P.E., 2000. Symmetry Methods for Differential Equations: A Beginner's Guide (Vol. 22). Cambridge University Press.
[2] Barenblatt, G.I. and Isaakovich, B.G., 1996. Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Vol. 14). Cambridge University Press.
[3] Hinch, E.J., 1991. Perturbation Methods. Cambridge University Press.
[4] Ockendon, H. and Ockendon, J.R., 1995. Viscous Flow (Vol. 13). Cambridge University Press. [5] Ockendon, J.R., Howison, S., Lacey, A. and Movchan, A., 2003. Applied Partial Differential Equations. Oxford University Press.
[6] Tayler, A.B., 2001. Mathematical Models in Applied Mechanics (Vol. 4). Oxford University Press.
Identifying similarity solutions. Invariants.
Similarity solution for the heat equation and for the Rayleigh problem.
Similarity solutions of the first kind.
Similarity solutions of the second kind.
2. Free-boundary problems (~4 lectures)
Stefan problems. One-phase Stefan problem. Connection back to similarity solutions section. Two-phase Stefan problem. Two-dimensional Stefan problem. Stability of interface. Connection to Hele–Shaw problem. Regularization and mushy regions. Examples, including electric welding.
Co-dimension 2 free boundary problems. Examples, including welding and electropainting.
Useful texts
[1] Hydon, P.E., 2000. Symmetry Methods for Differential Equations: A Beginner's Guide (Vol. 22). Cambridge University Press.
[2] Barenblatt, G.I. and Isaakovich, B.G., 1996. Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Vol. 14). Cambridge University Press.
[3] Hinch, E.J., 1991. Perturbation Methods. Cambridge University Press.
[4] Ockendon, H. and Ockendon, J.R., 1995. Viscous Flow (Vol. 13). Cambridge University Press. [5] Ockendon, J.R., Howison, S., Lacey, A. and Movchan, A., 2003. Applied Partial Differential Equations. Oxford University Press.
[6] Tayler, A.B., 2001. Mathematical Models in Applied Mechanics (Vol. 4). Oxford University Press.