Further Partial Differential Equations - Material for the year 2020-2021

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Prof. Ian Griffiths
Course Term: 
Course Lecture Information: 

8 lectures

Course Level: 
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 Syllabus: 

Similarity Solutions;
Free-boundary Problems;
Boundary-layer Theory.

Course Synopsis: 

1. Similarity solutions (~3 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 (~3 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.

3. Boundary-layer theory (~2 lectures)

  • Introduction to asymptotic analysis and singular perturbation problems.
  • Thermal boundary layers on a semi-infinite flat plate.
  • Connection to similarity solutions section.
  • Viscous boundary layer on a semi-infinite flat plate.
    Reading List: 
    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
    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.

    Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.