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18.306

Advanced Partial Differential Equations with Applications

3-0-9 units · Graduate

Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.

Fall 2009

The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.

Readings

  • There is no required textbook for this course. A list of a few recommended textbooks can be found below.
    Listed reading
  • Whitham, G. B. Linear and Nonlinear Waves . New York, NY: Wiley, 1999. ISBN: 9780471359425.
    Listed reading
  • Kevorkian, J. Partial Differential Equations: Analytical Solution Techniques . Texts in Applied Mathematics, vol. 35. 2nd ed. New York, NY: Springer, 2000. ISBN: 9780387986050.
    Listed reading
  • Hinch, E. J. Perturbation Methods . Cambridge Texts in Applied Mathematics. Cambridge, UK: Cambridge University Press, 1991. ISBN: 9780521378970.
    Listed reading
  • Guenther, Ronald B., and John W. Lee. Partial Differential Equations of Mathematical Physics and Integral Equations . New York, NY: Dover Publications, 1996. ISBN: 9780486688893.
    Listed reading
  • Debnath, Lokenath. Nonlinear Partial Differential Equations for Scientists and Engineers . Boston, MA: Birkhäuser, 1997. ISBN: 9780817639020.
    Listed reading
  • Carrier, George F., and Carl E. Pearson. Partial Differential Equations: Theory and Technique . 2nd ed. Boston, MA: Academic Press, 1988. ISBN: 9780121604516.
    Listed reading
  • Barenblatt, G. I. Scaling, Self-Similarity, and Intermediate Asymptotics . Cambridge Texts in Applied Mathematics, vol. 14. Cambridge, UK: Cambridge University Press, 1996. ISBN: 9780521435222.
    Listed reading
  • Levine, Harold. Partial Differential Equations . AMS/IP Studies in Advanced Mathematics, vol. 6. Providence, RI: American Mathematical Society, 1997. ISBN: 9780821807750.
    Listed reading
  • Drazin, P. G., and R. S. Johnson. Solitons: An Introduction . Cambridge Texts in Applied Mathematics. Cambridge, UK: Cambridge University Press, 1989. ISBN: 9780521336550.
    Listed reading
  • Evans, Lawrence C. Partial Differential Equations . Graduate Studies in Mathematics, vol. 19. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.
    Listed reading
  • DiBenedetto, Emmanuele. Partial Differential Equations . Boston, MA: Birkhäuser, 1995. ISBN: 9780817637088.
    Listed reading
  • Garabedian, Paul. Partial Differential Equations . Providence, RI: AMS Chelsea, 1998. ISBN: 9780821813775.
    Listed reading
  • There is no required textbook for this course. A list of a few recommended textbooks can be found in the Readings section. ↗
    Listed reading

Prerequisites & course information

Prereq: ( 18.03 or 18.032 ) and ( 18.04 , 18.075 , or 18.112 ) Acad Year 2027-2028: G (Spring) Acad Year 2028-2029: Not offered 3-0-9 units

Sources