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LECTURE NOTES ON APPLIED MATHEMATICS

LECTURE NOTES ONAPPLIED MATHEMATICSM ethods and ModelsJohn K. HunterDepartment of MathematicsUniversity of California, DavisJune 17, 2009 Copyrightc 2009 by John K. HunterContentsLecture 1. Introduction11. Conservation laws12. Constitutive equations23. The KPP equation3 LECTURE 2. Dimensional Analysis, Scaling, and Similarity111. Systems of units112. Scaling123. Nondimensionalization134. Fluid mechanics135. Stokes formula for the drag on a sphere186. Kolmogorov s 1941 theory of turbulence227. Self-similarity258. The porous medium equation279. Continuous symmetries of differential equations33 LECTURE 3. The Calculus of Variations431. Motion of a particle in a conservative force field442. The Euler-Lagrange equation493. Newton s problem of minimal resistance514. Constrained variational principles565. Elastic rods576. Buckling and bifurcation theory617.

Jun 17, 2009 · The conservation law (1.2) is not a closed equation for the density u. Typically, we supplement it with constitutive equations that relate the ux ~qand the source density ˙to uand its derivatives. While the conservation law expresses a gen-eral physical principle, constitutive equations describe the response of a particular system being modeled.

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