Summary.
This course is an introduction to applied mathematics with emphasis on the practical solution of real-world problems using differential equations, numerical methods and optimization techniques.

Textbooks.
a) J. D. Logan, Applied Mathematics, Wiley (1996)
b) P. DuChateau and D. Zachmann, Applied Partial Differential Equations, Dover (1989)
c) Numerical solution of partial differential equations, by K.W. Morton and D. F. Mayers, Cambridge Univ. Press (2002)

Grading.
The grade will be based on two midterm exams, a final exam and special projects.

1. Basic equations of mathematical physics, engineering and biology. Buck-
ingham's pi theorem. Dimensional analysis.
2. Classification of PDEs. Well and ill posed problems.
3. Fourier series, Sturm Liouville problems, L2 space. Nonlinear boundary
value problems.
4. Laplace and Poisson equations, parabolic equations and hyperbolic equa-
tions.
5. Laplace transform. Fourier transform. Solutions on unbounded domains.
Duhamel's principle.
6. Uniqueness and continuous dependence on initial and boundary data. En-
ergy inequalities.
7. Stability and bifurcation of solutions. Kelvin Helmholtz instability. Fin-
gering.
8. Maximum and minimum principles. Nonlinear diffusion. Population dy-
namics.
9. Conservation laws. Method of characteristics. Shocks. Entropy principle.
10. Finite difference methods: Global and local truncation errors. Error es-
timation. Proofs of Convergence and stability. Numerical diffusion. Up-
winding.
11. Regular and singular perturbations. Asymptotic expansions.
12. Calculus of variations. Euler equations. Isoparametric problems. Appli-
cations to optimization problems.