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I. Introduction to dimensional analysis and scaling
1. Applied mathematics.
2. A classical example.
3. Generalisation.
4. A unit free physical law.
5. The Pi-theorem.
6. One more example.
7. Characteristic scales.
8. A heat conduction problem.
9. Some population models.
10. Exercises..
II. Introduction to perturbation methods.
1. Introduction..
2. The main idea behind perturbation methods.
3. Motion in a nonlinear resistive medium.
4. One example.
5. Comparison with the exact solution.
6. A nonlinear oscillator.
7. Poincaré-Lindstedt’s method.
8. Ordo-notation.
9. Regular perturbation does not always work.
10. Inner and outer approximations.
11. Singular perturbation – when does regular perturbation not work?
12. The outer approximation.
13. The inner approximation.
14. Matching.
15. A final example.
16. WKB-approximation
17. Exercises. (ii)
III. Introduction to the calculus of variations.
1. Functions – extreme points.
2. Functionals – extremals.
3. Some function spaces.
4. Some examples of variational problems.
5. Euler’s equation for the simplest problem.
6. Two solved problems.
7. Simplification of Euler´s equation.
8. Proof of theorem 1.
9. Natural boundary conditions.
10. The Euler equation for some more general cases.
11. Normed linear spaces.
12. Local minimum of a functional.
13. Differentiation of functionals.
14. A necessary condition for extremum of a functional.
15. Exercises.
IV. Introduction to the theory of partial differential equations.
V. Introduction to Sturm-Liouville theory, the theory for the corresponding generalized Fourier series and some further methods for solving PDE.
VI. Introduction to transform theory with applications.
VII. Introduction to Hamiltonian theory and isoperimetric problems.
1. The Lagrangian
2. Hamilton’s principle.
3. The Hamiltonian.
4. Some examples.
5. Canonical formalism.
6. The general case.
7. Lagrange multipliers.
8. Isoperimetric problems.
9. Some more examples.
10. Exercises. vii
VIII. Introduction to the theory of integral equations.
IX. Introduction to the theory of dynamical systems, chaos, stability and bifurcations.
1. Introduction.
2. Diskrete dynamical systems.
3. The Lyapunov exponent.
4. Julia and Mandelbrot sets.
5. Continuous dynamical systems.
6. Some introductory examples.
7. Classification of critical points.
8. The general solution of a linear system.
9. Classification of equilibrium points in specific systems.
10. Exercises. ix
X. Introduction to discrete mathematics.
1. Set theory
2. Division
3. Euclids algorithm for the least common divisor
4. Diophantine equations
5. Congruences
6. Induction
7. Exercises
VII. Introduction to Hamiltonian theory and isoperimetric problems.
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