- Web course
- I. Introduction to dimensional analysis and scaling
- 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.
- 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.

We get the outer approximation bya putting . In our last example that implied that we should solve the equation

It has the solution

which well agrees with the exact solution

since

for small . If *t=O(1)* we have that

which implies that

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