v Checklists

Checklists to the lectures



Lecture 1, 20.08

1. Definintion of the Laplace transform.
2. Laplace transform of some elementary functions.
3. Properties of the Laplace transform:
4. Laplace transform of sin and cos.
5. Laplace transform of derivative. Relation to differential equations.
6. Inverse Laplace transform
7. Exercises

Lecture 2, 22.08

1. Laplace transform of derivatives.
2. Solution differential equations with constant coefficients by mean of the Laplace transform.
3. Exercises
4. Laplace transform of integrals
5. Application of this theorem for calculating the inverse Laplace transform.

Lecture 3, 27.08

1. Shift of functions
2. Heaviside function, its Laplace transform
3. Second shift theorem.
4. Example with electrical circuit.
5. Delta function and its Laplace transform.
6. Differential equations with delta function in input side.

Lecture 4, 29.08

1. Derivative of the Laplace transform
2. Integral of the Laplace transform.
3. Exercises
4. Convolution of two functions. machanical interpretation.
5. Properties of convolution
6. Laplace transform of convolution

Lecture 5, 03.09

1. Examples of taking inverse Laplace transform. Simple fractions expansions
2. Systems of linear differential equations.

Lecture 6, 05.09

1. Periodic functions
2. Periodical prolongation
3. Orthogonality relations
4. Fourier coefficients.
5. Fourier series.

Lecture 7, 10.09

1. Properties of Fourier coefficients.
2. Odd and even functions. Fourier coefficients of odd and even functions.
3. Convergence theorems for the Fourier series.
4. Examples.

Lecture 8, 12.09

1. Different possibilities for periodic prolongation of functions defined on a part of the period.
2. Halfrange expansions.
3. Refreshment: Complex numbers.

Lecture 9, 17.09

1. Complex Fourier series.
2. Summation of numerical series using Fourier series expansions.
3. Examples.

Lecture 10, 19.09

1. Fourier series for functions with arbitrary period.
2. Differential equations with constant coefficients. Use of linearity.
3. Differential equations with sine/cos functions in the righthandside (Refreshment from Ch. 2)
4. Differential equations with periodic righthandsides
5. Examples.

Lecture 11, 24.09

1. Differential equations with periodic righthandsides, end of examples. The same is in the complex form.
2. Trigonometrical polynomials. Formulation of the problem about approximation. Square error.
3. Evaluation of square error


Lecture 12, 26.09

1. Square error. Repetition.
2. Expressions for square errors for complex Fourier series.
3. Parseval identity. Bessel inequality
4. Example: summation of numerical series by using the Parseval identity.
5. Fourier integrals: cosine transform.

Lecture 13, 01.10

1. Fourier integrals: cosine transform, sine transform, complex form.
2. Inverse transforms.
3. Parseval identity for complex Fourier transform.
4. Examples
5. Linearity
6. Fourier transforms of derivatives

Lecture 14, 03.10

1. Fourier transform of derivatives (cnt).
2. Calculation Fourier transform using Fourier transforms of derivatives.
3. Convolution of two functions on the real axis
4. Fourier transform of the convolution.
5. Spectral interpretation. Low pass filters.
6. Survey of Fourier transforms.

Lecture 15, 08.10

1. Introduction to partial differential equations:
2. Deriving equation of oscillating string.
3. Idea of the method of separation of variables
4. Solution of equation of oscillating string by separating variables.

Lecture 16, 10.10


1. Solution of equation of oscillating string by separating variables (cont).
2.D'Alambert solution
3. General heat equation.
4. Boundary problem for the heat equation.
5. Solution to one-dimensional heat equation.
6. Analising solution.

Lecture 17, 15.10

1. Steady heat equation.
2. Various boundary problems (Dirichlet, Newmann, mixed).
3. Solution of the Newmann problem for rectangle by separating variables.
4. Examples.
Last modified: Tue Oct 16 10:04:50 MEST 2001