The aim of studying and bibliography

The main aim of study: To familiarize students with the basics of Laplace and Fourier operator theory and their applications to differential equations.

The general information about the module: The module is implemented in the third semester in the form of lectures (30 hours).

Bibliography required to complete the module

Bibliography used during lectures

1	G. Doetsch	Introduction to the Theory and Application of the Laplace Transformation	Springer-Verlag Berlin Heidelberg New York.	1974
2	J. Musielak	Wstęp do analizy funkcjonalnej	PWN.	1976
3	L.C. Evans	Równania różniczkowe cząstkowe	Wydawnictwo Naukowe PWN.	2008

Bibliography to self-study

1

W. Krysicki, L. Włodarski

Analiza matematyczna w zadaniach 2

Wydawnictwo Naukowe PWN.

2015

Basic requirements in category knowledge/skills/social competences

Formal requirements: Requirements accordant with Rules and Regulations of studies

Basic requirements in category knowledge: A student has mathematical knowledge which allows him/her to understand the lectured material.

Basic requirements in category skills: Ability to use fundamental mathematical tools and the knowledge obtained during the first and second level studies.

Basic requirements in category social competences: A student is prepared to undertake substantiated mathematical operations in order to solve a task.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with OEK
01	Knows the concepts of convolution and its basic properties.	lecture	test	K_W01+++ K_W02++ K_W03++ K_W04++ K_W05+++ K_W07+	X2A_W01 X2A_W02 X2A_W03 X2A_W06
02	Knows the basic properties of Laplace transforms. He can calculate the transforms of various functions.	lecture	test	K_W01+++ K_W02+ K_W04+ K_W05+	X2A_W01 X2A_W03
03	Student is able to solve some equations and systems of differential equations using the Laplace Transform	lecture	test	K_U01++ K_U02++ K_U03++ K_K01++ K_K04++ K_K07++	X2A_U01 X2A_U02 X2A_U03 X2A_U05 X2A_U07 X2A_K01 X2A_K03 X2A_K04 X2A_K06
04	Knows the basic properties of Fourier transforms. He can solve some partial equations.	lecture	test	K_W01+ K_W03++ K_W05+ K_W07++ K_K02+	X2A_W01 X2A_W02 X2A_W06 X2A_K01 X2A_K02

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Preliminary. Convolution and its properties, smoothing and approximation theorems.	W01-W04	MEK01
3	TK02	Laplace transform and its properties, theorems on shift, integration, differentiation, convolution, injectivity. Inverse transform. Transforms of basic functions. Examples of calculation of Laplace transforms.	W05-W15	MEK02
3	TK03	Applications of Laplace transform to solve equations and systems of differential equations.	W16-W20	MEK02 MEK03
3	TK04	Fourier transform in L^1 and L^2, its properties, inverse transform. Applications of Fourier transform to solve some equations of partial coefficients with constant coefficients (heat equation, wave equation).	W21-W30	MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 1.00 hours/sem.	contact hours: 30.00 hours/sem.	complementing/reading through notes: 3.00 hours/sem. Studying the recommended bibliography: 5.00 hours/sem.
Advice (sem. 3)	The preparation for Advice: 2.00 hours/sem.	The participation in Advice: 3.00 hours/sem.
Credit (sem. 3)	The preparation for a Credit: 10.00 hours/sem.	The written credit: 2.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	A credit for the lecture is based on the results of tests.
The final grade	The final grade is a credit for the lecture.

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: no