- Nazwa przedmiotu:
- Differential and Difference Equations
- Koordynator przedmiotu:
- Andrzej Fryszkowski
- Status przedmiotu:
- Obowiązkowy
- Poziom kształcenia:
- Studia I stopnia
- Program:
- Computer Science
- Grupa przedmiotów:
- Technical Courses
- Kod przedmiotu:
- EDDE
- Semestr nominalny:
- 3 / rok ak. 2015/2016
- Liczba punktów ECTS:
- 6
- Liczba godzin pracy studenta związanych z osiągnięciem efektów uczenia się:
- Lecture and tutorial attendance: 60 h;
preparation to lectures: 15 h;
preparation to tutorials: 15 h;
preparation to written tests: 30 h;
preparation to the final examination: 10h;
Total: 130 h
- Liczba punktów ECTS na zajęciach wymagających bezpośredniego udziału nauczycieli akademickich:
- 2
- Język prowadzenia zajęć:
- angielski
- Liczba punktów ECTS, którą student uzyskuje w ramach zajęć o charakterze praktycznym:
- 2
- Formy zajęć i ich wymiar w semestrze:
-
- Wykład30h
- Ćwiczenia30h
- Laboratorium0h
- Projekt0h
- Lekcje komputerowe0h
- Wymagania wstępne:
- Passing Introduction to Discrete Mathemetics (EIDM) and Mathematical Analysis (EMANA).
- Limit liczby studentów:
- 60
- Cel przedmiotu:
- - To convey and reinforce the knowledge on differential and difference equations, different types of equations and system of equations, methods of solving them.
- To acquire thorough understanding of basic concepts and methods, computational processes and to master skills of using them (labs) in applications and to master the skill of correct mathematical reasoning and inference.
- Treści kształcenia:
- Lectures:
1. The notion of differential equation (DE) (2h): ordinary DE (ODE) and partial DE (PDE), the order of DE. Solving DE – a particular solution, the general solution (GS). ODE of the first order – geometrical interpretation. Cauchy problems for ODE and their interpretation. The existence and uniqueness of solutions. The separable DE and solving by separation of variables.
2. Finding orthogonal trajectories (2h). Linear DE (LDE) of the 1st order – homogeneous and nonhomogeneous. Solving LDE by variation of parameter. Solving LDE by integrating factor.
3. Application of LDE (2h). The Bernoulli DE. Exact DE, solving by integrating factor. DE reducible to 1st order.
4. LDE of arbitrary order (2h): properties of solutions, fundamental solutions (FS), Wrońskian. Homogeneous LDE with constant coefficients – characteristic equation, fundamental solutions (FS) and GS.
5. Nonhomogeneous LDE with constant coefficients (2h): solving by variation of parameters and by the method of undetermined coefficients.
6. Euler Differential Equation (2h). Linear systems of DE (SDE). Solving by the method of elimination. Solving by variation of parameters and by matrix method.
7. The Laplace transform (2h): definitions and properties.
8. The Laplace transform applied to DE (2h) and systems of DE.
9. Difference calculus (4h): functions with discrete argument; difference operator; inverse difference operator; shift operator. The notion of difference equation (dE): ordinary dE (odE) , the order of dE; solving dE – a particular solution, the general solution (GS); the existence and uniqueness of solutions; mechanical interpretation of dE; Cauchy problems for odE and their interpretation.
10. Linear dE (ldE) of the n – th order (2h): homogeneous and nonhomogeneous: ldE of the 1st; general theory for homogeneous ldE; general theory for nonhomogeneous ldE; ldE with constant coefficients.
11. Linear systems of DE (2h): homogeneous systems; nonhomogeneous systems; systems with constant coefficients; systems with periodic coefficients.
12. Applications of dE in ODE (2h): Taylor series methods for solving ODE
13. The Riemann transform (2h): definitions & properties. Application for solving odE.
14. Numerical methods for solving odE and ODE (2h).
Tutorials:
1. The notion of differential equation (DE), ordinary DE (ODE) and partial DE (PDE), the order of DE. Solving DE – a particular solution, the general solution (GS). ODE of the first order – geometrical interpretation. Cauchy problems for ODE and their interpretation. The existence and uniqueness of solutions. The separable DE and solving by separation of variables. Finding orthogonal trajectories.
2. Linear DE (LDE) of the 1st order – homogeneous and nonhomogeneous. Solving LDE by variation of parameter. Solving LDE by integrating factor. Application of LDE. The Bernoulli DE. Exact DE, solving by integrating factor. DE reducible to 1st order.
3. LDE of arbitrary order – properties of solutions, fundamental solutions (FS), Wrońskian. Homogeneous LDE with constant coefficients – characteristic equation, fundamental solutions (FS) and GS.
4. Nonhomogeneous LDE with constant coefficients – solving by variation of parameters and by the method of undetermined coefficients.
5. Euler Differential Equation. Linear systems of DE (SDE). Solving by the method of elimination. Solving by variation of parameters and by matrix method.
6. The Laplace transform – definitions and properties.
7. The Laplace transform applied to DE and systems of DE.
8. Difference calculus: functions with discrete argument; difference operator; inverse difference operator; shift operator. The notion of difference equation (dE): ordinary dE (odE) , the order of dE; solving dE – a particular solution, the general solution (GS); the existence and uniqueness of solutions; mechanical interpretation of dE; Cauchy problems for odE and their interpretation.
9. Linear dE (ldE) of the n – th order – homogeneous and nonhomogeneous: ldE of the 1st; general theory for homogeneous ldE; general theory for nonhomogeneous ldE; ldE with constant coefficients.
10. Linear systems of DE: homogeneous systems; nonhomogeneous systems; systems with constant coefficients; systems with periodic coefficients.
11. Applications of dE in ODE: Taylor series methods for solving ODE.
12. The Riemann transform – definitions and properties. Application for solving odE.
13. Numerical methods for solving odE and ODE.
- Metody oceny:
- 3 tests (50 pts) + written final exam (50 pts)
Sum Grade
50 – 56 3.0
57 – 63 3.5
64 – 70 4.0
71 – 80 4.5
81 – 100 5.0
- Egzamin:
- tak
- Literatura:
- 1. N. Finizio, G. Ladas, Ordinary Differential Equations.
2. P.N.V. Tu, Dynamical Systems. An Introduction with Applications in Economics and Biology, Springer, 1994 (selected chapters).
3. K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos. An Introduction to Dynamical Systems, Springer 1996 (selected chapters).
4. H.-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals. New Frontiers of Science, Springer 1993.
- Witryna www przedmiotu:
- https://studia.elka.pw.edu.pl
- Uwagi:
Efekty uczenia się
Profil ogólnoakademicki - wiedza
- Efekt EDDE_W01
- Student knows basic types of differential equations and methods for solving them.
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_W01
Powiązane efekty obszarowe:
T1A_W01, T1A_W02, T1A_W03, T1A_W07
- Efekt EDDE_W02
- Knows the Laplace transformations and their applications to solving linear differential equations and linear systems of differential equations
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_W01
Powiązane efekty obszarowe:
T1A_W01, T1A_W02, T1A_W03, T1A_W07
- Efekt EDDE_W03
- Knows the linear difference equations of the first and second order and methods of solving them
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_W01
Powiązane efekty obszarowe:
T1A_W01, T1A_W02, T1A_W03, T1A_W07
Profil ogólnoakademicki - umiejętności
- Efekt EDDE_U01
- Student can recognize the type of differential equations and is able to solve them
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_U01, K_U05, K_U08
Powiązane efekty obszarowe:
T1A_U01, T1A_U05, T1A_U08, T1A_U09
- Efekt EDDE_U02
- Students can evaluate the Laplace transformations with the use of a table of the transformations
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_U01, K_U05
Powiązane efekty obszarowe:
T1A_U01, T1A_U05
- Efekt EDDE_U03
- Students can apply the Laplace transformation to solve linear differential equations and linear systems
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_U01, K_U05
Powiązane efekty obszarowe:
T1A_U01, T1A_U05
- Efekt EDDE_U04
- Student is able to solve the linear difference equation of the first and second order
Weryfikacja: Test and exam
Powiązane efekty kierunkowe:
K_U01, K_U05, K_U08
Powiązane efekty obszarowe:
T1A_U01, T1A_U05, T1A_U08, T1A_U09