Difference between revisions of "BSc: Differential Equations.f23"

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(Created page with "= Differential Equations = * '''Course name''': Differential Equations * '''Code discipline''': CSE205 * '''Subject area''': Math == Short Description == The course is designe...")
 
 
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* '''Subject area''': Math
  
* '''Subject area''': Math
 
== Short Description ==
  
== Short Description ==
  +
This course is an introduction to ordinary differential equations(ODEs) and their applications. Topics covered include first order ODEs, second order linear ODEs, Laypunov’s stability theory and numerical methods.The course will also introduce students to systems of linear equations and eigenvalue problems.
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The course is designed to provide Software Engineers and Computer Scientists by knowledge of basic (core) concepts, definitions, theoretical results and techniques of ordinary differential equations theory, basics of power series and numerical methods, applications of the all above in sciences. All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems.
  
   
 
== Course Topics ==
  
== Course Topics ==
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! Section !! Topics within the section
  
! Section !! Topics within the section
 
|-
  
|-
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| First-order equations and their applications||
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| Differential equations of the first order||
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# Introduction into differential equations. Origins and examples of the differential equations.
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# Separable equation
  
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# A geometrical interpretation of the differential equations of the first order. A field of directions and solutions of the differential equations as trajectories
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# Initial value problem
  
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# Theorem about existence of the solution of the differential equations. Proof of the theorem.
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# Homogeneous nonlinear equations
  
  +
# Equations with separable variables and linear equations of the first order.
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# Substitutions
  
  +
# Homogeneous equations and exact equations and integration factors.
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# Linear ordinary equations
  
  +
|-
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# Bernoulli & Riccati equations
  
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# Exact differential equations, integrating factor
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| Differential equations of the second order||
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# Linear equations of the second order. Phase portraits, trajectories and conservation laws. Singular points of the second order equations.
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# Examples of applications to modeling the real world problems
  
  ⚫
# Non-homogeneous equations and method of undetermined coefficients. Resonances.
  +
# Variations parameters for the second order equations.
  +
# Boundary value problems for the second-order equations and Green's function.
  ⚫
# Applications of Laplace transform.
  +
|-
  +
| Nonlinear equations and Lyapunov's stability ||
  +
# Nonlinear equations of the second order. Conservation laws and trajectories.
  +
# Lyapunov's stability. Lyapunov's function and Lyapunov stability theorems.
  +
# Chetaev's instability theorem and examples of chaotic systems.
  +
# Partial differential equations of the first order and method of characteristics.
 
|-
  
|-
  ⚫
| Systems of the differential equations ||
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| Introduction to Numerical Methods||
 
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# Linear systems of differential equations of the first order and matrix of fundamental solutions.
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# Method of sections (Newton method)
  
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# Method of variations parameters for the non-homogeneous linear systems.
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# Method of tangent lines (Euler method)
  
⚫
# Improved Euler method
  
⚫
# Runge-Kutta methods
  
 
|-
  
|-
  +
| Numerical methods ||
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| Higher-order equations and systems ||
 
  ⚫
# Euler's method.
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# Homogeneous linear equations
  
  ⚫
# Runge-Kutta method.
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# Constant coefficient equations
  
  +
# Stability and accuracy of numerical methods.
⚫
# A method of undetermined coefficients
  
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# A method of variation of parameters
  
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# A method of the reduction of order
  
⚫
# Laplace transform. Inverse Laplace transform.
  
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# Application of the Laplace transform to solving differential equations.
  
⚫
# Series solution of differential equations.
  
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# Homogeneous linear systems
 
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# Non-homogeneous systems
 
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# Matrices, eigenvalues and matrix form of the systems of ODE
  
 
|}
  
|}
  +
 
== Intended Learning Outcomes (ILOs) ==
  
== Intended Learning Outcomes (ILOs) ==
   
  +
=== Course objectives ===
  +
  +
Upon completion of this course, students should be able to:
  +
* Realize conditions of existence for the equations of the first order and solve first-order ordinary differential equations using various techniques such as separation of variables, integration factors.
  +
* Solve second-order linear differential equations with constant coefficients using techniques such as the characteristic equation and the method of undetermined coefficients and applications of Laplace transform for the linear equations.
  +
* Define the resonant conditions for the linear and nonlinear equations of the second order equation.
  +
* Apply the Lyapunov's stability theory for the linear and nonlinear systems.
  +
* Know the properties of the solutions of first-order partial differential equations.
  +
* Apply numerical methods to approximate solutions to differential equations.
  +
* Understand the concept of eigenvalues and eigenvectors and use them to solve systems of linear differential equations.
  +
<!--
 
=== What is the main purpose of this course? ===
  
=== What is the main purpose of this course? ===
   
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* predict the number of terms in series solution of the equation depending on the given accuracy,
  
* predict the number of terms in series solution of the equation depending on the given accuracy,
 
* implement a certain numerical method in self-developed computer software.
  
* implement a certain numerical method in self-developed computer software.
  +
-->
   
 
== Grading ==
  
== Grading ==
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| Midterm || 20
  
| Midterm || 20
 
|-
  
|-
−
| Interim Assessment|| 20
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| Interim Assessment|| 20 pts (2 tests by 10 pts)
 
|-
  
|-
 
| Final exam || 30
  
| Final exam || 30
 
|-
  
|-
−
| Computational assignment || 25
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| Computational assignment || 30
 
|-
  
|-
−
| In-class participation || 5
 +
| Attendance and In-class participation || 7
 
|}
  
|}
  +
<!--
−  
 
=== Recommendations for students on how to succeed in the course ===
  
=== Recommendations for students on how to succeed in the course ===
 
* Participation is important. Attending lectures is the key to success in this course.
  
* Participation is important. Attending lectures is the key to success in this course.
 
* Review lecture materials before classes to do well.
  
* Review lecture materials before classes to do well.
 
* Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.
  
* Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.
  +
-->
   
 
== Resources, literature and reference materials ==
  
== Resources, literature and reference materials ==
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=== Open access resources ===
  
=== Open access resources ===
 
* Elementary Differential Equations by William F. Trench. Brooks/Cole Thomson Learning, 2001 [https://digitalcommons.trinity.edu/mono/8/ link]
  
* Elementary Differential Equations by William F. Trench. Brooks/Cole Thomson Learning, 2001 [https://digitalcommons.trinity.edu/mono/8/ link]
  +
* Stephen L. Campbell and Richard Haberman, Introduction to differential equations with dynamical systems
−  
  +
* J.L.Brenner, Problems in DifferentialEquations(adapted from ”Problems in differential equations” by A.F.Filippov)
  +
* S.G.Glebov, O.M.Kiselev, N.Tarkhanov. Nonlinear equations with small parameter. Volume I:Oscillations and resonances
  +
<!--
 
== Activities and Teaching Methods ==
  
== Activities and Teaching Methods ==
 
{| class="wikitable"
  
{| class="wikitable"
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=== The retake exam ===
  
=== The retake exam ===
 
Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.
  
Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.
  +
-->

Latest revision as of 21:12, 31 August 2023

Differential Equations

  • Course name: Differential Equations
  • Code discipline: CSE205
  • Subject area: Math

Short Description

This course is an introduction to ordinary differential equations(ODEs) and their applications. Topics covered include first order ODEs, second order linear ODEs, Laypunov’s stability theory and numerical methods.The course will also introduce students to systems of linear equations and eigenvalue problems.

Course Topics

Course Sections and Topics
Section Topics within the section
Differential equations of the first order
  1. Introduction into differential equations. Origins and examples of the differential equations.
  2. A geometrical interpretation of the differential equations of the first order. A field of directions and solutions of the differential equations as trajectories
  3. Theorem about existence of the solution of the differential equations. Proof of the theorem.
  4. Equations with separable variables and linear equations of the first order.
  5. Homogeneous equations and exact equations and integration factors.
Differential equations of the second order
  1. Linear equations of the second order. Phase portraits, trajectories and conservation laws. Singular points of the second order equations.
  2. Non-homogeneous equations and method of undetermined coefficients. Resonances.
  3. Variations parameters for the second order equations.
  4. Boundary value problems for the second-order equations and Green's function.
  5. Applications of Laplace transform.
Nonlinear equations and Lyapunov's stability
  1. Nonlinear equations of the second order. Conservation laws and trajectories.
  2. Lyapunov's stability. Lyapunov's function and Lyapunov stability theorems.
  3. Chetaev's instability theorem and examples of chaotic systems.
  4. Partial differential equations of the first order and method of characteristics.
Systems of the differential equations
  1. Linear systems of differential equations of the first order and matrix of fundamental solutions.
  2. Method of variations parameters for the non-homogeneous linear systems.
Numerical methods
  1. Euler's method.
  2. Runge-Kutta method.
  3. Stability and accuracy of numerical methods.

Intended Learning Outcomes (ILOs)

Course objectives

Upon completion of this course, students should be able to:

  • Realize conditions of existence for the equations of the first order and solve first-order ordinary differential equations using various techniques such as separation of variables, integration factors.
  • Solve second-order linear differential equations with constant coefficients using techniques such as the characteristic equation and the method of undetermined coefficients and applications of Laplace transform for the linear equations.
  • Define the resonant conditions for the linear and nonlinear equations of the second order equation.
  • Apply the Lyapunov's stability theory for the linear and nonlinear systems.
  • Know the properties of the solutions of first-order partial differential equations.
  • Apply numerical methods to approximate solutions to differential equations.
  • Understand the concept of eigenvalues and eigenvectors and use them to solve systems of linear differential equations.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 90-100 -
B. Good 75-89 -
C. Satisfactory 60-74 -
D. Fail 0-59 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Midterm 20
Interim Assessment 20 pts (2 tests by 10 pts)
Final exam 30
Computational assignment 30
Attendance and In-class participation 7

Resources, literature and reference materials

Open access resources

  • Elementary Differential Equations by William F. Trench. Brooks/Cole Thomson Learning, 2001 link
  • Stephen L. Campbell and Richard Haberman, Introduction to differential equations with dynamical systems
  • J.L.Brenner, Problems in DifferentialEquations(adapted from ”Problems in differential equations” by A.F.Filippov)
  • S.G.Glebov, O.M.Kiselev, N.Tarkhanov. Nonlinear equations with small parameter. Volume I:Oscillations and resonances
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