This document discusses numerical methods for solving initial value problems. It introduces the Euler method and Runge-Kutta methods of orders 2 and 3 for approximating the solutions to differential equations. Examples are provided to demonstrate applying each method. The objectives are to understand applications of initial value problems and use the Euler and Runge-Kutta methods to solve them. Convergence of the numerical solutions is also examined.
How to handle Initial Value Problems using numerical techniques?
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Numerical Response for Initial Value Problems using Newmark Method.
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How to handle a system of initial value problems using Runge-Kutta method?
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Solve nonlinear equations using bracketing methods: Bisection and False Position
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What is interpolation?
How to interpolate a polynomial through a given set of data?
General approach, Newton method, Lagrange method
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How to find the roots of Nonlinear Equations?
Newton-Raphson method is not the only way!
How about a system of nonlinear equations?
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What is numerical differentiation?
What is finite difference?
How to apply that to boundary value problems?
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How to handle Initial Value Problems using numerical techniques?
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https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
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Numerical Response for Initial Value Problems using Newmark Method.
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https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
How to handle a system of initial value problems using Runge-Kutta method?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
Solve nonlinear equations using bracketing methods: Bisection and False Position
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations
What is interpolation?
How to interpolate a polynomial through a given set of data?
General approach, Newton method, Lagrange method
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Interpolation
How to find the roots of Nonlinear Equations?
Newton-Raphson method is not the only way!
How about a system of nonlinear equations?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations
What is numerical differentiation?
What is finite difference?
How to apply that to boundary value problems?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Boundary+Value+Problems+-+Finite+Difference
How to interpolate data with slopes?
How to interpolate data involving more than one independent variable?
A Generalized Approach to Interpolation
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FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
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How to best-fit a function to a given set of data?
Why is this different from interpolation?
Do you have more than one independent variable?
Learn about regression and least squares method!
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How to create and solve finite element models?
Application to 2nd Order Differential Equations!
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How to derive the finite element model using the stationary functional approach?
Application to bars and beams!
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What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
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FEM: Nonlinear Beam Deflection Model (with Temperature)Mohammad Tawfik
Derivation and solution of the nonlinear finite element model of a beam under thermal loading.
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How to interpolate data with slopes?
How to interpolate data involving more than one independent variable?
A Generalized Approach to Interpolation
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Interpolation
FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
How to best-fit a function to a given set of data?
Why is this different from interpolation?
Do you have more than one independent variable?
Learn about regression and least squares method!
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Regression
How to create and solve finite element models?
Application to 2nd Order Differential Equations!
#WikiCourses #FEM
https://wikicourses.wikispaces.com/TopicX+Element+Equations
How to derive the finite element model using the stationary functional approach?
Application to bars and beams!
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Stationary+Functional+Approach
https://eau-esa.wikispaces.com/Topic+Stationary+Functional+Approach
What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
#WikiCourses
https://wikicourses.wikispaces.com/Lect04+Multiple+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Multiple+Degree+of+Freedom+%28MDOF%29+Systems
FEM: Nonlinear Beam Deflection Model (with Temperature)Mohammad Tawfik
Derivation and solution of the nonlinear finite element model of a beam under thermal loading.
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics
https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
Probabilistic Models of Time Series and SequencesZitao Liu
Tutorial on Probabilistic Models of Time Series and Sequences. Hidden Markov Models. Linear Dynamical Systems. Forward/backward algorithm. Kalman Filtering. Kalman Smoothing. Viterbi algorithm. Baum-Welch algorithm. Learning HMM. Learning LDS.
Brief description of current state of drones and some future challenges.
The presentation is prepared for delivery in the "Interact with Today's World" conference held in Bibliotica Alexandria 5-6 August 2016
Finite Element Analysis of the Aeroelasticity Plates Under Thermal and Aerody...Mohammad Tawfik
At the flight condition of panel-flutter (usually supersonic flight conditions), the phenomenon is associated with elevated temperatures, produced from the aerodynamic heating through the boundary layer friction and the presence of shock waves. This heating adds to the complexity of the problem by introducing panel stiffness reduction and thermal loading, which might also be associated with post-buckling deflection. In the following, a literature review of panel-flutter analysis and control topics is presented.
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Get to learn the basic quantities and definitions used in aerodynamics.
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Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
A report on the finite element analysis of a beam under thermal loading. Nonlinear deflections and solution procedures covered.
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2. Initial Value Problems
Mohammad Tawfik
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Objectives
• Understand the applications of initial-value
problems
• Be able to apply the Euler method for
solving initial value problems
• Be able to apply the Runge-Kutta method
for solving initial value problem
3. Initial Value Problems
Mohammad Tawfik
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Example Problem
dt
dv
mmaF
cvmgFFF UD
cvmgvm
dt
dv
m
m
cvmg
v
mct
e
c
mg
tv /
1
5. Initial Value Problems
Mohammad Tawfik
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Approximate Solution
12
12
tt
vv
t
v
dt
dv
m
cvmg
tt
vv
12
12
m
cvmg
tt
vv 1
12
12
6. Initial Value Problems
Mohammad Tawfik
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Approximate Solution
m
cvmg
ttvv 1
1212
7. Initial Value Problems
Mohammad Tawfik
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Euler Method
• Given the differential
equation:
• We may write:
• Giving:
tyf
dt
dy
,
tyf
t
yy
t
y
dt
dy ttt
,
tytfyy ttt ,
8. Initial Value Problems
Mohammad Tawfik
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Example
• Given the differential
equation:
• The exact solution is:
• At t=0, y=2
• Find y(4) using Euler
method with step
t=1
ye
dt
dy t
5.04 8.0
tt
eety 5.08.0
08.108.3
9. Initial Value Problems
Mohammad Tawfik
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Solution
ye
dt
dy t
5.04 8.0
yetyy t
tttt 5.04 8.0
5142
5.04 0
0
01
yeyy
4.115.245
5.04
8.0
1
8.0
12
e
yeyy
5.254.11*5.044.11
5.04
6.1
2
2*8.0
23
e
yeyy
8.565.25*5.045.25
5.04
4.2
3
3*8.0
34
e
yeyy
11. Initial Value Problems
Mohammad Tawfik
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Runge-Kutta Methods
• The Runge-Kutta methods achieves the
Taylor series accuracy
• Many forms of the method are available;
we will use 2nd order and 3rd order
methods only
12. Initial Value Problems
Mohammad Tawfik
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2nd Order Runge-Kutta method
• For the DE:
• The 2nd order R.K. solution
is:
• Where:
• Note: in the textbook there
are 3 different methods, we
are using the first!
tyf
dt
dy
,
21
2
kk
t
yy ttt
tyfk t ,1
tttkyfk t ,12
13. Initial Value Problems
Mohammad Tawfik
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Example
• Given the differential
equation:
• The exact solution is:
• At t=0, y=2
• Find y(4) using 2nd
order R.K. method
with step t=1
ye
dt
dy t
5.04 8.0
tt
eety 5.08.0
08.108.3
14. Initial Value Problems
Mohammad Tawfik
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Solution
• At t=0
32*5.04 0
1 ek
4.61*325.04 108.0
2
ek
7.6
2
2101
kk
t
yy
• Repeat for all t
17. Initial Value Problems
Mohammad Tawfik
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3rd Order Runge-Kutta method
• For the DE:
• The 3rd order R.K.
solution is:
• Where:
tyf
dt
dy
,
321 4
6
kkk
t
yy ttt
tyfk t ,1
2
,
2
1
2
t
t
tk
yfk t
tttktkyfk t ,2 213
18. Initial Value Problems
Mohammad Tawfik
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Homework #4
• Solve:
• Given y(0)=1
1. Analytically
2. Using Euler method until t=2, with t=0.5
3. Repeat part 2 using 2nd order RK method
4. Repeat part 2 using 3rd order RK method
5. Repeat parts 2 through 4 using t=0.25
6. Compare results of all parts above
yyt
dt
dy
2.12