User:Bchethan

Simple Exercise
What is x2+2x+1 if x=1? Solution: 4

Sample Quiz
{What is Symmetric matrix + A matrix whose transpose matrix is same as the original matrix. - A matrix whose diagonal elements are non zero and other elements are zero - A zero matrix - None of those
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{ The determinant of transpose of $$\begin{bmatrix} 2&-1\\0&4\end{bmatrix}$$ is { 8_2 }.
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$$ \displaystyle \ y_1 = $$ { 4 } $$ \displaystyle \ y_2 = $$ { -1 } $$ \displaystyle \ y_3 = $$ { 6 }

Next, we have Ux = y

$$ \left[\begin{array}{c c c}3 & 4 &2\\2 & 1 & 2\\3 & 4 & 1\end{array} \right]$$ X $$\left[\begin{array}{c}x_1\\x_2\\x_3\end{array} \right] $$ = $$\begin{bmatrix} y_1\\y_2\\y_3\end{bmatrix} $$

Use backward substitution we have:

$$ \displaystyle \ x_1 = $$ { 1 } $$ \displaystyle \ x_2 = $$ { 1 } $$ \displaystyle \ x_3 = $$ { 1 }

Final Project
1)Solve differential equation y' = 3t2y in [0,1] using Euler's method with n=10,y(0)=1. Solution: We need to find the solution ODE y' = 3t2y using Euler's method.

yn+1=yn+hf(tn,yn)

We divide time span with number points to find the step size h.

h=tmax-tmin/n=1-0/10=0.1

y1=y0+hf(t0,y0)

=1.000000+.1x3x02x1

=1.000000

y2=y1+hf(t1,y1)

=1.000000+.1x3x0.12x1.000000

=1.003000

y3=y2+hf(t2,y2)

=1.003000+.1x3x0.22x1.003000

=1.0150360

y4=y3+hf(t3,y3)

=1.01503600+.1x3x0.32x1.0150360

=1.0424420

y5=y4+hf(t4,y4)

=1.0424420+.1x3x0.42x1.0424420

=1.0924792

y6=y5+hf(t5,y5)

=1.0924792+.1x3x0.52x1.0924792

=1.1744151

y7=y6+hf(t6,y6)

=1.1744151+.1x3x0.62x1.1744151

=1.3012520

y8=y7+hf(t7,y7)

=1.3012520+.1x3x0.72x1.3012520

=1.4925360

y9=y8+hf(t8,y8)

=1.4925360+.1x3x0.82x1.4925360

=1.7791029

y10=y9+hf(t9,y9)

=1.77910290+.1x3x0.92x1.7791029

=2.2114249

2)Solve differential equation y' = 2ty in [0,1] using RK2 method yn+1=yn+h/4(k1+3K2) where k1=f(tn,yn),k2=f(tn+2/3h,yn+2/3hk1)with n=5,y(0)=1.

Solution: We need to find the solution ODE y' = 2ty using RK second order method.

yn+1=yn+h/4(k1+3K2)

k1=f(tn,yn)

k2=f(tn+2/3h,yn+2/3hk1)

y1=y0+h/4(k1+3K2)

k1=f(t0,y0)

=2x.0x1

=0

k2=f(t0+2/3h,y0+2/3hk1)

=2x2/3x.2x1

=.26667

y1=y0+h/4(k1+3K2)

=1+.2/4(0+3x.26667)

=1.04

y2=y1+h/4(k1+3K2)

k1=f(t1,y1)

=2x.2x1.04

=.416

k2=f(t1+2/3h,y1+2/3hk1)

=f(.2+.13333,1.04+.0554667)

=.730304

y2=y1+h/4(k1+3K2)

=1.04+.2/4(0.416+3x.730304)

=1.17035

y3=y2+h/4(k1+3K2)

k1=f(t2,y2)

=2x.4x1.17035

=.93648

k2=f(t2+2/3h,y2+2/3hk1)

=.730304

y2=y1+h/4(k1+3K2)

=1.04+.2/4(0.416+3x.730304)

=1.17035

3)Solve differential equation y' = 2ty in [0,1] using Two step Adams Bashforth method with n=5,y(0)=1.

Solution: We need to find the solution ODE y' = 2ty using Two step Adams Bashforth method.

yn+2 = yn+1 + 3/2 x hf(tn+1,yn+1)-1/2 x hf(tn,yn)

Calculating h=1-0/5=0.2

We need two approxmations for calculating y2

Calculating y1 using RK second order method.

y1=1.04

y2 = y1 + 3/2 x hf(t1,y1)-1/2 x hf(t0,y0)

=1.04+3/2(.2)f(0.2,1.04)-1/2(0.2)f(0.0,1)

=1.1648

y3 = y2 + 3/2 x hf(t2,y2)-1/2 x hf(t1,y1)

=1.1648+3/2(.2)f(0.4,1.1648)-1/2(0.2)f(0.2,1.04)

=1.40275

y4 = y3 + 3/2 x hf(t2,y2)-1/2 x hf(t2,y2)

=1.40275+3/2(.2)f(0.6,1.40275)-1/2(0.2)f(0.4,1.1648)

=1.81456

y5 = y4 + 3/2 x hf(t4,y4)-1/2 x hf(t3,y3)

=1.81456+3/2(.2)f(0.8,1.81456)-1/2(0.2)f(0.6,1.40275)

=2.21333

y6 = y5 + 3/2 x hf(t5,y5)-1/2 x hf(t4,y4)

=2.21333+3/2(.2)f(1,2.21333)-1/2(0.2)f(0.8,1.81456)

=3.25098

Introduction
My final project is about adding exercises for different ODE methods to Wikiversity page.I have added exercises for Euler method,Multistep method and Runge Kutta method.

Initial Experience
Prior to the final project I worked on small project as part of my homework.That was adding material to wikipidea page on Taylor's Approximation.I have gone through many books and decided to add few points to the existing information.I was corrected by Dr.Mohlenkamp on few of issues, after that I have modified the changes accordingly.After that I have realized that the changes were not required.So i have done more work for my final project and made sure that I won't repeat previous mistakes.

Motivation
I found useful material  on Wikipedia related to ODE methods but there were no solutions for examples.So I thought it will be useful to put some exercises and solution of it completely.I have gone through books and with the inspiration of it wrote few problems which are very easy but gives clear idea of method and solved it step by step.

Changes
I have added Wikipedia pages on ODE methods to Numerical Analysis Wikiversity page and after that created exercise for Euler Method, Multistep Method and Runge Kutta method.If the user is not able to get solution they can refer solution which covered each step in detail.I have also created quiz which covers error order of ODE methods and stability of ODE methods.

Reference
Guide to Numerical Analysis by Peter R. Turner