User:EML5526.S11.Team3.risher/homework2

Problem 2.1
From notes 6-2: Derive heat problem in 1-D, i.e, (1) p.4-2

$$ \frac{\partial}{\partial x} \left[ A(x)k(x)\frac{\partial u}{\partial x} \right] + f(x,t) = A \rho c \frac{\partial u}{\partial x} $$

$$ m = \rho A $$

Problem 2.2
From notes 7-3 and 7-4:

1.) Find $$det \left[ b_{jk} \right]$$

2.) Find $$ \underline{ \Gamma } (\underline{b}_{1},\underline{b}_{2},\underline{b}_{3}) = \underline{K}, det \underline{ \Gamma }$$

3.) $$ \underline{F} = \left[ F_{i} \right] = \left[ \underline{b}_{i} * \underline{v} \right]   $$

4.) Solve (5) on p. 7-2 for $$ \underline{d} = \left[ v_{j} \right] $$

(5) on p. 7-2 : $$ \underline{K} \underline{d} = \underline{F} $$

5.) Use $$ \underline{w}_{i} * \underline{P} (\underline{v}) = 0 $$, $$ \forall_{i} = 1, ..., n $$ to find $$ \bar { \underline{K}} \underline{d} = \bar{ \underline{F}} $$

What is $$ \bar{ \underline{K} } $$ and $$ \bar{ \underline{F} } $$ ? Note: $$ \underline{d} = \left[ v_{j} \right] $$

6.) Solve for $$ \underline{d} $$; compare to $$ \underline{d} $$ in part 4.)

7.) Observe symetric properties of $$ \underline{K} $$ and $$ \bar{ \underline{K} } $$

Discuss pros and cons of the 2 methods

Note: (5) p. 7-2 → Bubnov-Galerkin

$$ \underline{K} \underline{d} = \underline{F} $$

Note: (2) p. 7-4 → Petrov-Galerkin

$$ \bar { \underline{K}} \underline{d} = \bar{ \underline{F}} $$

Problem 2.3
From notes 8-3:

Using $$ \left[ \underline{a}_{i} \right] $$ (basis), let $$ \underline{w} = \Sigma_{i} \beta_{i} \underline{a}_{i} $$

Then (1) p. 8-1 is equivalent to:

$$ \underline{w} * \underline{P} (\underline{v}) = 0 $$

$$ \forall \left[ \beta_{1}, ..., \beta_{n} \right] \in \mathbb{R}^n$$ at $$ \underline{w} = \Sigma_{i} \beta_{i} \underline{a}_{u} $$

Show that $$ \underline{w} * \underline{P} (\underline{v}) = 0 $$ is equivalent to:

$$ \underline{a}_{i} * \underline{P} (\underline{v}) = 0 $$

$$ \forall_{i} = 1, ..., n $$

Problem 2.4
From notes 9-2 and 10-1:

Show $$ \int_{} \frac{x^2}{1+x}dx = \frac{x^2}{2} - x +log(1+x) + k $$

1.) $$ \int_{} log(x)dx = xlog(x) - x $$ (HINT: Integration by parts)

2.) $$ \int_{} xlog(x)dx = \frac{1}{2}x^2 \left[ log(x) -\frac{1}{2} \right] $$

3.) Find $$ \int_{} \frac{x^2 dx}{1+cx} $$ (in particular c=1)

4.) find $$ \int_{} \frac{x^2 dx}{a+bx} $$

5.) Find exact solution u(x) for Pb. (3) p. 9-2

$$ \frac{\partial}{\partial x} \left[ (2+3x)\frac{\partial u}{\partial x} + 5x = 0 \right] $$ $$ \forall x \in ] 0,1 [ $$ $$ u(1)=4, -\frac{\partial u(x=0)}{\partial x} =6 $$

6.) Plot u(x)

Note: Use WolframAlpha to help check and solve some problems (very helpful tool) http://www.wolframalpha.com/

e.g., (debt usa)/(gdp usa)

integrate log(x), etc

Problem 2.5
From notes 10-1:

Complete part B of proof of (3) p. 8-2, i.e., show (2) p. 7-2 → (2) p. 8-2

(2) p. 7-2: $$ \underline{b}_{i} * \underline{P} (\underline{v}) = 0 $$ $$ i=1, ..., n $$

(2) p. 8-2:

$$ \underline{w} * \underline{P} (\underline{v}) = 0 $$ $$ \forall \left[ \alpha_{1}, ..., \alpha_{n} \right] \in \mathbb{R}^n$$ at $$ \underline{w} = \Sigma_{i} \alpha_{i} \underline{b}_{u} $$

Problem 2.6
From notes 10-2 and 10-3

Consider $$ F = \left[ 1, cos(iwx), sin(iwx) \right] $$ on interval [0,T], i.e., i = 1,2

$$ \left[ 1, cos(wx), cos(2wx), sin(wx), sin(wx), sin(2wx) \right] = \left[ b_{1}(x), b_{2}(x), b_{3}(x), b_{4}(x), b_{5}(x) \right] $$

1.) Constr. $$ \underline{\Gamma}(F) $$; obs. prop. of $$ \underline{\Gamma} $$

2.) Find $$ det \underline{\Gamma}(F) $$

3.) Conclude F is orthogonal basis, i.e., $$ \Gamma_{ij} = {<}b_{i},b_{j}{>} =\delta_{ij} $$ $$ \delta_{ij} = 1  $$ for  i = j   $$ \delta_{ij} = 0  $$ for  i (not equal to) j

Problem 2.7
From notes 10-3

Similarly for $$ F = [1, x , x^2 , x^3 , x^4] $$

$$ \Omega = \left[ 0,1 \right]$$ in Part 3.) of Problem 2.6

Is F orthog. family?

Problem 2.8
From notes 11-1:

Show (1) p. 10-4 ←→ (2) p. 10-4

Problem 2.9
From notes 12-1

My Solutions
Solutions for homework #2 by Risher

Risher's WikiHome
Go to Risher's WikiHome