Science:Math Exam Resources/Courses/MATH307/December 2012/Question 02 (a)

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MATH307 December 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q2 (a)  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q3 (e)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q4 (d)  •  Q4 (e)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  Q6 (f)  •  

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Question 02 (a)

The boundary value problem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \begin{align} &f''(x) + xf(x) = 1, \quad 0 < x < 1 \\ &f(0) = 1, \quad f'(1) = 1 \end{align} } can be approximated by an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle (N + 1) \times (N + 1)} system of linear equations of the form $(L+(\Delta x)^{2}Q)\mathbf {F} =\mathbf {b}$ (a) Write down L, Q, b and Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta x} when N = 4.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

With N = 4, we will consider f(x) at 5 evenly spaced values of x between 0 to 1, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x_{n} = \frac{n}{N},\ 0 < n < N} . This gives us 4 segments joining the points. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle f(x_{n})} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle 0<n<N} , we can approximate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle f′′(x_{n}) = \frac{(f(x_{n+1}) - f(x_{n})) - (f(x_{n}) - f(x_{n-1}))}{(\Delta x)^{2}}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle 0<n<N} This is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \frac{1}{(\Delta x)^{2}}\begin{bmatrix} 1 && -2 && 1 && 0 && 0 \\ 0 && 1 && -2 && 1 && 0 \\ 0 && 0 && 1 && -2 && 1 \end{bmatrix} \begin{bmatrix} f(x_{0})\\f(x_{1})\\ f(x_{2})\\f(x_{3})\\f(x_{4})\end{bmatrix}} Since we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle f(0) = 1} and $f'(1)=1$, we can add a row to this matrix to include these boundary conditions to get L Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle L = \frac{1}{(\Delta x)^{2}}\begin{bmatrix} (\Delta x)^{2} && 0 && 0 && 0 && 0 \\1 && -2 && 1 && 0 && 0 \\ 0 && 1 && -2 && 1 && 0 \\ 0 && 0 && 1 && -2 && 1 \\ 0 && 0 && 0 && -\Delta x&& \Delta x \end{bmatrix}} To find Q, we will consider the second half of the ODE: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle xf(x)} . We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle f''(x_{n})} for Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0<n<N} , so we will find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle x(f(x_n)} for these points. We can ignore n = 0 and n = N because here, we are considering their boundary conditions instead. Since in the equation, Q is multiplied by $(\Delta x)^{2}$, we divide by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle (\Delta x)^{2}} . Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q={\frac {1}{(\Delta x)^{2}}}{\begin{bmatrix}0&0&0&0&0\\0&\Delta x&0&0&0\\0&0&2\Delta x&0&0\\0&0&0&3\Delta x&0\\0&0&0&0&0\end{bmatrix}}={\frac {1}{(\Delta x)}}{\begin{bmatrix}0&0&0&0&0\\0&1&0&0&0\\0&0&2&0&0\\0&0&0&3&0\\0&0&0&0&0\end{bmatrix}}} We will combine L and Q to determine b Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle (L + (\Delta x)^{2}Q)\vec{F} = \vec{b}} We have arranged rows 1 to 3 of the matrices to correspond to the left hand side of the ODE when plugged into this equation, so the corresponding rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \vec{b}} will have entries equal to 1. Row 0 and 4 will have entries equal to their corresponding boundary conditions, so both are equal to one as well. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \vec{b} = \begin{bmatrix} 1\\1\\1\\1\\1\end{bmatrix}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \Delta x} is the spacing between the points. Since the points go from Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0<x<1} , are evenly spaced and we have 5 of them, Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta x={\frac {1}{4}}=0.25} We can plug Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \Delta x = 0.25} into our expressions for Q and L above to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle L =16\begin{bmatrix} \frac{1}{16} && 0 && 0 && 0 && 0 \\1 && -2 && 1 && 0 && 0 \\ 0 && 1 && -2 && 1 && 0 \\ 0 && 0 && 1 && -2 && 1 \\ 0 && 0 && 0 && \frac{-1}{4}&&\frac{1}{4} \end{bmatrix}} Q = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle 4\begin{bmatrix} 0 & 0 & 0 &0 &0 \\ 0& 1 & 0 & 0 &0 \\ 0 & 0& 2 & 0 & 0\\ 0 &0& 0& 3 & 0\\ 0 & 0& 0 &0& 0 \end{bmatrix}}

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