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

The boundary value problem can be approximated by an system of linear equations of the form (a) Write down L, Q, b and when N = 4. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/December 2012/Question 02 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. With N = 4, we will consider f(x) at 5 evenly spaced values of x between 0 to 1, where . This gives us 4 segments joining the points. For where , we can approximate Failed to parse (syntax error): {\displaystyle f′′(x_{n}) = \frac{(f(x_{n+1})  f(x_{n}))  (f(x_{n})  f(x_{n1}))}{(\Delta x)^{2}}} for This is equivalent to
Since we know that and , we can add a row to this matrix to include these boundary conditions to get L
To find Q, we will consider the second half of the ODE: . We have for , so we will find 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 , we divide by .
We will combine L and Q to determine 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 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.
is the spacing between the points. Since the points go from , are evenly spaced and we have 5 of them, We can plug into our expressions for Q and L above to get
Q = 