Find a formula for [ f(x)dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? auplinit Crank Nicholson and implicit
Find a formula for [ f(x)dx in terms of the functional values f(x₁) and f'(x2), which is exact for linear poly- nomials. For what values of x₁ or x2 is the discretization formula superaccurate? auplinit Crank Nicholson and implicit
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Numerical Integration and Interpolation
#### 3. Discretization and Superaccuracy
A discretization formula is called superaccurate when it uses fewer functional values \( f(x_i) \) than expected. Show that if \( x_1 = \frac{1}{2} \) in the first case in Problem 2, then \( f(x_2) \) does not have to be used. In this case, the discretization formula is called the Midpoint Rule. Find the values of \( x_i \) for which the other two formulas in Problem 2 are superaccurate.
#### 4. Formula Derivation
Find a formula for
\[ \int_0^1 f(x) \, dx \]
in terms of the functional values \( f(x_1) \) and \( f'(x_2) \), which is exact for linear polynomials. For what values of \( x_1 \) or \( x_2 \) is the discretization formula superaccurate?
#### 5. Functional Method and Heat Equation
Use the functional method to derive the explicit, Crank-Nicholson, and implicit schemes for the heat equations with vanishing BC. Show that the errors are \( O( \Delta t, (\Delta x)^2 ) \), \( O( (\Delta t)^2, (\Delta x)^2 ) \), respectively (use the interpolation theorem 10.1). You can do this by first holding \( x \) and then \( t \) fixed, explain why.
#### 6. Newton's Polynomial and Interpolation
Write a scheme to find the nth degree Newton's polynomial that solves the interpolation problem. Then write a scheme to implement the functional method to find the second derivative at a point. Let the accuracy be a choice. Make sure that your matrix is upper diagonal (this is why we use Newton's polynomial) so that you can solve the final system by backward substitution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c584eaa-2414-458e-b7ae-bfe37f5a3e56%2F513c164d-e60d-41c6-945b-59491bce1f16%2Fys2mi2e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Numerical Integration and Interpolation
#### 3. Discretization and Superaccuracy
A discretization formula is called superaccurate when it uses fewer functional values \( f(x_i) \) than expected. Show that if \( x_1 = \frac{1}{2} \) in the first case in Problem 2, then \( f(x_2) \) does not have to be used. In this case, the discretization formula is called the Midpoint Rule. Find the values of \( x_i \) for which the other two formulas in Problem 2 are superaccurate.
#### 4. Formula Derivation
Find a formula for
\[ \int_0^1 f(x) \, dx \]
in terms of the functional values \( f(x_1) \) and \( f'(x_2) \), which is exact for linear polynomials. For what values of \( x_1 \) or \( x_2 \) is the discretization formula superaccurate?
#### 5. Functional Method and Heat Equation
Use the functional method to derive the explicit, Crank-Nicholson, and implicit schemes for the heat equations with vanishing BC. Show that the errors are \( O( \Delta t, (\Delta x)^2 ) \), \( O( (\Delta t)^2, (\Delta x)^2 ) \), respectively (use the interpolation theorem 10.1). You can do this by first holding \( x \) and then \( t \) fixed, explain why.
#### 6. Newton's Polynomial and Interpolation
Write a scheme to find the nth degree Newton's polynomial that solves the interpolation problem. Then write a scheme to implement the functional method to find the second derivative at a point. Let the accuracy be a choice. Make sure that your matrix is upper diagonal (this is why we use Newton's polynomial) so that you can solve the final system by backward substitution.
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