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Question 1 15 Points) It is always desirable to have/ use the finite difference approximation wit...
1. U se Taylors formula to derive the forward, backward and center difference for- mulas for the derivative /"(x) a...
1. U se Taylors formula to derive the forward, backward and center difference for- mulas for the derivative /"(x) at a point x Use the reminder in Taylors formula to determine the order (truncation error) of the numerical approximation of the derivative in each case.
1. U se Taylors formula to derive the forward, backward and center difference for- mulas for the derivative /"(x) at a point x Use the reminder in Taylors formula to determine the order (truncation error)...
Use forward and backward difference approximations of O(h)
Use forward and backward difference approximations of O(h) and a centered difference approximation of O(h2) to estimate the first derivative of the following function:f(x)=25x³-6x²+7x-88Evaluate the derivative at x=2 using a step size of h=0.25. Compare your results with the true value of the derivative. Interpret your results on the basis of the remainder term of the Taylor series expansion.
5. Following an approach similar to what was performed in lecture for the first forward finite-divided-difference...
5. Following an approach similar to what was performed in lecture for the first forward finite-divided-difference equation with local truncation error Oſhº), please derive the following expression for the first backward finite-divided-difference equation with location truncation error O(h?). f'(x) – 3 f(xi) – 4 f(xi-1) + f(xi-2) 2 h
3. A five-point centered finite difference approximation to the first derivative is given by - f...
3. A five-point centered finite difference approximation to the first derivative is given by - f (x + 2h) +8f (x + h) – 8f (x – h) + f(x – 2h) 12h (a) What is the error term associated with this formula? (b) Numerically verify the order of approximation using f(x) = (1 + x2)-1 at x = 1 using the values h = 0.1, 0.01, 0.001.
4. Higher order method via higher order finite difference formula 4. Higher order method via higher...
4. Higher order method via higher order finite difference
formula
4. Higher order method via higher order finite difference formula 1. Prove the finite difference formula 2. Use this finite difference formula to derive a numerical method to solve the ODE y' = f(y,t), y(0) = 10. 3. What is the local truncation error of this method?
For the following set of data, calculate the derivative using the higher order finite-difference approximations for...
For the following set of data, calculate the derivative using the higher order finite-difference approximations for each data point, as shown in Figures 21.43-21.5. Round your answers to 2 decimal places, if needed. 0 0.5 1.0 1.5 2.0 2.5 X f(x) 33 72 80 10 25 58 Using the forward finite-difference approximation: f'(0) ~ f'(0.5) Using the centered finite-difference approximation: f'(1.0) f'(1.5) Using the backward finite-difference approximation: f'(2.0) f'(2.5) 은 8
(b) Derive a numerical differentiation formula of order O(h4) by applying Richardson ertrapolation to 2 MI...
(b) Derive a numerical differentiation formula of order O(h4) by applying Richardson ertrapolation to 2 MI 2h 120 Richardson extrapolation is a useful way to obtain higher precision from either an approximate formula or a code, provided you know precisely what the truncation error of the approximation is. The truncation error is of order h2 above. So what we will do is use the above formula over rth/2 to derive another approximation for the first derivative. The add/subtract these two...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation for u'(E) using the values u(0),u (1) and u(2). Choose the coefficients such that the formula is exact for polynomials with degree less or equal to 2. Can you use these ecoefficients to get an approximation for a first derivative based on function values v(r),v(x+h)and v(x +2h)? At which point z and for which functions v(a) is this approximation equal to '()? Determine the...
✓ 4. Derive an order h2 accurate one-sided finite difference approximation to the first derivative dø/dx...
✓ 4. Derive an order h2 accurate one-sided finite difference approximation to the first derivative dø/dx at the point x = tj. Suppose that the numerical solution di is available at the set of points {C;} and assume constant grid spacing, i.e. h = i+1 - Ti is the same for all i. The derivative must be calculated using ; and only points to its left (i.e. Xj, 2j-1, *;-2, ... use as many as you need). Hint: assume that...
1. U se Taylors formula to derive the forward, backward and center difference for- mulas for the derivative /"(x) at a point x Use the reminder in Taylors formula to determine the order (truncation error) of the numerical approximation of the derivative in each case.
1. U se Taylors formula to derive the forward, backward and center difference for- mulas for the derivative /"(x) at a point x Use the reminder in Taylors formula to determine the order (truncation error)...
5. Following an approach similar to what was performed in lecture for the first forward finite-divided-difference equation with local truncation error Oſhº), please derive the following expression for the first backward finite-divided-difference equation with location truncation error O(h?). f'(x) – 3 f(xi) – 4 f(xi-1) + f(xi-2) 2 h
3. A five-point centered finite difference approximation to the first derivative is given by - f (x + 2h) +8f (x + h) – 8f (x – h) + f(x – 2h) 12h (a) What is the error term associated with this formula? (b) Numerically verify the order of approximation using f(x) = (1 + x2)-1 at x = 1 using the values h = 0.1, 0.01, 0.001.
4. Higher order method via higher order finite difference
formula
4. Higher order method via higher order finite difference formula 1. Prove the finite difference formula 2. Use this finite difference formula to derive a numerical method to solve the ODE y' = f(y,t), y(0) = 10. 3. What is the local truncation error of this method?
For the following set of data, calculate the derivative using the higher order finite-difference approximations for each data point, as shown in Figures 21.43-21.5. Round your answers to 2 decimal places, if needed. 0 0.5 1.0 1.5 2.0 2.5 X f(x) 33 72 80 10 25 58 Using the forward finite-difference approximation: f'(0) ~ f'(0.5) Using the centered finite-difference approximation: f'(1.0) f'(1.5) Using the backward finite-difference approximation: f'(2.0) f'(2.5) 은 8
(b) Derive a numerical differentiation formula of order O(h4) by applying Richardson ertrapolation to 2 MI 2h 120 Richardson extrapolation is a useful way to obtain higher precision from either an approximate formula or a code, provided you know precisely what the truncation error of the approximation is. The truncation error is of order h2 above. So what we will do is use the above formula over rth/2 to derive another approximation for the first derivative. The add/subtract these two...
Calculate the first nonzero term in the Taylor series of the truncation error Tr(h) for the finite difference formula defined by the second row of Table 5.2. Table 5.2. Weights for forward finite difference formulas (p 0 in (5.4.2). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q=0. The term order of accuracy is explained in Section 5.5. Order of Node location 2h 3h 4h accuracy 1 2...
1. Use the method of undetermined coefficients to compute the coefficients of a finite difference approximation for u'(E) using the values u(0),u (1) and u(2). Choose the coefficients such that the formula is exact for polynomials with degree less or equal to 2. Can you use these ecoefficients to get an approximation for a first derivative based on function values v(r),v(x+h)and v(x +2h)? At which point z and for which functions v(a) is this approximation equal to '()? Determine the...
✓ 4. Derive an order h2 accurate one-sided finite difference approximation to the first derivative dø/dx at the point x = tj. Suppose that the numerical solution di is available at the set of points {C;} and assume constant grid spacing, i.e. h = i+1 - Ti is the same for all i. The derivative must be calculated using ; and only points to its left (i.e. Xj, 2j-1, *;-2, ... use as many as you need). Hint: assume that...
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