AE/ME 330 Spring 2013 Homework VI Due Friday, April

AE/ME 330 Spring 2013, Homework VI, Due Friday, April

Aeme 330 Spring 2013 Homework Vi Due Friday April 191 Use 3rd And

AE/ME 330 Spring 2013, Homework VI, Due Friday, April. Use 3rd and 6th degree Lagrange interpolation polynomials to approximate the function f(x) = sin(ex – 2) using equally spaced data on the interval 0 ≤ x ≤ 2. For each case, plot the exact function f(x), the approximation to the function by the Lagrange polynomials (fappx(x)), and the error (f(x) – fappx(x)) over the interval. Provide the error values at x=0.1, 0.9, 1.5, and 1.9.

Write a computer routine to approximate the function from Question 1 using a natural cubic spline with eleven equally spaced data points on [0, 2]. Plot the function, the spline approximation (fappx(x)), and the error across the interval. Include the error at x=0.1, 0.9, 1.5, and 1.9.

Derive the central finite difference formula (df/dx)_i = h (–f_{i+2} + 8f_{i+1} – 8f_{i–1} + f_{i–2}) to approximate the first derivative at xi. Show that its order of accuracy is O(h^4).

Use this finite difference formula to compute the first derivative of the function from Question 1 at xi=0.9 with h=10^–k for k=1 to 10. Calculate and report the absolute errors using the exact derivative value. Comment on the behavior of the errors. Then, using a first-order forward difference approximation, compute the derivative at the same point with the same h values and analyze the errors similarly.

For a non-uniform mesh spacing shown in a figure (not provided here), derive:

• (a) a second-order accurate one-sided finite difference formula for (df/dx) at x=x0.
• (b) a first-order accurate one-sided approximation for (d²f/dx²) at x=x0.

Show the derivations step by step and identify the leading term in the truncation error for each approximation.

Paper For Above instruction

The assignment encompasses multiple advanced topics in numerical analysis, including polynomial interpolation, spline approximation, finite difference formulas, and error analysis. These methods are essential tools for approximating derivatives and functions where exact solutions are difficult to obtain, particularly when dealing with real-world data and computer simulations.

Interpolation Using Lagrange Polynomials

The first part of the assignment involves approximating the function f(x) = sin(e^x – 2) using third and sixth-degree Lagrange interpolation polynomials. This approach constructs a polynomial passing through a set of equally spaced data points within the interval [0, 2]. The Lagrange interpolation polynomial for a set of n+1 points (x_i, y_i) is defined as:

   P_n(x) = Σ_{i=0}^n y_i * L_i(x),

where L_i(x) are the Lagrange basis polynomials. The accuracy of the interpolation depends on the degree of the polynomial and the distribution of data points. Higher-degree polynomials can approximate smooth functions more closely but may introduce Runge's phenomenon if not carefully managed.

Plotting the exact function alongside the interpolations reveals how well the polynomials approximate the function across the interval, while error analysis at specific points helps quantify local accuracy.

Spline Approximation

Next, the task involves implementing a natural cubic spline with eleven evenly spaced points. Cubic splines achieve smooth approximations by constructing piecewise cubic polynomials that ensure continuity of the function, its first derivative, and second derivative across subintervals, with boundary conditions forcing the second derivatives at the ends to zero. The spline’s smoothness often provides more accurate approximations than high-degree polynomials, especially with non-uniform data spacing.

Graphical and quantitative error analyses at selected points help illustrate the spline’s effectiveness in approximating the original function, offering insights into its local and global accuracy.

Finite Difference Derivative Formulas and Error Analysis

The derivation of the central finite difference formula given:

(df/dx)_i = h (–f_{i+2} + 8f_{i+1} – 8f_{i–1} + f_{i–2}),

aims to achieve a fourth-order accuracy, as demonstrated through Taylor series expansions. Showing this involves expanding each term about xi, then combining and simplifying to identify the leading error term, which confirms the O(h^4) accuracy.

Applying this finite difference for the derivative of the function at x=0.9 with decreasing h values illustrates how the error diminishes at a rate consistent with the theoretical order, provided the function behaves smoothly.

Contrastingly, the first-order forward difference approximation, which only considers f(xi) and f(xi+h), results in higher truncation errors for the same h values. Error analysis reveals the disparity in accuracy between higher and lower-order methods, especially as h becomes very small.

Plotting the errors on a log-log scale facilitates visualization of convergence rates and supports theoretical expectations.

Finite Difference Formulas on Non-Uniform Meshes

Finally, deriving one-sided finite difference formulas for non-uniform meshes involves solving approximation equations that incorporate variable spacing. For second derivatives or first derivatives at boundary points where data points are unevenly spaced, methods typically involve fitting polynomial interpolants locally and differentiating these polynomials to find suitable formulas.

The derivations aim to achieve specified orders of accuracy—second order for the first derivative and first order for the second derivative—by matching Taylor series expansions and canceling leading error terms. The leading truncation error corresponds to the next neglected term in the expansion, providing insight into the diminishing accuracy as the mesh spacing becomes unequal.

Conclusion

Overall, this assignment bridges fundamental concepts in numerical approximation. From polynomial interpolation and splines to finite difference derivatives and error analysis, it underscores the importance of understanding both the mathematical derivations and practical implications of numerical methods. Proper application and error evaluation ensure the development of accurate, reliable computational models for scientific and engineering problems.

References

• Atkinson, K. E. (2012). An Introduction to Numerical Analysis (2nd ed.). Springer.
• Burden, R. L., & Faires, J. D. (2010). Numerical Analysis (9th ed.). Brooks Cole.
• Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers (7th ed.). McGraw-Hill Education.
• Quarteroni, A., Sacco, R., & Saleri, F. (2007). Numerical Mathematics. Springer.
• Stoer, J., & Bulirsch, R. (2002). Introduction to Numerical Analysis. Springer.
• Weisstein, E. W. (n.d.). Lagrange Polynomial. Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/LagrangePolynomial.html
• Faires, J. D., & Patane, J. L. (2014). Finite Difference Methods. In Numerical Analysis and Scientific Computing (pp. 215–234). Cengage.
• Hamming, R. W. (1986). Numerical Methods for Scientists and Engineers. Dover Publications.
• Oden, J. T., & Demkowicz, L. (2011). Applied Functional Analysis. CRC Press.
• LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM.