1 CAAM 453: Numerical Analysis I Problem Set 5 Due: October 24st 2014 by 9am (beginning of class). Note: All M ATLAB functions mentioned in this homework assignment can be found on the CAAM453 homepage, or come with M ATLAB . You can use the M ATLAB codes posted on the CAAM453 web-page. If you modify these codes, please turn in the modified code. Otherwise you do not have to turn in printouts of the codes posted on the CAAM453 web-page. Turn in all M ATLAB code that you have written and turn in all output generated by your M ATLAB functions/scripts. M ATLAB functions/scripts must be commented, output must be formatted nicely, and plots must be labeled. Problem 1 (40 points) Legendre Polynomials Pn (x) ∈ Pn are solutions of the Legendre equation: d 2 d (1 − x ) Pn (x) + n(n + 1)Pn (x) = 0. dx dx (1) Each Legendre polynomial Pn (x) is an nth-degree polynomial. It may be expressed using Rodrigues’ formula: 1 dn 2 n Pn (x) = n (x − 1) . (2) 2 n! dxn You are going to demonstrate that those Pn ’s are (to a constant) the orthogonal polynomials we discussed in class. • Show that the Pn ’s defined through Rodrigue’s formula (2) satisfy (1). • Show that Legendre Polynomials Pn (x) can be computed using the following recursion formula (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) with P0 = 1 and P1 = x. • Show that 0 0 Pn+1 (x) − Pn−1 (x) = (2n + 1)Pn (x). • Show that Pn (1) = 1, Pn (−1) = (−1)n , Pn (−x) = (−1)n Pn (x). • Orthogonality: show that Z 1 −1 Pn (x)Pm (x)dx = 2 δmn . 2n + 1 For m 6= n, use the fact that both Pn and Pm satisfy (1). For m = n, use Rodrigue’s formula (2). 2 • Compute Z 1 0 Pn (x)dx. • Compute Z 1 −1 Pn0 (x)Pm (x)dx. • Show that P2n+1 (0) = 0 and P2n (0) = (−1)n (2n)! 22n (n!)2 . • In one single MATLAB figure, draw Pn (x), x ∈ [−1, 1], for n = 0, . . . , 5. Problem 2 (30 points) The steady temperature distribution T (x, y, z) inside the sphere of radius a satisfies the Laplace equation ∂2 T ∂2 T ∂2 T ∇2 T = 2 + 2 + 2 = 0. ∂x ∂y ∂z Spherical coordinates are defined as x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, r ∈ [0, a], θ ∈ [0, π] and ϕ ∈ [0, 2π[. Laplace equation can be written in spherical coordinates as 2 ∂2 ∂ 1 ∂ 1 2 ∂ ∂ 2 ∇ T (r, θ, ϕ) = T + sin θ T + + T. ∂r2 r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂ϕ2 • Assume that the temperature inside the sphere is independant of ϕ. In this case, the method of separation of variables allows us to write T as a series of Legendre polynomials ∞ Bn n T (r, θ) = ∑ An r + n+1 Pn (cos θ). (3) r n=0 In (3), An and Bn are parameters. Equation (3) is the solution of Laplace equation for any choice of An and Bn . If we heat the surface of the sphere i.e. we impose a fixed distribution of temperature on its surface: T (a, θ) = f (θ), (4) 3 find the unique set of coefficients An and Bn for which the solution (3) on r = a and the boundary condition (4) match. Hint: T should remain finite for r = 0. Hint: posing x = cos θ, the orthogonality relation of Legendre polynomials transforms to Z 1 Z 0 2 δmn . Pm (x)Pn (x)dx = Pm (cos θ)Pn (cos θ)(− sin θ)dθ = 2n + 1 −1 π Coefficients An can then be obtained computing Z 0 Z 0 T (a, θ)Pm (cos θ)(− sin θ)dθ = π f (θ)Pm (cos θ)(− sin θ)dθ π • Let us heat the north hemisphere and leave the southern half cold. So, f (θ) = 1 for 0 ≤ θ ≤ π/2 and f (θ) = 0 for π/2 < θ ≤ π. Compute explicitely T (r, θ). For that, use some of the results demonstrated in problem 1. • Calculate the temperature at the center of the sphere. • Using MATLAB, produce color plots of the temperature T in the r, θ plane using a finite number of modes, i.e. using equation (3) with n = 1, . . . , N. Use N = 2, N = 10 and N = 100 modes. Problem 3 (30 points) this problem is pledged Write a MATLAB program that computes a polynomial interpolation at order n of a function f using n + 1 interpolation points {x j }nj=0 . The calling sequence should be p = polynomial_interp(x,f); Input: x is a vector of size n+1 that contains interpolation points f is a handle to the function Output: p is a vector of size n+1 that contains the coefficents of the nth order polynomial that interpolates f at x 4 You are allowed to use the in-house Matlab vander function (type help vander for help in Matlab interface) and the backslash command to solve linear systems. Consider the now famous Runge function f (x) = 1 . 1 + x2 Compute a polynomial interpolant to Runge’s function at order n in the range x ∈ [−5, 5] using the following systems of interpolation points: • Use n + 1 equi-spaced points, • Use the n + 1 roots of the degree-(n+1) Chebyshev polynomial, • Use the n − 1 roots of the degree-(n-1) Legendre polynomial, plus the two endpoints of the interval. Compare the three interpolants and the true function graphically by superimposing the graphs of the three polynomial interpolants and f on the same plot (be sure to include a legend). Use n = 11 and then n = 21 (one plot for each n). Then, compute k f − pn kL∞ [−5,5] and report that value on a graph for n = 1, . . . , 21 using the three systems of interpolation points.