numerical analysis with mathematica wolfram (hw6)
Numerical Methods (Math 385/685) Homework 6 May 6, 2019 Due: Sunday, May 19th Problems: 1. Prove Theorem 3.4.2 by following the given outline: (a) Determine the maxt∈[0,1] |pi(t)| for the four b-spline basis func- tions. (b) Verify that ∑4 i=1 pi(t) = 1 (c) Let σi = (σi,1(t), σi,2(t)) be a B-spline segment for σp and take t0 ∈ [0, 1]. Prove that if x0 = σi,1(t0), then |f(x0)− σi,2(t0)| ≤ 3∑ k=0 |pk(t0)||f(x0)− f(xi+k)|. (d) Complete the proof by identifying a constant C depending on f that satisfies |f(x0)− σi,2(t0)| ≤ C∆ where ∆ = maxi(xi+1 − xi) 2. Let f(x) = xe−x−1 and set the interval to [1, 4]. Compute the integral of f using the two point Gaussian quadrature. Compare these results with those obtained by Trapezoidal method in Homework 5. 3. The growth of a tumor can be modeled by the following ordinary differential equation dU dt = aUα − bUβ 1 where α and β are chosen with respect to the tumor geometry,i.e Uα is the subpopulation of dividing cells and Uβ is the aging sub- population. The mitosis rate and death rate are give by a and b. A tumor initially consisting of 10 cells has the following parameters, a = 5, b = 0.01, α = 1 and β = 2. Estimate the number of cells present after 10 iterations with a time step of 0.1 and (a) Forward Euler (b) A single corrector (c) Midpoint scheme. (d) Calculate the absolute error at the 10th iteration of your predic- tion in each case (a-c) using the exact solution U(t) = 5000e 5t 500+10(e5t−1) . Discuss the accuracy of each scheme. 4. Consider the first order wave equation ∂u ∂t = α ∂u ∂x (a) Using forward difference in time and forward difference in space, render this PDE in finite difference format. In this case, it is usual to write C = α∆t∆x . C is called the Courant number. (b) Execute FDM for the following setting. α = −1/2 and α = 1/2, Region : [0, 300], Time interval : [0, 5]. Initial conditions: u(x, 0) = { 0 if 0 ≤ x ≤ 50 or 110 ≤ x ≤ 300 100 sin[π (x−50)60 ] if 50 ≤ x ≤ 110 and boundary condition: u(0, t) = 0 and u(300, t) = 0. Set ∆x = 5,∆t = 0.00015 (c) Use B-splines to display the output at times 1,3 and 5. (d) Discuss the quality of your solutions for the two different values of α at t = 5 5. Consider the wave equation in problem 3, 2 (a) Execute Neumann stability analysis for α < 0="" and="" α=""> 0 (b) What is the difference between the two cases and how does it compare with the computed solutions from problem 4. Graduate Students (a) Prove that the Crank Nicolson FDM applied to the 1D heat equa- tion is unconditionally stable. (b) use the discrete Fourier interpolation to get expressions for |cn+1k | |cn+1/2k | , |cn+1/2k | |cnk | (c) Multiply the two expressions to get an expression for |cn+1k | |cnk | 3