Given the model equation $$frac{partial u}{partial t} = alpha frac{partial^2 u}{partial x^2}$$ use the DuFort-Frankel method for this PDE and apply the von Neumann stability analysis. Your amplification factor should be in the form of a single equation, not a matrix. (That is, do not follow the procedure in the book.)
Added by Daniel B.
Close
Step 1
Given a model equation, we will use the DuFort-Frankel method to solve the PDE and then apply the von Neumann stability analysis to find the amplification factor in a single equation form. Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 57 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
3. Consider the uncontrolled system model as ș = 6x + u The performance is measured over time interval (0,1) as V = ∫[x² + 2u²] dt a. Determine the state-function of Pontryagin H. b. Determine the optimal input uₒ. c. Determine the equations governing the controlled system in terms of x and λ. d. Given that the solution for the equations in part(c) is [x(t) λ(t)] = [e⁶ᵗ 0.021(e⁻⁶ᵗ - e⁶ᵗ); 0.17(e⁻⁶ᵗ - e⁶ᵗ) e⁻⁶ᵗ] [x(0) λ(0)] find uₒ(t) for the boundary conditions x(0) = 1 and x(1) = 2.
Madhur L.
We would like to use the second-order Runge-Kutta method to compute an approximation yn+1 of the solution at time step tn+1, given yn, tn, and h. The formula for the approximation is: yn+1 = yn + h/2 * (f(tn, yn) + f(tn+1, yn + h * f(tn, yn))) where h = tn+1 - tn. By looking at the test problem y(t) = Ay(t), A < 0, 0 < t < 1, y(0) = 1, we can show that the above Runge-Kutta method is absolutely stable under the condition: h^2 * A^2 + h * A + 1 < 1. ii. For which A should the method be stable here? iii. How would you define h to ensure stability?
Sri K.
1. Find a PDE whose solution is z = f(x² + y²) 2. Find a PDE satisfied by z = f(x)g(y)
Ekaveera K.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD