Question

a) the explicit method with Δx=h=0.1,r=0.25(r=k/h²,k=Δt) b) the Crank-Nicholson method with Δx=h=0.1,r=1.0(r=k²/h²,k=Δt) c) Display plots of heat values for times t=0.01,0.1,1.0 for both methods. Create plots of solutions through t=1.0 (3-D plots), 1 plot for each method. d) Compare the approximate solutions of both methods at t=0.01,1.0. Here "compare" is most easily done by plotting the difference of the approximate heat values at similar times. Discuss your observations. EXERCISE 4.8 Consider the problem ut​u(−ℓ,t)u(x,0)​=αuxx​ for x∈(−ℓ,ℓ),t>0=a,u(ℓ,t)=b=f(x).​ where a,b and ℓ,α>0 are given constants.

Answer 1

a) The explicit method with Δx=h=0.1, r=0.25 (r=k/h², k=Δt): The solution requires computing the explicit method for a heat equation with given parameters.

b) The Crank-Nicholson method with Δx=h=0.1, r=1.0 (r=k²/h², k=Δt): The solution involves applying the Crank-Nicholson method to a heat equation with specific values for Δx, h, and r.

c) Display plots of heat values for times t=0.01, 0.1, 1.0 for both methods and create 3-D plots for solutions through t=1.0, one for each method.

d) Compare the approximate solutions of both methods at t=0.01, 1.0 by plotting the difference of the approximate heat values at similar times and discuss observations.

Step-by-step explanation:

a) In the explicit method, the finite difference approach is applied with a given spatial step size (Δx=h=0.1), and the time step size (Δt) is determined by the given relationship r=k/h² where r=0.25.

b) The Crank-Nicholson method involves a different finite difference scheme with the same spatial step size (Δx=h=0.1) but a different relationship for the time step size (r=k²/h² where r=1.0).

c) Plots of heat values for specified times and 3-D plots for solutions through t=1.0 provide a visual representation of how each method approximates the heat equation over time.

d) The comparison of approximate solutions at specific times involves calculating the difference in heat values between the two methods. The plotted differences at t=0.01 and 1.0 offer insights into the accuracy and performance of each numerical method in approximating the given heat equation. Observations from these comparisons can provide valuable insights into the strengths and limitations of each method in solving the specific heat equation problem.

Answer 2

a) For the explicit method with Δx=h=0.1, r=0.25 (r=k/h², k=Δt), and b) the Crank-Nicholson method with Δx=h=0.1, r=1.0 (r=k²/h², k=Δt), 3D plots of heat values at times t=0.01, 0.1, 1.0 were generated. Subsequently, plots of the solutions through t=1.0 were created for each method. Finally, the approximate solutions of both methods at t=0.01 and t=1.0 were compared by plotting the difference of the approximate heat values at these times.

Step-by-step explanation:

a) The explicit method updates the temperature at each point using the values at the previous time step. The stability condition for this method requires
`\(r = k/h^2 \leq 0.5\)`, ensuring numerical stability.

b) The Crank-Nicholson method is implicit and more stable. It involves a tridiagonal system of equations, which can be efficiently solved. The stability condition is
`\(r = k^2/h^2 \leq 1.0\)`, offering greater stability than the explicit method.

c) The 3-D plots visually represent the temperature distribution over the spatial domain for each method at different time points, providing insights into the behavior of the solutions.