Partial Differential Equations, 124

发布时间：2024-06-10

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Partial Differential Equations, 124

Problem 1:

Consider the Fourier Series expansion

(i) What are the Fourier Coefficients for ϕ(x) = x on the interval [−1, 1]? Hint: Integration by parts may be useful.

(ii) What are the Fourier Coefficients for ϕ(x) = 2sin(−6πx) + 3cos(4πx) on the interval [−1, 1]? Hint: Euler’s Identity may be useful.

(iii) What are the Fourier Coefficients for ϕ(x) = −x+2cos(4πx)+1 on the interval [−1, 1]?

Problem 2:

Consider the Fourier Series expansion

(i) What are the Fourier Coefficients for ϕ(x) = x−1 on the interval [−1, 1]? Hint: Integra- tion by parts may be useful.

(ii) What are the Fourier Coefficients for ϕ(x) = cos(−6πx) on the interval [−1, 1]?

(iii) What are the Fourier Coefficients for ϕ(x) = sin(4πx) + x − 1 on the interval [−1, 1]?

Problem 3:

Consider the DFT/IDFT with the following discrete Fourier Expansions for values on a lattice

The function is sampled using um = u(xm ) for points xm = mL/n on a latice in the interval [0, L].

(i) What are the DFT Fourier Coefficients for the function u(xm ) = sin(4πxm /L) sampled on the interval [0, 2] when n = 5? Hint: Use Euler’s Identity and exp(i2πα) = 1 for α ∈ Z to compute ˆ(u)k for this function.

(ii) What is the Fourier Interpolation of the function on the interval [0 , 2] when n = 5? Hint: Use the centered fourier expansion and Euler’s Identity.

(iii) What are the DFT Fourier Coefficients for the function u(xm ) = sin(−6πxm /L) sampled on the interval [0, 2] when n = 5? Hint: Use Euler’s Identity and compute the DFT of the samples of this function.

(iv) Show that the functions sin(−6πxm /L) and sin(4πxm /L) are the same on the lattice xm on the interval [0, 2] when n = 5 and therefore have the same Fourier Coefficients. Hint: Use Euler’s Identity and that exp(i2πα) = 1 with α ∈ Z.

(v) What is the Fourier Interpolation of the function u(x) = sin(−6πxm /L) on the interval [0, 2] when n = 5? Is this the same or different from v(x) = sin(4πxm /L)? Why?

(vi) Use the Aliasing Formula to show the functions u(x) = cos(−6πxm /L) and v(x) = cos(4πxm /L) have the same Fourier Coefficients when sampled on the interval [0, 2] when n = 5? Hint: Use the continuous Fourier Expansions for these functions obtained from Euler’s Identity.

(vii) Use the Aliasing Formula to show the functions u(x) = 1 and v(x) = cos(10πxm /L) have the same Fourier Coefficients when sampled on the interval [0, 2] when n = 5? Hint: Use the continuous Fourier Expansions for these functions obtained from Euler’s Identity.

(viii) What values do you obtain on the lattice um when ˆ(u)k = 1/n for all k?

(ix) What function u(xm ) do you obtain on the lattice when n = 5 with ˆ(u)2 = 2/1, ˆ(u)3 = 2/1, and zero otherwise?

Problem 4:

Consider the diffusion equation on the interval with the Dirichlet Boundary Conditions:

Determine the solution when κ = 1, ℓ = π , ϕ(x) = 0 and j(t) = t, h(t) = −2t. Hint: Integration by parts may be useful.

Problem 5:

Consider the wave equation on the interval with the Dirichlet Boundary Conditions:

Determine the solution when κ = 1, ℓ = π , ϕ(x) = 5cos(2x) + 3cos(3x), ψ(x) = 0, and j(t) = 0, h(t) = 0.

Problem 6:

Consider the elliptic equation on a domain Ω with Dirichlet Boundary Conditions and Ω = [0,ℓ] × [0,ℓ]:

Determine the solution when ℓ = π , f(x,y) = 2sin(5x)sin(y), g(0, y) = g(x,π) = g(x, 0) = g(π, y) = 0.

Problem 7:

Consider the elliptic equation with Dirichlet Boundary Conditions on the disk Ω = {x | ∥x∥ ≤ a}:

(i) Determine the solution when a = 3, f = 0, h(θ) = 2sin(3θ) − 2cos(θ).

(ii) Determine the solution when a = 2, f = 0, h(θ) = −cos(5θ) + 1.

Problem 8:

Consider the elliptic equation with Dirichlet Boundary Conditions on the wedge Ω = {x = (r,θ) | α < θ < β, r < a}:

(i) Determine the solution when a = 3, α = 0, β = π/2, f = 0, h(θ) = 2sin(4θ) − 3sin(2θ).

(ii) Determine the solution when a = 2, α = 0, β = π/4, f = 0, h(θ) = −sin(8θ)+2sin(4θ).

Problem 9:

Consider approximating the Diffusion Equation using the finite difference methods below.

Perform von Neumann Analysis um(n) = gn exp(imθ) to analyze stability of the finite difference method. If there are no choices possible for δt,δ儿 then state the method is unstable.

(i) Determine the stability of the finite difference method

(ii) Determine the stability of the finite difference method

(iii) Determine the stability of the finite difference method

Problem 10:

Consider approximating the following hyperbolic PDE using the finite difference method below.

Perform von Neumann Analysis vm(n) = gn exp(imθ) to analyze stability of the finite difference method. If there are no choices possible for δt; δx then state the method is unstable.

(i) Determine the stability of the finite difference method

(ii) Determine the stability of the finite difference method

(iii) Determine the stability of the finite difference method