MATH704 Linear Partial Differential Equations.

Problem Solving Questionnaire 1 Due on Friday 30 April 2021, 4 pm

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Question  1.  1) Separate the PDE u_xx 3u_t + 2u = 0 into a system of ODEs.   线性偏微分方程代写

2) Find all product solutions of this equation.

Question 2. Find all eigenvalues and eigenfunctions of the following eigenvalue problem:

X^jj + λX = 0, X^j(0) = X(1) = 0.

Question 3. 1) The temperature u(x, t) of a narrow metal rod of length L = π is modelled by the heat equation:

u_t = 9u_xx.   线性偏微分方程代写

Solve the equation if the initial temperature u(x, 0) = 2 cos x + 3 cos(3x), and the ends of the rod are insulated:

u_x(0, t) = u_x(π, t) = 0, t > 0.

2) Plot the answer from 1) in MATLAB as a surface u = u(x, t) in three-dimensional space. Include in your solution the Matlab code and picture of the surface.

Question 4. 1) Solve the following initial-boundary value problem modelling the vibration of a string with length L = 3 and fixed ends.

2) Plot the answer from 1) in MATLAB as a surface u = u(x, t) in three-dimensional space. Include in your solution the Matlab code and picture of the surface.

Question 5. The velocity potential of an incompressible flow satisfies Laplace equation:   线性偏微分方程代写

u_xx + u_yy = 0.

Solve the equation with the following boundary conditions:

Question 6. The temperature u(x, t) of a narrow metal rod of length L = 1 is modelled by the heat equation:

u_t = 9u_xx

Solve the equation if the initial temperature u(x, 0) = 1, and the temperature at the ends is kept at 0 degrees:

u(0, t) = u(1, t) = 0, t > 0.    线性偏微分方程代写

Hint : find Fourier sine series for the function f (x) = 1 in the initial condition.

Question 7. Solve the following initialboundary value problem modelling the vibrationof a string with length L = 2 and fixed ends.

Question 8. The distribution of heat in a metal rod is modelled by the following non- homogeneous heat equation:

u_t = 2u_xx + 6 sin x.

Solve the equation if the initial temperature u(x, 0) = x, and the end temperatures are kept constant:

u(0, t) = u(π, t) = 0, t > 0.

Question 9. The following non-homogeneous heat equation    线性偏微分方程代写

u_t = 4u_xx + 2 − x

models the distribution of heat in a metal rod. Solve the equation if the initial temperature u(x, 0) = 2 − x, and the end temperatures are kept constant:

u(0, t) = u(2, t) = 0, t > 0.

Question 10. Solve the following initial-boundary value problem modelling the vibration of a string with length L = π and fixed ends.

u_tt = 4u_xx + x u

(x, 0) = 0

u_t(x, 0) = 3 sin x

u(0, t) = u(π, t) = 0, t > 0.

Question 11. The following non-homogeneous Laplace equation (Poisson equation) mod- els the distribution of electrical potential when an outside charge is present:   线性偏微分方程代写

u_xx + u_yy = 1 − x.

Solve the equation subject to the following boundary conditions:

u(x, 0) = u(x, 1) = 0,

u(0, y) = u(1, y) = 0.

Question 12. The temperature u(x, t) of a narrow metal rod of length L = π with a heat source is modelled by the following non-homogeneous heat equation:

u_t = u_xx + π − x.

Solve the equation if the initial temperature u(x, 0) = 2π + x + sin x, and the ends are kept at constant temperatures as follows:

u(0, t) = 2π, u(π, t) = 3π, t > 0.

Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for w = u − c₁x − c₂.