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线性偏微分方程代写 MATH704代写

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线性偏微分方程代写

MATH704 Linear Partial Differential Equations.

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

线性偏微分方程代写 Start each question on a new page. Write thenumber of each question in the middle of a row.Do not copy the questions.

Attach this cover page to your questionnaire Do not attach any other covers

FAMILY NAME  
First

Name

线性偏微分方程代写
Student ID  
Lecturer’s Name Dr Farida Kachapova
Question Marks possible Marks given
1. 1) 2  
2) 2  
2 4  
3. 1) 6  
2) 3  
4. 1) 6  线性偏微分方程代写
2) 3  
5 6  
6 9  
7 9  
8 10  
9 10  

 线性偏微分方程代写

10 10  
11 10  
12 10  
Total 100  

Instructions: 线性偏微分方程代写

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Question  1.  1) Separate the PDE uxx 3ut + 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:

Xjj + λX = 0, Xj(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:

ut = 9uxx.   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:

ux(0, t) = ux(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.

线性偏微分方程代写
线性偏微分方程代写

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

uxx + uyy = 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:

ut = 9uxx

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:

ut = 2uxx + 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

ut = 4uxx + 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.

utt = 4uxx + x u

(x, 0) = 0

ut(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

uxx + uyy = 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:

ut = uxx + 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.

线性偏微分方程代写
线性偏微分方程代写

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