﻿ 线性偏微分方程代写 MATH704代写 - 作业代写, 数学代写

# 线性偏微分方程代写 MATH704代写

2021-10-25 09:43 星期一 所属： 作业代写 浏览：34 ## Problem Solving Questionnaire 1 Due on Friday 30 April 2021, 4pm

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 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

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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.

### Question5.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. 