PARTIAL DIFFERENTIAL EQUATIONS AND VECTOR CALCULUS B

偏微分方程代写

Answer THREE of the FOUR questions.

If more than three questions are attempted, then credit will be given for the FIRST THREE answers.

Answer THREE of the FOUR questions.

If more than three questions are attempted, then credit will be given for the FIRST THREE answers.

Electronic calculators may be used, provided that they cannot store text.

1. 偏微分方程代写

a) Showthat the eigenvalues of the eigenvalue problem

X′′ (x) − λX(x) = 0,< x < 1,

(where X′′ (x) denotes the second derivative of X(x) with respect to x), subject to the boundary condition

X(0) = 0, Xr(1) = 0,

are

偏微分方程代写
偏微分方程代写

Find the corresponding eigenfunctions Xn(x), n = 0, 1, 2, . . .

[8 marks]

 

b) Using the method of separation of variables, and your answer to part a), show that the solution u(x, t) to the one-dimensional heat equation

偏微分方程代写
偏微分方程代写

subject to the boundary conditions

and the initial condition u(x, 0) = f (x), can be written in the form

偏微分方程代写
偏微分方程代写

Determine expressions for Xn(x) and Tn(t) and give integral expressions for any arbitrary constants. Show ALL steps in your calculation but do not evaluate any integrals. (You may assume without proof that the eigenfunctions Xn(x) are orthogonal on the interval [0, 1].)

[12 marks]

 

2. 偏微分方程代写

a) Show that the function

偏微分方程代写
偏微分方程代写

is a solution to Laplace’s equation

(1)

[4 marks]

b) Use the chain rule for partial derivatives to show that

偏微分方程代写
偏微分方程代写

where r, θ denote the usual polar coordinates. Using this expression, and applying the chain rule again,derive an expression for
in terms of the derivative

[12 marks]

c) Laplace’s equation (1), rewritten in polar coordinates is 偏微分方程代写

(2)

Convert the function  into a function in polar coordinates and show that it satisfies Laplace’s equation (2) in polar coordinates.

[4 marks]

3.We want to approximate the solution u(x) of the convection-diffusion equation:

subject to the boundary conditions 偏微分方程代写

u(0) = 1, u(1) = 0.

Let Uj denote an approximation to u(xj), where xj = jh are uniformly spaced grid points, with j = 0, 1, . . . , N and

偏微分方程代写
偏微分方程代写

a) Define the first centered finite difference δu(xj)at a point x xj and derive the second centered finite difference δ2u(xj). Hence derive the following approximation to the second derivative:

偏微分方程代写
偏微分方程代写

[4 marks]

 

b) By combining the following approximation

with the one given in part a), derive the following finite difference scheme for the convection-diffusion equation

偏微分方程代写
偏微分方程代写

State the boundary conditions that the approximations must satisfy.

[4 marks]

c) Choose N = 4 intervals and let f = 1. Derive (but do not solve) the linear system of equations that must be solved to obtain the approximations U1U2U3.

[3 marks] 偏微分方程代写

d) Define the term truncation errorTj at a point xj and show that, using suitable Taylor series expansions,

[9 marks]

4.

a) For the scalar function g(x, y, z) = x + 3y2z + 2y and the vector function G = 2zy2i zj + 3xyk, calculate the gradient ∇g, and the curl ∇ × G.

[4 marks]

b) Prove that, for any scalar function f f (x, y, z), ∇ × (∇f ) = 0. 偏微分方程代写

[4 marks]

c) Byusing spherical coordinates show that

where V denotes the sphere of radius two, centered at the origin.

[5 marks]

d) Use the standard formula to construct a unit vector nˆ that points in the outward normal direction to the surface S of the sphere in part c). Hence, evaluate the surface integral

where F xi yj zk. (You may use without proof the standard formula for the surface area of a sphere.) 偏微分方程代写

[4 marks]

e) State the Divergence Theorem and, using your answers to parts c) and d), show that it is satisfied for the vector function F xi yj zk, where V is the volume enclosed by a sphere of radius

[3 marks]

 
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