York University

MATH3271

FW02

Final Examination Saturday, Dec. 14, 8:30 - 11:30

· Note the formula sheet at the end of this test paper.

· There are a total of 11 questions on this paper and the marks are given at the beginning of each question.

· The total number of marks on this paper is 120.

1. (4 marks) Answer the following questions in the answer booklet.

(a) The graph below represents a periodic function. What is its period?

Is it even or odd or neither even nor odd ?

(b) The graph below represents a periodic function. What is its period?

Is it even or odd or neither even nor odd ?

2, (6 marks) If f(x) and g(x) represent arbitrary T-periodic functions:

(a) Prove f(x)g(x) is a T-periodic function;

(b) Are the following statements TRUE or FALSE? Put your answer in the answer booklet.

i) f(px) has period pT;

ii) ;

iii) .

3. (11 marks)

(a) Find the Fourier series for the function f(x) = cos²(x) on the interval -p < x < p.

(b) If F(x) is the Fourier series for the function

what is the value of F(0)?

(c) If C(x) represents the half-range cosine series for x(1-x) on the interval 0 < x < 1 what does C(x) represent on the interval -1 < x < 0 ?

4. (12 marks) The d'Alembert solution of the one-dimensional wave equation

subject to the boundary conditions u(0, t) = 0, u(a, t) = 0 and the initial conditions u(x, 0) = f(x), u_t(x,0) = g(x) is given by

where f*(x) and g*(x) are the odd extensions of f and g, respectively.

(a) Show that this solution satisfies the boundary condition u(0, t) = 0;

(b) Show that this solution satisfies the initial condition u(x, 0) = f(x);

(c) If g(x) = x sin(p x/a) for 0 < x < a define g*(x) for all x.

5. (7 marks) Find the steady-state temperature distribution in a bar of length L if the one end is kept at 20 degrees while the other end is insulated.

6. (20 marks)

(a) Evaluate the following expressions using the vector operators in cylindrical polar coordinates:

i) Ñ (r cos j);

ii) Ñ ´ (sin j u_r + cos j u_f).

(b) Evaluate the following expressions using the vector operators in spherical polar coordinates:

i) Ñ× u_r ;

ii) Ñ ² (r² sin² q).

7. (8 marks)

(a) Below is a plot of the the first four Legendre polynomials P₀, P₁, P₂ and P₃. The curves are labelled from 1 to 4. Indicate in the answer booklet which label is associated with each of the four Legendre polynomials.

(b) From the above graph, what can you deduce about the following functional values:

(i) P_n(1);

(ii) P_n(-1);

(iii) P_n(0)?

8. (20 marks)

(a) Using separation of variables, show that the eigenfunctions of the Helmholtz equation for a circular disk

subject to the boundary condition u(a, j) = 0, 0 < j < 2 p are given by

J_m(l_mnr) cos(m j) and J_m(l_mnr) sin(m j).

Give an explicit expression for the eigenvalues l_mn.

(b) Use the results of part (a) to solve the Poisson equation

subject to the boundary condition u(a, j) = 0, 0 < j < 2 p.

Give an explicit expression for the generalized Fourier coefficients.

9. (8 marks)

(a) The spherical Bessel functions j_n(lr) and y_n(lr) of order n are solutions of the parametric differential equation

r² R¢¢ + 2r R¢ +(l² r² - n(n+1))R = 0

Use the method of Frobenius to analyse the behaviour of the solutions of this equation near r = 0 when n is a positive integer.

(b) What is the relationship between the spherical Bessel functions j and the regular Bessel functions J?

(c) If the solution in part (a) which is finite at r = 0 is subject to the boundary condition R(a) = 0 write down the explicit expression for the eigenvalues l.

10. (13 marks) The Helmholtz equation in a sphere

subject to the boundary condition v(a, q, j) = 0 has eigenfunctions j_n(l_njr)Y_n,m(q, j) and eigenvalues k = l²_nj where the spherical harmonics Y_n,m(q, j) are orthonormal functions on the surface of a sphere. Use these eigenfunctions to find the solution of the wave equation in a sphere

with boundary condition u(a, q, j, t) = 0

and initial conditions u(r, q, j, 0) = f(r, q, j) and u_t(r, q, j, 0) = g( r, q, j).

Give explicit expressions for the generalized Fourier coefficients.

11. (11 marks)

A Sturm-Liouville problem is of the form

[p(x)y¢]¢ + [q(x) + lr(x)] y = 0, a < x < b

subject to the boundary conditions

c₁y(a) + c₂y¢(a) = 0

d₁y(b) + d₂y¢(b) = 0

(a) Put the differential equation for the spherical Bessel functions given in question 9 (including the boundary conditions of part (c)) into the form of a Sturm-Liouville problem. Clearly identify each of the functions and parameters of this problem.

b) Is this a regular or a singular Sturm-Liouville problem? Give reasons for your answer.

(c) Using the results of part (a) write down the orthogonality relationship for the spherical Bessel functions of order n.

THE END