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.

(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;

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), ut(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 ur + cos j uf).

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

i) Ñ× ur ;

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) Pn(1);

(ii) Pn(-1);

(iii) Pn(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

Jm(lmnr) cos(m j) and Jm(lmnr) sin(m j).

Give an explicit expression for the eigenvalues lmn.

(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 jn(lr) and yn(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 jn(lnjr)Yn,m(q, j) and eigenvalues k = l2nj where the spherical harmonics Yn,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 ut(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