Mathematics 322Handout

Through February 27, 2005

Assignment 3: (Due Tuesday February 27)4.7: Q5 4.8: Q2, Q7Problems 3–5 in this set.

1. Consider the Sturm-Liouville problem: y′′ + λy = 0, y(π) = y(−π) and y′(π) = y′(−π).

a. Determine all eigenvalues and eigenfunctions of the Sturm-Liouville problem. What prop-erty do these eigenfunctions have?

b. (Fourier Series) Using the eigenfunctions determined above, obtain an orthogonal expansionfor the following function:

f(x) ={

x if − π < x < π0 elsewhere

c. Discuss the convergence of the Fourier series obtained in (b) if f(x) is extend to the real

line as a 2π-periodic functions. Determine also the value of the series∞∑1

(−1)n+1

2n− 1.

2. Obtain the Fourier series expansion for the 2π-periodic function:

f(x) ={

0 if − π < x ≤ 0x if 0 < x < π

Discuss the convergence of the Fourier series obtained.

3. Consider the Legendre’s equation: (1− x2)y′′ − 2xy′ + n(n + 1)y = 0 where n ∈ N.

a. Write the Legendre’s equation as a Sturm-Liouville equation.

b. The eigenfunction Pn(x) corresponding to the eigenvalue λn = n(n + 1) of the Sturm-Liouville problem in part (a) are called Legendre polynomials. They are given by the formula:

Pn(x) =[n/2]∑m=0

(−1)m (2n− 2m)!2nm!(n−m)!(n− 2m)!

xn−2m (n = 0, 1, 2, 3, . . . )

(Refer to the first few we discussed in class).Here [y] is the greatest integer less than or equal to y. The Legendre polynomials have the

orthogonal property∫ 1

−1Pm(x)Pn(x) dx =

(2

2m + 1

)δmn (m, n = 0, 1, 2, 3, . . . ).

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Find a Fourier-Legendre Series for the following function:

f(x) ={

1 if − 1/2 < x < 1/20 elsewhere

Homework:4.7: Q3 - Q9 4.8: Q1 - Q7 10.2 Q1 - Q16

Recall the Sturm-Liouville problem:

[r(x)y′]′ + [q(x) + λp(x)]y = 0 (1)

k1y(a) + k2y′(a) = 0 (2a)

l1y(b) + l2y′(b) = 0 (2b)

Here k1 and k2 are given constants, not both zero, and so are l1 and l2. On the intervala ≤ x ≤ b, we assume that p, q, r, r′ are real-valued continuous and also p(x) > 0. Then wehave the following results:

a. (Existence and reality) Eigenvalues λ exist and are real values. Each eigenvalue λn giveseigenfunctions yn.

b. (Orthogonality with respect to p(x)) If yn and ym are eigenfunctions corresponding todifferent eigenvalues. Then yn and ym are orthogonal on a ≤ x ≤ b with respect to p(x):∫ b

ap(x) · yn(x)ym(x)dx = 0

c. If r(a) = 0 then condition 2(a) can be dropped. If r(b) = 0 then condition 2(b) can bedropped. We call such Sturm-Liouville problem singular (as opposed to regular). In such cases,we require y and y′ to be bounded at the points for which r vanishes.

3. (a) Use the transformation x = et to show that the eigenfunctions of the Sturm-Liouville

problem: (xy′)′ +λ

xy = 0, y′(1) = 0, y(

√e) = 0 are given by:

yn(x) = cos((2n− 1)π lnx) (n = 1, 2, 3, . . . )

(b) What are the eigenvalues?Write down the orthogonality relations and the square of the norms of the functions in thefamily {yn : n ∈ N} defined above.

4. Write each of the following problems in the Sturm-Louiville form described above.

a. Legendre’s equation: (1− x2)y′′ − 2xy′ + p(p + 1)y = 0

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b. Bessel’s equation: x2y′′ + xy′ + (x2 − p2)y = 0 Hint: Multiply by 1/x.

c. Hermite’s equation: y′′ − 2xy′ + 2py = 0 Hint: Multiply by e−x2.

d. Chebyshev’s equation: (1− x2)y′′ − xy′ + p2y = 0 Hint: Multiply by (1− x2)−1/2.

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e. Laguerre’s equation: xy′′ + (1− x)y′ + py = 0 Hint: Multiply by e−x.

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5. Consider the Sturm-Liouville problem: (x3y′)′ + λxy = 0 y(1) = 0 y(e) = 0

It is known that the eigenfunctions of the above problem are given by yn(x) =1x

sin(nπ lnx)for n = 1, 2, 3, . . .

a. State the interval on which the problem is defined:

b. Find the norm of yn for n = 1, 2, 3, . . .

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