Sturm-Liouville Problem and Orthogonality Properties of Bessel Functions and Legendre

Sturm-Liouville problem and Orthogonality properties of Bessel functions and Legendre polymonials

Objective

To study Sturm-Liouville problem.

To study orthogonality properties of Bessel functions and Legendre polymonials.

To Study the Fourier-Bessel expansion and Fourier-Legendre expansion of continuous functions

Modules

Module I- Sturm-Liouville problem

Module II- Orthogonality property of Bessel functions

Module III- Orthogonality property of Legendre Polynomials

Module I- Sturm-Liouville problem

A differential equation of the form

(1)

is known as Sturm-Liouville equation where is a real constant.

Instead of initial conditions, this equation is usually subjected to the boundary conditions on the interval [x1, x2] as

(2)

(3)

where a1, a2, b1 and b2 are constants such that either a1 or a2 is not zero and either b1 or b2 is not zero.

For the problem (1) subject to (2) and (3), it turns out that the non-trivial solutions exist only for specific values of . These values are termed as eigen values or characteristic values of equation (1). The non-trivial solutions of (1) corresponding to the eigen values are termed as eigen functions or characteristic functions. If be the ith eigen value, then the non-trivial solution yi corresponding to is the ith eigen function.

Orthogonality property of eigen functions

Let ym and yn be two eigen functions of the sturm-liouville problem corresponding to eigen values and respectively then the eigen functions are orthogonal with respect to the weight function r(x) over the interval [x1, x2].

Proof.

Since the two distinct eigen values and the corresponding eigen functions are the solutions of the sturm-liouville problem (1), we have

Multiplying the first equation by yn and the second by ym and then subtracting, we obtain

Integrating from x1 to x2

, integrating by parts

, using boundary conditions

` ,

Example 1. For the Sturm-Liouville problem , find the eigen functions.

Solution.

Let , then the general solution is

Using boundary conditions we obtain c1 = c2 = 0. Hence y(x) = 0is not an eigen function.

Let , then the general solution is

Since , we have .

Since , we have

Thus the eigen values are and taking c2 = 1, we obtain the eigen functions as

By the above theorem, the eigen functions sin x, sin 2x, sin 3x,… are orthogonal on the interval with respect to the weight function r(x) = 1.

Remark: Under suitable transformations all the special differential equations (i.e., Bessel equation, Legendre equation etc.) can be reduced to Sturm-Liouville problem.

Consider the Bessel equation

Let x = kt, then and .

Substituting and in the Bessel equation, we get with y = y(kt)

Replacing the variable t by x, we obtain

which is a Sturm-Liouville equation with

and .

Consider the Legendre equation

which can be written as

, where

Therefore Legendre equation is itself Sturm-Liouville equation with and .

Consider the general equation of the form

Let be an integrating factor of the first two terms.

Multiplying the given equation by the integrating factor we obtain the Sturm-Liouville form

where q(x) = Q(x).p(x) and r(x) = R(x).p(x).

Module II- Orthogonality property of Bessel functions

If and are roots of Jn(x) = 0, then

Proof.

Let and which will satisfy the Bessel equations

(1)

and (2)

Multiplying (1) by v/x and (2) by u/x and subtracting, we get

or .

Integrating both sides from 0 to 1, we obtain

(3)

Since and , we have

and .

Substituting these values in (3), we get

(4)

Let , then , since and are roots of the equation Jn(x) = 0

Hence from (4), we have

Now consider .

Then the right hand side of equation (4) gives form. To apply L’Hospital rule, let us consider as a constant and as a variable approaching. Then we have

, since .

Hence the proof.

Fourier-Bessel expansion

If f(x) be a continuous function having finite number of oscillations in (0, a), then we can expand f(x) as

where are the positive roots of the equation Jn(x) = 0.

To determine cj, multiply both sides by and integrate from 0 to a. From the orthogonality property of Bessel functions, all integrals on the right hand side will vanish except the one containing cj and we have

Therefore .

Putting j = 1, 2, 3, … we can find c1, c2, c3, … and hence the function f(x).

Example 2. Expand f(x) = x in [0, 2] in terms of , where are determined by .

Solution.

Let the Fourier-Bessel expansion of f(x) be

Multiplying both sides by and integrating with respect to x from 0 to 2, we obtain

Hence the required Fourier-Bessel expansion is

Example 3. If are the positive roots of the equation J0(x) = 0, prove that

Solution.

We know that if , then

Taking f(x) = 1, a = 1 and n = 0, we get

Therefore, we have

or .

Module III- Orthogonality property of Legendre Polynomials

The Legendre polynomial Pn(x) satisfy the following orthogonal property

Proof.

Case 1. When

Let the Legendre polynomials Pm(x) and Pn(x) satisfy the differential equations

(1)

and (2)

Multiplying (1) by Pn(x) and (2) by Pm(x) and subtracting, we get

Combining the first two terms, we get

Integrating from -1 to 1, we get

Case 2. When m = n

We have the generating function

Squaring both sides and then integrating with respect to x from -1 to 1, we get

(1)

Now

(2)

Also,

, using the first part

(3)

Using (2) and (3) in (1), we get

Equating the coefficient of z2n from both sides, we obtain

Example 4. Prove that .

Solution.

We have

(n + 1)Pn+1(x) = (2n + 1)xPn(x) – nPn-1(x).

Multiplying both sides by Pn-1(x), we get

(n + 1)Pn+1(x). Pn-1(x) = (2n + 1)xPn(x) .Pn-1(x) – nPn-12(x).

Integrating from -1 to 1, we get

, using orthogonality property

Fourier-Legendre expansion

If f(x) be a continuous function having continuous derivatives over the interval [-1, 1], then we can expand f(x) as

(1)

To determine cn, multiply both sides by and integrate from -1 to 1, then

The series (1) converges uniformly in [-1, 1] and is known as Fourier-Legendre expansion of f(x).

Example 5. Find the Fourier-Legendre series of the following function

f(x) = x3 – 3x + 4 in [-1, 1]

Solution.

Let

where

and so on.

Therefore

Summary

In the session we have discussed The Sturm-Lioville problem, orthogonality property of eigen functions and orthogonality property of Bessel functions and Legendre polynomials. We have seen that these special equations can be reduced to Sturm-Lioville problem by suitable transformations. Also we have discussed Fourier-Bessel expansion and Fourier-Legendre expansion.

Assignments

1. Find the eigen functions of the following Sturm-liouville problem and verify their orthogonality:

ii.

2. State and prove orthogonality property of Bessel functions.

3. Expand f(x) = x2 in the interval 0 < x < 1 in terms of functions where are determined by J2(x) = 0.

4. State and prove orthogonality property of Legendre polynomials.

5. Express f(x) = 5x3 – x in terms of Legendre polynomials.

Reference

1. Advanced Engineering Mathematics by E. Kreyszig, John Wylie & Sons, New York (1999).

2. Differential Equations by F. Simmons, Tata McGraw-Hill, New Delhi (1996).

3. Advanced Engineering Mathematics by Michael D. Greenberg, Pearson Education pvt. ltd. Delhi (2002).

Quiz

1. The eigen functions of Sturm-Liouville problem are

2. The Bessel equation can be reduced into Sturm-Liouville problem by the substitution

a. x2 = kt b. x = kt c. x = kt2

3. is true if

a. b. c.

4. The value of is

a. b. c.

5. The value of for is

a. 1 b. 0 c.

6. The value of is

a. 2/15 b. 2/5 c.0

7. If , then

Answers

1.a 2.b 3.c 4.c 5.b 6.a 7.b

Glossary

Differential equation: An equation containing ordinary differential coefficients is called an ordinary differential equation.

Solution: The relation between independent variable and the dependent variable from which a differential equation is formed is called the solution of the differential equation.

Orthogonal: Two curves are said to be orthogonal if their tangents cross at right angles at every point where the curves intersect.

Integration by parts: .

L’Hospital rule: Suppose that f(a) = g(a) = 0and that f and g are differentiable on an open interval containing a. Suppose also that on I if . Then , if the limit on the right exists (or is or ).

Bessel function: The differential equation is known as Bessel’s equation of order n. Here n is not necessarily an integer, and the particular solutions of Bessel’s equation are called Bessel functions of order n. The Bessel function of the first kind of order n is denoted by and is.

Legendre polynomial: The Legendre polynomials of degree n are given by , n = 0, 1, 2, …, where or according as n is even or odd.

Converges: By associating with a given series, a sequence, where Sn denotes the sum of first n terms of the series, i.e., Sn = u1 + u2 + … + un, for all n, Then the sequence is called the sequence of partial sums of the series and the partial sums, S1 = u1, S2 = u1 + u2, S3 = u1 + u2 + u3 and so on, may be regarded as approximations to the full infinite sum of the series. If the sequence of partial sum converges, then the series is regarded as convergent and lim Sn is said to be the sum of the series. If however, does not tend to a limit, we must take it that the sum of the infinite series does not exist. We express this fact by saying that the series does not converge.

Continuous function: A function f is continuous at an interior point x = c of its domain if .

FAQs

1. Explain Orthogonality property of eigen functions of Sturm-liouville problem.

Answer.

Proof.

Since the two distinct eigen values and the corresponding eigen functions are the solutions of the sturm-liouville problem (1), we have

Multiplying the first equation of yn and the second ym and then subtracting, we obtain

Integrating from x1 to x2

, integrating by parts

, using boundary conditions

` ,

2. Find the eigen functions of the periodic Sturm-liouville problem

and verify their orthogonality.

Answer.

Let , then the general solution is

Using boundary conditions we obtain and .

These gives

Thus the eigen values are and taking c1 = c2 = 1, we obtain the eigen functions as

and

i.e., 1, cos x, sin x, cos 2x, sin 2x, …

These eigen functions are orthogonal on the interval with respect to the weight function r(x) = 1.

3. If and are roots of Jn(x) = 0, then prove that

Answer.

Let and which will satisfy the Bessel equations

(1)

and (2)

Multiplying (1) by v/x and (2) by u/x and subtracting, we get

or .

Integrating both sides from 0 to 1, we obtain

(3)

Since and , we have

and .

Substituting these values in (3), we get

(4)

Let , then , since and are roots of the equation Jn(x) = 0

Hence from (4), we have

Now consider .

Then the right hand side of equation (4) gives form. To apply L’Hospital rule, let us consider as a constant and as a variable approaching. Then we have

, since .

Hence the proof.

4. Prove that Prove that .

Solution.

We have (2n + 1)xPn(x) = (n + 1)Pn+1(x) + nPn-1(x).

Replacing n by n + 1 and n – 1, we get

(2n + 3)xPn+1(x) = (n + 2)Pn+2(x) + (n+1)Pn(x) (1)

(2n - 1)xPn-1(x) = nPn(x) + (n -1)Pn-2(x). (2)

Multiplying (1) and (2),

(2n + 3) (2n - 1)x2 Pn+1(x). Pn-1(x) = n (n + 2)Pn+2(x). Pn(x) + (n + 2)(n - 1)Pn+2 Pn-2+ n(n+1)Pn2(x) + (n + 1)(n - 1)PnPn-2

Integrating from -1 to 1, we get

, using orthogonality property