lecture 4 10 short

8/12/2019 Lecture 4 10 Short

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Orthogonality Sturm-Liouville problems Eigenvalues and eigenfunctions

Introduction to Sturm-Liouville Theory

Ryan C. Daileda

Trinity University

Partial Differential EquationsApril 10, 2012

Daileda Sturm-Liouville Theory
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Inner products with weight functions

Suppose that w(x) is a nonnegative function on [a, b]. Iff(x) and

g(x) are real-valued functions on [a, b] we define their innerproduct on [a, b] with respect to the weight w to be

f, g= ba

f(x)g(x)w(x) dx.

We say f and g areorthogonal on [a, b] with respect to theweight w if

f, g= 0.

Remarks:The inner product and orthogonality depend on the choice ofa, band w.

When w(x)1, these definitions reduce to the ordinaryones.



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Examples

1 The functionsfn(x) = sin(nx) (n= 1, 2, . . .) are pairwiseorthogonal on [0, ] relative to the weight function w(x)1.

2 LetJm be the Bessel function of the first kind of order m, andlet mn denote its nth positive zero. Then the functionsfn(x) =Jm(mnx/a) are pairwise orthogonal on [0, a] withrespect to the weight function w(x) =x.

3 The functions

f0(x) = 1, f1(x) = 2x, f2(x) = 4x2 1, f3(x) = 8x3 4x,

f4(x) = 16x4

12x2

+ 1, f5(x) = 32x5

32x3

+ 6x

are pairwise orthogonal on [1, 1] relative to the weightfunction w(x) =

1 x2. They are examples ofChebyshev

polynomials of the second kind.

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Series expansions

We have frequently seen the need to express a given function as a

linear combination of an orthogonal set of functions. Ourfundamental result generalizes to weighted inner products.

Theorem

Suppose that

{f1, f2, f3, . . .

}is an orthogonal set of functions on

[a, b] with respect to the weight function w . If f is a function on[a, b] and

f(x) =n=1

anfn(x),

then the coefficients an are given by

an =f, fnfn, fn =

ba

f(x)fn(x)w(x) dx

b

a f2n(x)w(x) dx

.


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Remarks

The series expansion above is called a generalized Fourierseries forf, andan are thegeneralized Fourier coefficients.

It is natural to ask:

Where do orthogonal sets of functions come from?

To what extent is an orthogonal set complete, i.e. whichfunctions fhave generalized Fourier series expansions?

In the context of PDEs, these questions are answered bySturm-Liouville Theory.


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Sturm-Liouville equations

A Sturm-Liouville equation is a second order linear differential

equation that can be written in the form

(p(x)y)+ (q(x) + r(x))y= 0.

Such an equation is said to be in Sturm-Liouville form.

Here p, qand rare specific functions, and is a parameter.Because is a parameter, it is frequently replaced by othervariables or expressions.

Many familiar ODEs that occur during separation of

variables can be put in Sturm-Liouville form.

Example

Show that y+ y= 0 is a Sturm-Liouville equation.

We simply take p(x) =r(x) = 1 and q(x) =0.Daileda Sturm-Liouville Theory

Orthogonality Sturm Liouville problems Eigenvalues and eigenfunctions


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Example

Put the parametric Bessel equation

x2y+xy+ (2x2 m2)y = 0

in Sturm-Liouville form.

First we divide by x to get

xy+y (xy)

+

2xm

2

x

y= 0.

This is in Sturm-Liouville form with

p(x) =x, q(x) =m2

x , r(x) =x,

provided we write the parameter as 2.


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Example

PutLegendres differential equation

y 2x1 x2 y

+

1 x2 y= 0


First we multiply by 1 x2 to get(1 x2)y 2xy

((1x2)y)

+y= 0.


p(x) = 1 x2, q(x) = 0, r(x) = 1,

provided we write the parameter as .


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Example

Put Chebyshevs differential equation

(1 x2)y xy+n2y= 0


First we divide by

1

x2 to get

1 x2 y x

1 x2y

(

1x2 y)

+ n21 x2 y= 0.


p(x) =

1 x2, q(x) = 0, r(x) = 11 x2 ,

provided we write the parameter as n2

.Daileda Sturm-Liouville Theory

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Sturm-Liouville problems

A Sturm-Liouville problem consists ofA Sturm-Liouville equation on an interval:

(p(x)y)+ (q(x) + r(x))y = 0, a


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Orthogonality Sturm Liou ille problems Eigen alues and eigenfunctions

Regularity conditions

A regular Sturm-Liouville problem has the form

(p(x)y)+ (q(x) + r(x))y = 0, a


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g y p g g

Example

The boundary value problem

y+ y = 0, 0


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Example

The boundary value problem

x2y+xy+ (2x2 m2)y = 0, 0< x


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Eigenvalues and eigenfunctions

A nonzero function ythat solves the Sturm-Liouville problem

(p(x)y)+ (q(x) + r(x))y = 0, a


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Example

Find the eigenvalues of the regular Sturm-Liouville problem

y+ y = 0, 0


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Example

Find the eigenvalues of the regular Sturm-Liouville problem

y+ y = 0, 0< x


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The previous examples demonstrate the following generalproperties of a regular Sturm-Liouville problem

(p(x)y)+ (q(x) + r(x))y= 0, a


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Another general property is the following.

Theorem

Suppose that yjand ykare eigenfunctions corresponding todistinct eigenvaluesj andk. Then yjand ykare orthogonal on[a, b] with respect to the weight function w(x) =r(x). That is

yj, yk

=

b

a

yj(x)yk(x)r(x) dx = 0.

This theorem actually holds for certain non-regularSturm-Liouville problems, such as those involving Besselsequation.

Applying this result in the examples above we immediatelyrecover familiar orthogonality statements.

This result explains why orthogonality figures so prominentlyin all of our work.


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Examples

Example

Write down the conclusion of the orthogonality theorem for

y+ y = 0, 0


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Example

If m0, write down the conclusion of the orthogonality theoremfor

x2y+xy+ (2x2 m2)y = 0, 0< x

lecture 4 10 short

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