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Chapter 5 Series solutions of ODEs, Special function

Page history last edited by sidjaggi 13 years, 4 months ago

Chapter 5 Series solutions of ODEs, Special functions


1 Power series


A power series is an infinite series of the form


                                                            Formula

 

where

 

  • Formula, Formula, Formula, Formula are constants, the coefficients of the series.
  • Formula is a also a constant, the center of the series.

 

The Taylor series of a function Formula is also a power series. It is of the form


                                                            Formula

 

A Maclaurin series is a Taylor series with center Formula.


Example 1.1.

 

Maclaurin series


                                                            Formula

 

Exercise 1.1.

 

Find the Maclaurin series of Formula.


2 The power series method for solving ODEs


Given

 

                                                            Formula

 

  • First represent Formula, Formula and Formula by power series in powers of Formula (or Formula if solution in powers of Formula is wanted).
  • Next, assume a solution Formula in form of a power series of Formula with unknown coefficients:


                                                            Formula

 

  • Thus,


                                                            Formula

 

                              And substitute the power series from of Formula, Formula and Formula into (1) to solve for the unknown coefficients Formula, Formula, Formula, Formula, by comparing coefficients of Formula in the L.H.S. and R.H.S. of (1).

 

Example 2.

 

The power series method to solve Formula.

 

Solution.

 

First let the solution by of the form


                                                            Formula

Hence,

                                                            Formula

 

Then substitute into the ODE,


                                                            Formula

 

Grouping terms of Formula, we have


                                                            Formula

 

By comparing the coefficients of Formula in the L.H.S. and R.H.S. we gives


                                                            Formula

 

Thus,


                                                            Formula

 

Exercise 2.

 

Solve the ODE Formula by using the power series method.

 

3 Theory of the power series method


3.1 Basic concepts


Given a power series


                                                            Formula

 

  • Assume Formula is variable

 

  • Formula is the center
  • The coefficients Formula are real
  • Its Formulath partial sum is


                                                            Formula

          where Formula.

  • The remainder of Formula is

 

                                                            Formula

 

          For example, consider the geometric series


                                                            Formula

 

          we have


                                                            Formula

 

  • Thus, associated with Formula, there is a sequence of partial sums Formula
  • If for some Formula this sequence converges, say,

 

                                                            Formula


          then Formula is convergent at Formula, the number Formula is called the value or sum of Formula at Formula. And we have


                                                            Formula

 

  • For any positive Formula, Formula (which depends on the value of Formula) s.t.


                                                           Formula

 

3.2 Convergence interval and radius of convergence


With respect to the convergence of the power series there are three cases.
Case 1. The series always converges at Formula, because for Formula all its terms are 0, perhaps except for the first one, Formula. In exceptional cases Formula may be the only Formula for which converges. Such a series is of no practical interest.
Case 2. If there are further values of Formula for which the series converges, these values form an interval, called the convergence interval. If this interval is finite, its midpoint is Formula, so that it is of the form

 

                                                           Formula

 

and the series converges for all Formula such that Formula and diverges for all Formula such that Formula. This Formula is called the radius of convergence and it can be found by the ratio test or the root test, as below.


                                                           Formula

 

provided these limits exist and are not zero.
Case 3. The convergence interval may sometimes be infinite, i.e., the power series converges Formula. For instance, if either of the limits above is 0 (in other words, if Formula), this case occurs.


Example 3.2.1.

 

For Case 1, consider a series


                                                           Formula

 

we have Formula, and by the ratio test,


                                                           Formula

 

It only converges at Formula. Such a series is useless.


Example 3.2.2.

 

For Case 2, for the geometic series we have


                                                           Formula

 

In fact, Formula, and from the ratio test we obtain Formula. That is, the geometric series converges and represents Formula when Formula.

 

Try the following MAXIMA program to see how the Taylor series converges within the given interval, and diverges elsewhere....

 

load(draw);

tay(n, x) := block(
   [ts : taylor(1/(1-x__), x__, 0, n)],
   subst(x__=x, ts)
)$
with_slider(
   /* first two arguments are the parameter and parameter values */
   n, makelist(i, i, 1, 20),
   /* the rest of arguments are for plot2d command */
   [1/(1-x), '(tay(n, x))],
   [x, -2, 2],
   [y, -10, 10]
)$

 

Next, try the following MAXIMA program to see how the series centred around 3 converges/diverges...

 

load(draw);

tay(n, x) := block(
   [ts : taylor(1/(1-x__), x__, 3, n)],
   subst(x__=x, ts)
)$
with_slider(
   /* first two arguments are the parameter and parameter values */
   n, makelist(i, i, 1, 20),
   /* the rest of arguments are for plot2d command */
   [1/(1-x), '(tay(n, x))],
   [x, 0, 6],
   [y, -10, 10]
)$

 

Example 3.2.3.

 

For Case 3, in the case of the series


                                                           Formula

 

we have Formula. Hence by the ratio test,


                                                           Formula

 

so that the series converges for all Formula.

 

Example 3.2.4.

 

Hint for some problems
Find the radius of convergence of the series


                                                           Formula

 

Solution.

This is a power series of Formula with coefficients Formula, so that in (3b),


                                                           Formula

 

Thus Formula. Hence the series converges for Formula, i.e. Formula.


Exercise 3.2.

 

Determine the radius of convergence.

 

                                                           Formula

 

3.3 Existence of power series solutions and Real analytic functions


3.3.1 Real analytic function


A real function Formula is called analytic at a point Formula if it can be represented by a power series in powers of Formula with radius of convergence Formula.


3.3.2 Existence of power series solutions


Consider


                                                           Formula

 

If Formula, Formula and Formula in (4) are analytic at Formula, then every solution of the ODE is analytic at Formula and can thus be represented by a power series in powers of Formula with radius of convergence Formula.
For an ODE


                                                           Formula

 

the same is true if Formula and Formula are all analytic at Formula and Formula.


Exercise 3.3.

 

Consider the ODE

 

                                                           Formula

 

Does it have a power series solution? If yes, find it in the power of Formula.

 

 

 

 

 

 

ADVANCED TOPICS

 

4 Legendre's equation


This is Legendre's equation


                                                           Formula

 

where Formula is a given constant. A typical application of Legendre's Equation is for problems with spherical symmetry.

 

4.1 Solving Legendre's equation


Dividing (5) by the coefficient Formula of Formula, we see


                                                           Formula

 

The coefficients Formula and Formula of the new equation are analytic at Formula.
Hence Legendre's equation has power series solutions of the form


                                                           Formula

 

Substituting it into the ODE and comparing the coefficients of the powers of Formula of both sides, we obtain the general formula


                                                           Formula

 

This is called a recurrence relation or recursion formula. It gives each coefficient in terms of the second one preceding it, except for Formula and Formula, which are left as arbitrary constants.


                                                           Formula


and so on. By inserting these expressions for the coefficients into (6) we obtain


                                                           Formula


where


                                                           Formula

 

These series converge for Formula and linearly independent solutions. Hence it is a general solution of (5) on the interval  Formula.

 

Exercise 4.1.

 

Prove
(a) Legendre's ODE has a power series solution.
(b) The recursion formula above.

 

4.2 Legendre polynomials Formula

 

In various applications,

 

  • The power series solutions of ODEs reduce to polynomials, i.e. they terminate after finitely many terms.
  • For Legendre's equation this happens when Formula is a non-negative integer because then the right side of the recursion formula is zero for Formula, so that Formula, Formula, Formula, Formula. Hence if Formula is even, Formula reduces to a polynomial of degree Formula. If Formula is odd, the same is true for Formula.

 

These polynomials, multiplied by some constants, are called Legendre polynomials, Formula. The standard choice of a constant is


                                                           Formula

 

(and Formula if Formula).
Then solving for Formula in terms of Formula, that is,


                                                           Formula

 

Note: The choice of constants above makes Formula for every Formula. This motivates the choice.
For Formula we then obtain


                                                           Formula

 

Exercise 4.2.1.

 

Try to write down Formula, Formula and Formula. Find the general terms when Formula.

 

 

 

In general,

 

                                                           Formula

 

The resulting solution of Legendre's differential equation is called the Legendre polynomial of degree Formula and is denoted by Formula.
From the above we obtain


                                                           Formula

 

where Formula or Formula, whichever is an integer. The first few of these functions are


                                                           Formula

 

Figure 1: Legendre polynomials


Exercise 4.2.

 

Obtain Formula and Formula.

 

4.3 Applications of Legendre polynomials in physics1                              1http://en.wikipedia.org/wiki/Legendre_polynomials


Figure 2: Electric potenital


Consider the electric potential at point Formula,


                                                           Formula

 

Then, using the cosine law,


                                                           Formula

 

Since the Maclaurin series of


                                                           Formula

 

Let


                                                           Formula

 

                                                           Formula


For Formula, let


                                                           Formula

 

                                                           Formula

 

Consider the Legendre polynomials Formula and comparing to Formula and Formula, we have


                                                           Formula


Exercise 4.3.

 

Kreyszig 9th Edition Problem set 5.3 question 14(b)
Potential theory.

 

 

Let Formula and Formula be two points in space. Show that


                                                           Formula

 

This formula has applications in potential theory.

 

5 Frobenius method

 

Any ODE of the form


                                                           Formula


where Formula and Formula are analytic at Formula, has AT LEAST ONE SOLUTION that can be represented in the form


                                                           Formula


where the exponent Formula can be any real or complex number and Formula is choose such that Formula.
The equation also has a second solution s.t. the two solutions are linearly independent and the 2nd solution may be similar to the first one with a different Formula and different coefficients, or may contain a logarithmic term.


5.1 Indicial equation, Indicating the form of solutions


Multiple the standard form by Formula, we have


                                                           Formula


Expanding Formula and Formula in power series,


                                                           Formula


Differentiating this term by term


                                                           Formula


Substituting into the original ODE, and equating the sum of coefficients of each power of Formula to zero, we have a system of equations with coefficient Formula as unknowns. In particular, consider the coefficient of Formula


                                                           Formula


Since Formula, this gives


                                                           Formula


This is called the indicial equation of the ODE.
The Frobenius method yields a basis of solutions. One of the two solutions will always be of the from (12), where Formula is a root of the indicial equation. The other solution will be of a form indicated by the indicial equation. There are three cases:
Case 1. Distinct roots Formula and they do not differing by an integer. A basis is


                                                           Formula


and


                                                           Formula

 

Case 2. A double root Formula. A basis is


                                                           Formula


and


                                                           Formula


Case 3. Distinct roots Formula and differing by an integer. A basis is


                                                           Formula


and


                                                           Formula


Example 5.1.1.

 

For the Euler-Cauchy equation


                                                           Formula


substitution of Formula gives the auxiliary equation


                                                           Formula


which is the indicial equation and Formula is a very special form of (12). For different roots Formula, Formula we get a basis Formula, Formula, and for a double root Formula we get a basis Formula, Formula.
Accordingly, for this simple ODE, Case 3 plays no extra role.


Example 5.1.2.

 

illustration of Case 2.
Solve the ODE


                                                           Formula


Solution. Write the ODE in standard form.


Ex 5.1.1.

 

What do Formula and Formula become?


By inserting (12) and its derivatives into it, we obtain


                                                           Formula


The smallest power is Formula.

 

Ex 5.1.2.

 

Write down the indicial equation in this example and solve it.


By equating the sum of its coefficients to zero we have


                                                           Formula


Hence this indicial equation has the double root Formula.
We insert this value Formula into (15) and equate the sum of the coefficients of the power Formula to zero, obtaining


                                                           Formula


thus Formula. Hence Formula, and by choosing Formula we obtain the solution


                                                           Formula


To obtain the second independent solution Formula we use the method of reduction of order. Or we can use the above discussion, let Formula. Then using substitution to find Formula.
We obtain


                                                           Formula


(Please see section 2.5 of chapter 2 and try youself.)


Example 5.1.3.

 

Illustration of Case 3.
Solve the ODE


                                                           Formula


Solution. Substituting the Mclaurin series of the solution and and its derivatives into it, we have


                                                           Formula

 

                                                           Formula

 

In the first series we set Formula and in the second Formula, thus Formula. Then


                                                           Formula


Since Formula in the second series, the lowest power is Formula and the indicial equation is


                                                           Formula


The roots are Formula and Formula. They differ by an integer. This is Case 3.
From the recurrence relation above, with Formula we have


                                                           Formula


This gives the recurrence relation


                                                           Formula


Hence Formula, Formula, Formula successively. Taking Formula, we get as a first solution


                                                           Formula


Again, we use the method of reduction of order and we obtain


                                                           Formula

 

Or we use the above discussion, let Formula and use substitution to find Formula.

 

Exercise 5.1.

 

Find a basis of solutions for the following ODE by the Frobenius method.


                                                           Formula

 

6 Bessel's equation


Bessel's equation


                                                           Formula

 

  • Formula is real and non-negative
  • It can be solved by the Frobenius method
  • Bessel's equation often relates to problems of potentials which show cylindrical symmetry, e.g. heat conduction, electrical fields or membrane vibrations in cylinders.


Accordingly, we substitute the series


                                                           Formula


and its derivatives into the ODE. This gives


                                                           Formula


Take Formula in the first, second and fourth series, and to Formula in the third series. Then equating the sum of coefficients of Formula to zero, we have


                                                           Formula


From this we obtain the indicial equation


                                                           Formula


The roots are Formula and Formula. For Formula, (20b) reduces to Formula. Hence Formula since Formula.
From (c) in the recurrence relation above, since Formula, we have Formula. So we only need to deal with even-numbered coefficients.
Let Formula, we have


                                                           Formula


Solving for Formula gives the recursion formula


                                                           Formula


Hence


                                                           Formula


and so on, and in general


                                                           Formula

 

6.1 Bessel's function Formula for integer Formula


For Formula, the recurrence relationship becomes


                                                           Formula


For ease of further manipulation, we set


                                                           Formula


So,


                                                           Formula


With these coefficients and Formula we get a particular solution of the ODE as


                                                           Formula


Formula is called the Bessel function of the 1st kind of order Formula and it converges for all Formula.


Example 6.1.

 

For Formula we obtain from above, the Bessel function of order Formula


                                                           Formula

which looks similar to cosine.
For Formula we obtain the Bessel function of order Formula


                                                           Formula


which looks a little bit like a decaying sinusoidal function.

 

Figure 3: Bessel functions of the first kind Formula and Formula

 

6.2 Bessel Function Formula for Any Formula


By defining the Gamma function


                                                           Formula


By integration by parts we obtain


                                                           Formula


Now consider Formula,


                                                           Formula


Then consider Formula, we obtain FormulaFormulaFormula and in general


                                                           Formula


This shows that the gamma function does in fact generalize the fractorial function.
Now we take


                                                           Formula


Then (21) becomes


                                                           Formula


Since we have Formula,


                                                           Formula


so that


                                                           Formula


Hence


                                                           Formula


With these coefficients and Formula we get a particular solution of the ODE as


                                                           Formula


Formula is called the Bessel function of the 1st kind of order Formula and it also converges for all Formula.

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