2008/12/16

Series Solutions of DE. Special Functions

Chih-Ping LinNational Chiao Tung Univ.cplin@mail.nctu.edu.tw

(Section 4.1 and 4.?)

Series Solutions of DE-Special Functions

Homogeneous Linear ODEw/ constant coefficients

Solved by linear algebraic method

Solutions are elementaryFunctions known from calculus

Homogeneous Linear ODEw/ variable coefficients

Solved by Series Method

Solutions are special functionNot discussed in calculus

Example of homogeneous linear ODE w/ variable coefficients

Lenendre’s eqnHypergeometric eqnBessel’s eqn.

Study of Sturm-Liouville problem leads to orthogonality of functions and orthogonal expansion of function(Study of eigen-value problem leads to orthogonality of vectors and orthogonal expansion of vector)

In general can be written in a general form called “Sturm-Liouville eqn”

OutlinePower series methodLegendre’s eqn. Lengendre Polynomial Pn(x)Frobenius methodBessel’s eqn. Bessel’s Function Jv(x)Bessel’s function of second kind Yv(x)Sturm-Liouville problems. Orthogonal FunctionsOrthogonal Eigenfunction expansion

Ideas & Questions Power Series Method - Introduction(4.1)

The standard basic method for solving linear ODE w/ variable coefficients.Power series in power of (x-x0)

Power series in powers of x

Familiar examples of power series and the Maclaurin series

Idea of Power Series MethodFor a given linear ODEy’’+p(x)y’+q(x)y=0

First represent p(x) and q(x) by power series in powers of x (or x-x0 if solutions in power of x-x0 are wanted)Next we assume a solution in the form of a power series w/ unknown coefficients

Substitution and collect like powers of x and equate the sum of the coefficients of each occurring power of x to 0

Example of simple equations Theory of the Power Series Method(4.2)

A few relevant facts on Power Seires from Calculus

Operations on Power Series needed in the method

Existence of Power Series solutions

Basic Concept

Power Series of the form

The nth partial sum is

The remainder is

……. (*)

•If for some x=x1, this sequence converges, say

then, the series is called “convergent” at x1, S(x1) is called the value of sum of the power series at x1

• If the sequence diverge at x=x1, the series is called“divergent” at x=x1

•In the case of convergence, for any positive ε, thereis an N (depending on ε) such that

Convergence intervals, Radius of Convergence1. The series (*) always converge at x=x0. This is of no

practical interest2. If there are further values of x for which (*) converges,

these values from an interval, called the convergenceinterval of the form

R can be obtained from either of the formulas

Ex1 (Case1 of convergence only at the center)

Ex2 (Case 2 of convergence in a finite interval)

Ex3 (Case 3 of convergence for all x)

Operations on Power SeriesTerm-wise differential

Term-wise addition

Term-wise multiplication

Vanishing of all coefficientsIf a power series has a positive radius of convergence and a sum that is identically zero through out its interval of convergence, then each coefficient of the series must be zeroShifting summation indices

Existence of Power Series Solutions

Consider y’’+p(x)y’+q(x) = r(x) …………….(**)If the coefficients p and q and r on the right side have power series representations, then (**) has power series solutionsNOTE: A real function is called analytic at point x =x0 if it

can be represented by a power series in powers of x-x0 w/ R>0

Theorem (Existence of Power Series Solutions)If p, q, r in (**) are analytic at x=x0, then every solution of (**) is analytic at x=x0, and can thus be represented by a power series in powers of x-x0 with radius of convergence R > 0.

To answer Q1

Legendre’s Eqn, Legender Polynomial Pn(x)Legendre’s equation

This eqn arises in numerous problem, particularly in boundary value problem for spheres. The parameter n is a given real number

rewrite (1) as

p and q are analytic at x=0, thus, we may apply power series method

Lengendre PolynomialThe resulting solution of Legendre’s eqn is called Legendre polynomial of degree n denoted by Pn(x)

Frobenius Method (4.4)

Indical Eqn. indicating the form of solutions

Our method will yield a basis of solutions• One of the two solutions will always be of the form (2),

where r is a root of (3)• The form of the other solution will be indicated by the

indical equation. There are three cases1) Distinct roots not differing by an integer2) A double root3) Roots differing by an integer 1, 2, 3, …

Suppose that the diff. eqn. (1) satisfies the assumption intheorem 1. Let r1 and r2 be the roots of the indical eqn. (3).Then we have the following three casesCase 1. Distinct roots not differing by an integer

A basis is y1(x) = xr1(a0+a1x+a2x2+…)y2(x)=xr2(A0+A1x+A2x2+…)

Case 2. Double roots r1=r2=rA basis is y1(x) = xr(a0+a1x+a2x2+…)y2(x)=y1(x)lnx+xr(A0+A1x+A2x2+…)

Case 3 Roots differing by an integery1(x) = xr1(a0+a1x+a2x2+…)y2(x)=ky1(x)lnx+xr2(A0+A1x+A2x2+…)

Where the roots are denoted that r1-r2>0, and k may turnout to be zero.

Typical ApplicationsTechnically, the Frobenius method is similar to power series method, once roots of the indical eqn. have been determined. However, Theorem 2 merely indicates the general form of a basis, and a second solution can often be obtained more rapidly by reduction of order (sec. 2.1)

Bessel’s Equation. Bessel Function