Session 11
Boundary Value Problems
for
Ordinary Differential Equations

Ernesto Gutierrez-Miravete

December 2, 1999

1 Linear Shooting

A boundary value problem (BVP) involving an ordinary differential equation of order two consists of determining the function y(x) for x Î [a,b] such that

y˘˘ = f(x,y,y˘)

subject to the boundary conditions

y(a) = a

and

y(b) = b

Theorem 11.1 (p. 625) If the function f(x,y,y˘) as well as śf/śy and śf/śy˘ are continuous in the set D = {(x,y,y˘) : a Ł x Ł b, -Ľ Ł y +Ľ, -Ľ Ł y˘+Ľ} and if śf/śy > 0 for all (x,y,y˘) Î D and |śf/śy| Ł M for all (x,y,y˘) Î D where M is a constant, then the BVP has a unique solution.

Example 11.1.1 (p. 626).

The shooting method is based on the idea of replacing the original BVP by an equivalent set of initial value problems (IVPs) which are solved using standard IVP methods..

Corollary 11.2 (p. 626) If the BVP is linear , i.e.

y˘˘ = p(x) y˘+ q(x) y + r(x), a Ł x Ł b, y(a) = a, y(b) = b

and satisfies

i) p(x), q(x), and r(x) are continuous on [a,b],

ii) q(x) > 0 on [a,b],

then the problem has a unique solution which is given by

y(x) = y₁(x) +

b- y₁(b)

y₂(b)

y₂(x)

where y₁(x) and y₂(x) are, respectively, the solutions to the following IVPs

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

y˘˘ = p(x) y˘+ q(x) y, a Ł x Ł b, y(a) = 0, y(a)˘ = 1

Algorithm 11.1 (p. 627) Linear Shooting.

Example 11.1.2 (p. 629).

2 Nonlinear Shooting

If the BVP is nonlinear, its solution cannot be obtained by linear combination of the solutions of the associated IVPs. Instead one must produce a sequence of solutions y(x,t) corresponding to an IVP of the form

y˘˘(x,t) = f(x,y(x,t),y˘(x,t)), a Ł x Ł b, y(a,t) = a, y˘(a,t) = t

where t is a parameter of the method whose values must be selected so that

y(b,t) - b = 0

If the preceeding differential equation is differentiated with respect to t and z(x,t) = (śy/śt) (x,t) the second associated IVP is obtained

z˘˘(x,t) =

śf

śy

z(x,t) +

śf

śy˘

z˘(x,t), a Ł x Ł b, z(a,t) = 0, z˘(a,t) = 1

Therefore, nonlinear shooting proceeds by solving the above two IVP's and Newton's method to determine subsequent iterate values for t_k as follows

t_k = t_k-1 -

y(b,t_k-1) - b

z(b,t_k-1)

Algorithm 11.2 (p. 635) Nonlinear Shooting.

Example 11.2.1 (p. 637).

3 Finite Difference Methods for Linear Problems

Consider the linear BVP

y˘˘ = p(x) y˘+ q(x) y + r(x), a Ł x Ł b, y(a) = a, y(b) = b

Introduce a mesh in [a,b] by dividing the interval into N+1 equal subintervals of size h. This produces two boundary mesh points x₀ and x_N+1, and N interior mesh points x_i, i = 1,2,..,N. The values y(x₀) = y(a) and y(x_N+1) = y(b) are known but the values y(x_i), i = 1,2,..,N must be determined.

From the Taylor expansions for y(x_i+1) and y(x_i-1) the following centered difference formula is obtained

y˘˘(x_i) =

h²

[y(x_i+1) - 2 y(x_i) + y(x_i-1)] +

h²

y⁽⁴⁾(x_i)

for some x_i Î (x_i-1,x_i+1).

Similarly, an approximation for y˘(x_i) is

y˘(x_i) =

2 h

[y(x_i+1) - y(x_i-1)] -

h²

y⁽³⁾(h_i)

for some h_i Î (x_i-1,x_i+1).

If the higher order terms are discarded from the above formulae and the approximations are used in the original differential equation, the second order accurate finite difference approximation to the BVP becomes a system of simulataneous linear algebraic equations

- (

w_i+1 - 2 w_i + w_i-1

h²

) + p(x_i) (

w_i+1 - w_i-1

2 h

) + q(x_i) w_i = -r(x_i)

or, alternatively

-(1 +

p(x_i)) w_i-1 + (2 + h² q(x_i)) w_i -(1-

p(x_i)) w_i+1 = - h² r(x_i)

for i = 1,2,..,N.

In matrix notation the system becomes

A w = b

where A is a tridiagonal matrix.

Theorem 11.3 (p. 641) If p,q, and r are continuous and q(x) > 0 on [a,b] the problem A w = b has a unique solution provided h < 2/L where L = max_{a Ł x Ł b} |p(x)|. Further, if y⁽⁴⁾ is continuous on [a,b] the truncation error is O(h²).

Algorithm 11.3 (p. 641) Linear Finite Difference Method.

Example 11.3.1 (p. 642).

Example 11.3.2 (p. 643). Same as 11.3.1 but with Richardson extrapolation.

4 Finite Difference Methods for Nonlinear Problems

Let the BVP

y˘˘ = f(x,y,y˘), y(a) = a, y(b) = b

be such that

1) f, śf/śy = f_y and śf/śy˘ = f_y˘ are continuous on D,

2) śf/śy ł d > 0

3) Constants k and L exist such that k = max |śf/śy| and L = max |śf/śy˘|

Then, a unique solution exists. Introducing finite differences and since f is nonlinear, the following system of nonlinear algebraic equations is obtained

- (

w_i+1 - 2 w_i + w_i-1

h²

) + f(x_i,w_i,

w_i+1 - w_i-1

2 h

) = 0

for each interior mesh point i = 1,2,..,N. The nonlinear system has a unique solution as long as h < 2/L.

The Jacobian matrix associated with this system is tridiagonal and it is given by

J(w₁,...,w_N)_ij =

ě
ď
ď
ď
ď
ď
í
ď
ď
ď
ď
ď
î

-1 +

f_y˘

i = j-1, j = 2,..,N

2 + h² f_y

i = j, j = 1,..,N

-1 +

f_y˘

i = j+1, j = 1,..,N-1

Newton's method applied to this problem requires first the solution of the linear system

J v^(k) = b

to produce v^(k) which is then added to w^(k-1) to obtain w^(k), i.e.

w^(k) = w^(k-1) + v^(k)

for each i = 1,2,..,N.

Algorithm 11.4 (p. 648) Nonlinear Finite Difference Method.

Example 11.4.1 (p. 649).

5 Rayleigh-Ritz Method

Instead of using finite difference approximation to the derivatives in the original differential equation one can invoke variational principles.

Theorem 11.4 Let p Î C¹[0,1], q,f Î C[0,1] with p(x) ł d > 0 and q(x) ł 0 for 0 Ł x Ł 1. Then, the function y(x) Î C²[0,1] with y(0) = y(1) = 0 is the solution to the differential equation

( p(x)

) + q(x) y = f(x)

if and only if y(x) is the unique function which minimizes the integral

I[y] =

ó
ő

1

0

{ p(x) [ y˘(x)]² + q(x) [ y(x)]² -2 f(x) y(x) } dx

In practice one searches for the solution among a set of functions produced by linear combination of a set of linearly independent and well defined basis functions f_i(x), i = 1,2,...,n. The solution y(x) is first approximated by ĺ_iⁿ c_i f_i(x) and then the coefficients c_i are chosen so that I[y] = I[ĺ_iⁿ c_i f_i(x)] is minimized, i.e.

śI

śc_j

= 0

for each j = 1,2,..,n.

Substituting y(x) ť ĺ_iⁿ c_i f_i(x) into I[y], differentiating and introducing the minimization condition leads to

Ac = b

where

a_ij =

ó
ő

1

0

[ p(x) f_i˘(x) f_j˘(x) + q(x) f_i(x) f_j(x) ] dx

are the entries of the n ×n stiffness matrix A,

b_i =

ó
ő

1

0

f(x) f_i(x) dx

is the load vector and c = (c₁,c₂,..,c_n)^t is the vector of unknown coefficients.

The simplest kind of basis functions is the set of piecewise linear polynomials defined by first introducing in [0,1] a collection of nodes x₀,x₁,...,x_n+1 with h_i = x_i+1 - x_i and then making

f_i(x) =

ě
ď
ď
ď
ď
ď
ď
í
ď
ď
ď
ď
ď
ď
î

0 Ł x Ł x_i-1

x - x_i-1

h_i-1

x_i-1 < x Ł x_i

x_i+1 - x

h_i

x_i < x Ł x_i+1

x_i+1 < x Ł 1

for each i = 1,2,..,n. This approximation is continuous but it is not differentiable. Note that f_i(x) f_j(x) = 0 and f_i˘(x) f_j˘(x) = 0 except when j is equal to i-1, i, or i+1 and this makes A tridiagonal. Evaluation of the matrix entries requires the calculation of 6 integrals for each node. One can approximate the functions p, q, and f by first order interpolating polynomials and then integrate numerically.

Algorithm 11.5 (p. 658) Piecewise Linear Rayleigh-Ritz.

Example 11.5.1 (p. 659).

Differentiability of the approximation can be obtained by using spline functions.

Algorithm 11.6 (p. 663) Cubic Spline Rayleigh-Ritz.

Example 11.5.2 (p. 665)

File translated from T_EX by T_TH, version 2.34.
On 29 Nov 1999, 18:48.
Updated: 1999-11-29, 18:49

Session 11 Boundary Value Problems for Ordinary Differential Equations

Ernesto Gutierrez-Miravete

December 2, 1999

1 Linear Shooting

2 Nonlinear Shooting

3 Finite Difference Methods for Linear Problems

4 Finite Difference Methods for Nonlinear Problems

5 Rayleigh-Ritz Method

Session 11
Boundary Value Problems
for
Ordinary Differential Equations