Chapter 17

Boundary Value Problems

Standard Form of Two-Point Boundary Value Problem

1 2

1 1 1 1 2 1

2 2 2 1 2 2

( )( , , , , ), 1, 2, ,

At point : ( , , , , ) 0, 1, 2, ,

At point : ( , , , , ) 0, 1,2, ,

ii N

j N

k N

dy xf x y y y i N

dxx B x y y y j n

x B x y y y k n

In total, there are n1+n2=N boundary conditions.

Example, Eigenvalue Problem

2

2

( )( )

with (0) 0, (1) 0

d y xy x

dxy y

1 2 3

1 2

2 3 1

3

1 1 3 3

Let , ',

y

0

(0) 0, (1) 0, (0) (1)

y y y y y

y

y y y

y

y y y y

is also unknown.

Shooting Method

Use Newton-Raphson to get the target

The Shooting Method (start)

• At the starting point x1 we have n1

conditions to satisfy, thus we have n2=N-n1 freely variable starting parameters

• Let

be the initial values of y which is parametrized by n2 V-values without constraint.

21 1 1 2( ) ( ; , , , ) 1, 2, ,i i ny x y x V V V i N

The Shooting Method (discrepancy)

• Using any standard ODE solver to find the solution at x2. Compute a difference between the required boundary condition and actual value:

• Our objective is to search the root of F with respect to V.

2 2 2 2( ) ( , ( )) 1, 2, ,k kF B x x k n V y

Newton-Raphson for Root2

2

1

1

1 1

0 ( ) ( ) ( )

or 0 where

Update by

Compute by finite diference

( , , , ) ( , , , )

ni

i i jj j

iij

j

i j j i ji

j j

FF F V O V

V

FJ

V

F V V V F V VF

V V

V V V

F J V

V V V V V J F

J

An Example2

1 2

1 2

1 1

2 1

1 2

1 2 1

2 1 1

3, (0) 4, (1) 1

2Let , , rewrite as

(0) 4, (1) 13

21, 2

So we let (0), (0) 4

( ) ( ) (1) 1

y y y y

y y y y

y yy y

y y

n n N

V V y y

F V B y y

Relaxation Methods

Work with finite differences

Difference Method

• Consider

• Discretize the interval xj=a+jh and equation

( ) ( ), ( ) , ( )y q x y g x y a y b

0

1 12

1

,

2( ) ( ), 1, 2, ,

i i ii i i

n

y

y y yq x y g x i n

hy

The difference equations form a linear system Ay = b if the equation is linear.

Reviews

• Errors in numerical calculations• Linear systems• Interpolations• Integrations of definite integrals and differential

equations• Random number and Monte Carlo• Least squares and optimizations• Root finding• Sorting, computational complexity• FFT

Topics not Covered

• Eigenvalue problems, Ax=x• Evaluation of special functions

• Integral equations

• Partial differential equations (PDE)

Review Problem 1Mix and Match

Problems

Solve Ax = b

Det(A)

Approximate f(x) by polynomial

Integrate

Fit a straight line

Find minima

Solve ODE or PDE

Estimate error of fit

Compute condition number

Traveling salesman

Nonlinear equation

Methods

Crout’s Newton-Raphson Relax

Gaussian quadrature Trapzoidal rule

Romberg method LU SVD FFT

Lagrange formula Neville’s shooting

Gauss-Jordan elimination Euler

Back/forward substitution Bulirsch-Stoer

Spines Steepest descent Symplectic

Conjugate gradient Secant Metropolis

Golden section Bisection Heapsort

Simulated annealing Bit-reversal

Wavelet Variance Normal equation

Levenberg-Marquardt Runge-Kutta

Review Problem 2

• Error in numerical calculation, catastrophic cancellation

• Discuss the pitfalls of solving the quadratic equation by the standard formula

2

2

1,2

0

4

2

ax bx c

b b acr

a

Read the IEEE 754 webpage article “What every computer scientist should know about floating-point arithmetic”.

Review Problem 3

• To interpolate or extrapolate with polynomials, we do Neville’s algorithm with Lagrange interpolation formula. Discuss what is required (computationally) if we consider rational functions (This is known as Padé approximation).

20 1 2

20 1 2

NND

D

a a x a x a x

b b x b x b x

Review Problem 4

• Work out the steps for Conjugate Gradient and Steepest Descent for minimum of the following function

1( )

2where

3 2 2,

2 6 8

T Tf x x Ax b x

A b

The minimum is at

(2,2).

Review Problem 5

• Discuss the general features (trajectories in phase space) of the ordinary differential equation of a pendulum. What method should be best used to solve it numerically?

2

2sin( ) 0

d

dt

Other Routine Problems

• Gauss elimination• Solve LU decomposition with Crout’s• Do interpolation with Neville’s• The errors in well-known integration rules• Do quick sort or heap sort on data• Newton Raphson iteration formula• Normal equation for least-square• Euler/midpoint methods for ODE

Conceptual Type of Problems

• Existence of solutions

• O(N?) of an algorithm, why fast or slow

• Accuracy of methods O(h?)

• Basic analysis techniques (e.g. Taylor series expansion)

• Key idea in an algorithm or a method (e.g., conjugate gradient, gaussian quadrature, Romberg, quick sort, FFT, etc)