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## BVP of ODE

1

### NTNU

Tsung-Min Hwang December 20, 2003

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(2)

## BVP of ODE

2

1 – Mathematical Theories . . . 4

2 – Finite Difference Method For Linear Problems . . . 15

2.1 – The Finite Difference Formulation . . . 15

2.2 – Convergence Analysis . . . 19

3 – Shooting Methods . . . 22

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(3)

## BVP of ODE

3

The two-point boundary-value problems (BVP) considered in this chapter involve a

second-order differential equation together with boundary condition in the following form:

00

0

### y(a) = α, y(b) = β

(1)

The numerical procedures for finding approximate solutions to the initial-value problems can not be adapted to the solution of this problem since the specification of conditions involve two different points,

and

### x = b

. New techniques are introduced in this chapter for handling problems (1) in which the conditions imposed are of a boundary-value rather than an initial-value type.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(4)

## BVP of ODE

3

The two-point boundary-value problems (BVP) considered in this chapter involve a

second-order differential equation together with boundary condition in the following form:

00

0

### y(a) = α, y(b) = β

(1)

The numerical procedures for finding approximate solutions to the initial-value problems can not be adapted to the solution of this problem since the specification of conditions involve two different points,

and

### x = b

.

New techniques are introduced in this chapter for handling problems (1) in which the conditions imposed are of a boundary-value rather than an initial-value type.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(5)

## BVP of ODE

3

The two-point boundary-value problems (BVP) considered in this chapter involve a

second-order differential equation together with boundary condition in the following form:

00

0

### y(a) = α, y(b) = β

(1)

The numerical procedures for finding approximate solutions to the initial-value problems can not be adapted to the solution of this problem since the specification of conditions involve two different points,

and

### x = b

. New techniques are introduced in this chapter for handling problems (1) in which the conditions imposed are of a boundary-value rather than an initial-value type.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(6)

## BVP of ODE

4

### 1 – Mathematical Theories

Before considering numerical methods, a few mathematical theories about the two-point boundary-value problem (1), such as the existence and uniqueness of solution, shall be discussed in this section.

Theorem 1 Suppose that

### f

in (1) is continuous on the set

0

0

and that

and

### ∂y

0 are also continuous on

. If

1.

0

for all

0

, and

2. a constant

exists, with

0

0

,

0

### ) ∈ D

,

then (1) has a unique solution.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(7)

## BVP of ODE

5

When the function

0

### )

has the special form

0

0

### + q(x)y + r(x),

the differential equation become a so-called linear problem. The previous theorem can be simplified for this case.

Corollary 1 If the linear two-point boundary-value problem

00

0

satisfies

1.

, and

### r(x)

are continuous on

, and

2.

on

### [a, b]

,

then the problem has a unique solution.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(8)

## BVP of ODE

6

Many theories and application models consider the boundary-value problem in a special form as follows.

00

### y(0) = 0, y(1) = 0

We will show that this simple form can be derived from the original problem by simple

techniques such as changes of variables and linear transformation. To do this, we begin by changing the variable. Suppose that the original problem is

00

(2)

where

. Now let

### λ = b − a

and define a new variable

### λ (x − a).

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(9)

## BVP of ODE

7

That is,

. Notice that

corresponds to

, and

corresponds to

. Then we define

with

. This gives

0

0

### (x)

and, analogously,

00

0

2

00

2

2

### f (a + λt, z(t)).

Likewise the boundary conditions are changed to

and

### z(1) = y(b) = β.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(10)

## BVP of ODE

8

With all these together, the problem (2) is transformed into

00

2

(3)

Thus, if

### y(x)

is a solution for (2), then

### z(t) = y(a + λt)

is a solution for the boundary-value problem (3). Conversely, if

### z(t)

is a solution for (3), then

λ1

### (x − a))

is a solution for (2).

Example 1 Simplify the boundary conditions of the following equation by use of changing variables.

00

2

### y(1) = 3, y(4) = 7

Solution: In this problem

, hence

### λ = 3

. Now define the new variable

13

, hence

, and let

### z(t) = y(x) = y(1 + 3t)

. Then

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(11)

## BVP of ODE

9

2

2

###  ,

and the original equation is reduced to

00

2

### z(0) = 3, z(1) = 7

To reduce a two-point boundary-value problem

00

### z(0) = α, z(1) = β

to a homogeneous system, let

then

00

00

, and

and

### u(1) = z(1) − β = 0

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(12)

## BVP of ODE

10

Moreover,

### g(t, z) = g(t, u + α + (β − α)t) ≡ h(t, u).

The system is now transformed into

00

### u(0) = 0, u(1) = 0

Example 2 Reduce the system

00

t

### z(0) = −7, z(1) = −5

to a homogeneous problem by linear transformation technique.

Solution: Let

### u(t) = z(t) − [−7 + (−5 + 7)t] = z(t) − 2t + 7.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(13)

## BVP of ODE

11

Then

, and

00

00

t

t

### = [5u + sin(3u)]e

t

The system is transformed to

00

t

### u(0) = u(1) = 0

Example 3 Reduce the following two-point boundary-value problem

00

2

2

### + xy y(3) = 7, y(5) = 9

to a homogeneous system.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(14)

## BVP of ODE

12

Solution: In the original system,

. Let

### λ = b − a = 2

and

define a new variable

Let the function

. Then

00

2

00

2

2

2

2

2

### − 12t − 6]

The original problem is first transformed into

00

2

2

Next let

or equivalently,

### z(t) = u(t) + 2t + 7.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(15)

## BVP of ODE

13

Then

00

2

2

2

2

### + 36t + 64].

The original problem is transformed into the following homogeneous system

00

2

2

### u(0) = u(1) = 0

Theorem 2 The boundary-value problem

00

### y(0) = 0, y(1) = 0

has a unique solution if

### ∂y

is continuous, non-negative, and bounded in the strip

and

### −∞ < y < ∞

.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(16)

## BVP of ODE

14

Theorem 3 If

### f

is a continuous function of

in the domain

and

such that

1

2

1

2

### |, (K < 8).

Then the boundary-value problem

00

### y(0) = 0, y(1) = 0

has a unique solution in

### C[0, 1]

.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(17)

## BVP of ODE

15

### 2 – Finite Difference Method For Linear Problems

We consider finite difference method for solving the linear two-point boundary-value problem of the form

00

0

### y(a) = α, y(b) = β.

(4)

Methods involving finite differences for solving boundary-value problems replace each of the derivatives in the differential equation by an appropriate difference-quotient approximation.

### 2.1 – The Finite Difference Formulation

First, partition the interval

into

### n

equally-spaced subintervals by points

0

1

n

n

### = b

. Each mesh point

### x

i can be computed by

i

with

where

### h

is called the mesh size. Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(18)

## BVP of ODE

15

### 2 – Finite Difference Method For Linear Problems

We consider finite difference method for solving the linear two-point boundary-value problem of the form

00

0

### y(a) = α, y(b) = β.

(4)

Methods involving finite differences for solving boundary-value problems replace each of the derivatives in the differential equation by an appropriate difference-quotient approximation.

### 2.1 – The Finite Difference Formulation

First, partition the interval

into

### n

equally-spaced subintervals by points

0

1

n

n

### = b

. Each mesh point

### x

i can be computed by

i

with

where

### h

is called the mesh size. Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(19)

## BVP of ODE

15

### 2 – Finite Difference Method For Linear Problems

We consider finite difference method for solving the linear two-point boundary-value problem of the form

00

0

### y(a) = α, y(b) = β.

(4)

Methods involving finite differences for solving boundary-value problems replace each of the derivatives in the differential equation by an appropriate difference-quotient approximation.

### 2.1 – The Finite Difference Formulation

First, partition the interval

into

### n

equally-spaced subintervals by points

0

1

n

n

### = b

. Each mesh point

### x

i can be computed by

i

with

where

### h

is called the mesh size. Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(20)

## BVP of ODE

15

### 2 – Finite Difference Method For Linear Problems

We consider finite difference method for solving the linear two-point boundary-value problem of the form

00

0

### y(a) = α, y(b) = β.

(4)

Methods involving finite differences for solving boundary-value problems replace each of the derivatives in the differential equation by an appropriate difference-quotient approximation.

### 2.1 – The Finite Difference Formulation

First, partition the interval

into

### n

equally-spaced subintervals by points

0

1

n

n

.

Each mesh point

### x

i can be computed by

i

with

where

### h

is called the mesh size. Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(21)

## BVP of ODE

15

### 2 – Finite Difference Method For Linear Problems

We consider finite difference method for solving the linear two-point boundary-value problem of the form

00

0

### y(a) = α, y(b) = β.

(4)

Methods involving finite differences for solving boundary-value problems replace each of the derivatives in the differential equation by an appropriate difference-quotient approximation.

### 2.1 – The Finite Difference Formulation

First, partition the interval

into

### n

equally-spaced subintervals by points

0

1

n

n

### = b

. Each mesh point

### x

i can be computed by

i

with

where

### h

is called the mesh size.

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(22)

## BVP of ODE

16

At the interior mesh points,

i, for

### i = 1, 2, . . . , n − 1

, the differential equation to be approximated satisfies

00

i

i

0

i

i

i

i

### ).

(5)

The central finite difference formulae

0

i

i+1

i−1

2

(3)

i

(6)

for some

### η

i in the interval

i−1

i+1

, and

00

i

i+1

i

i−1

2

2

(4)

i

(7)

for some

### ξ

i in the interval

i−1

i+1

### )

, can be derived from Taylor’s theorem by expanding

i.

Let

### u

i denote the approximate value of

i

i

. If

4

### [a, b]

, then a finite difference method with truncation error of order

2

### )

can be obtained by using the approximations Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(23)

## BVP of ODE

16

At the interior mesh points,

i, for

### i = 1, 2, . . . , n − 1

, the differential equation to be approximated satisfies

00

i

i

0

i

i

i

i

### ).

(5)

The central finite difference formulae

0

i

i+1

i−1

2

(3)

i

(6)

for some

### η

i in the interval

i−1

i+1

,

and

00

i

i+1

i

i−1

2

2

(4)

i

(7)

for some

### ξ

i in the interval

i−1

i+1

### )

, can be derived from Taylor’s theorem by expanding

i.

Let

### u

i denote the approximate value of

i

i

. If

4

### [a, b]

, then a finite difference method with truncation error of order

2

### )

can be obtained by using the approximations Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(24)

## BVP of ODE

16

At the interior mesh points,

i, for

### i = 1, 2, . . . , n − 1

, the differential equation to be approximated satisfies

00

i

i

0

i

i

i

i

### ).

(5)

The central finite difference formulae

0

i

i+1

i−1

2

(3)

i

(6)

for some

### η

i in the interval

i−1

i+1

, and

00

i

i+1

i

i−1

2

2

(4)

i

(7)

for some

### ξ

i in the interval

i−1

i+1

### )

, can be derived from Taylor’s theorem by expanding

i.

Let

### u

i denote the approximate value of

i

i

. If

4

### [a, b]

, then a finite difference method with truncation error of order

2

### )

can be obtained by using the approximations Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(25)

## BVP of ODE

16

At the interior mesh points,

i, for

### i = 1, 2, . . . , n − 1

, the differential equation to be approximated satisfies

00

i

i

0

i

i

i

i

### ).

(5)

The central finite difference formulae

0

i

i+1

i−1

2

(3)

i

(6)

for some

### η

i in the interval

i−1

i+1

, and

00

i

i+1

i

i−1

2

2

(4)

i

(7)

for some

### ξ

i in the interval

i−1

i+1

### )

, can be derived from Taylor’s theorem by expanding

i.

Let

### u

i denote the approximate value of

i

i

.

If

4

### [a, b]

, then a finite difference method with truncation error of order

2

### )

can be obtained by using the approximations Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(26)

## BVP of ODE

16

At the interior mesh points,

i, for

### i = 1, 2, . . . , n − 1

, the differential equation to be approximated satisfies

00

i

i

0

i

i

i

i

### ).

(5)

The central finite difference formulae

0

i

i+1

i−1

2

(3)

i

(6)

for some

### η

i in the interval

i−1

i+1

, and

00

i

i+1

i

i−1

2

2

(4)

i

(7)

for some

### ξ

i in the interval

i−1

i+1

### )

, can be derived from Taylor’s theorem by expanding

i.

Let

### u

i denote the approximate value of

i

i

. If

4

### [a, b]

, then a finite difference method with truncation error of order

2

### )

can be obtained by using the approximations

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(27)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

, respectively.

Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

. Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

. In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved. The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(28)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

### )

, respectively. Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

. Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

. In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved. The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(29)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

### )

, respectively. Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

.

Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

. In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved. The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(30)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

### )

, respectively. Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

. Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

.

In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved. The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(31)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

### )

, respectively. Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

. Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

. In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved.

The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(32)

## BVP of ODE

17

0

i

i+1

i−1

and

00

i

i+1

i

i−1

2

for

0

i

and

00

i

### )

, respectively. Furthermore, let

i

i

i

i

i

i

### ).

The discrete version of equation (4) is then

i+1

i

i−1

2

i

i+1

i−1

i

i

i

### , i = 1, 2, . . . , n − 1,

(8)

together with boundary conditions

0

and

n

### = β

. Equation (8) can be written in the

form

i

i−1

2

i

i

i

i+1

2

i

(9)

for

. In (8),

1

2

### , . . . , u

n−1 are the unknown, and there are

### n − 1

linear equations to be solved. The resulting system of linear equations can be expressed in the matrix form

### Au = f,

(10)

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(33)

## BVP of ODE

18

where A =

−2 − h2q1 1 −

h 2 p1 1 + h

2p2 −2 − h2q2 1 −

h 2p2

. .. . .. . ..

. .. . .. . ..

1 + h

2pn−2 −2 − h2qn−2 1 −

h

2pn−2 1 + h

2pn−1 −2 − h2qn−1

 ,

1

2

.. .

n−2

n−1

and

2

1

h2

1

2

2

.. .

2

n−2

2

n−1

h2

n−1

### 

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(34)

## BVP of ODE

19

Since the matrix

### A

is tridiagonal, this system can be solved by a special Gaussian elimination in

2

### )

flops.

Theorem 4 Suppose that

, and

### r(x)

in (4) are continuous on

, and

on

### [a, b]

. Then (10) has a unique solution provided that

, where

a≤x≤b

.

### 2.2 – Convergence Analysis

We shall analyze that when

### h

converges to zero, the solution

### u

i of the discrete problem (8) converges to the solution

### y

i of the original continuous problem (5).

### y

i satisfies the following system of equations

i+1

i

i−1

2

2

(4)

i

i

i+1

i−1

2

(3)

i

i

i

i

(11) for

### i = 1, 2, . . . , n − 1

. Subtract (8) from (11) and let

i

i

i, the result is

i+1

i

i−1

2

i

i+1

i−1

i

i

2

i

### , i = 1, 2, . . . , n − 1,

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(35)

## BVP of ODE

19

Since the matrix

### A

is tridiagonal, this system can be solved by a special Gaussian elimination in

2

### )

flops.

Theorem 4 Suppose that

, and

### r(x)

in (4) are continuous on

, and

on

### [a, b]

. Then (10) has a unique solution provided that

, where

a≤x≤b

.

### 2.2 – Convergence Analysis

We shall analyze that when

### h

converges to zero, the solution

### u

i of the discrete problem (8) converges to the solution

### y

i of the original continuous problem (5).

### y

i satisfies the following system of equations

i+1

i

i−1

2

2

(4)

i

i

i+1

i−1

2

(3)

i

i

i

i

(11) for

### i = 1, 2, . . . , n − 1

. Subtract (8) from (11) and let

i

i

i, the result is

i+1

i

i−1

2

i

i+1

i−1

i

i

2

i

### , i = 1, 2, . . . , n − 1,

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(36)

## BVP of ODE

19

Since the matrix

### A

is tridiagonal, this system can be solved by a special Gaussian elimination in

2

### )

flops.

Theorem 4 Suppose that

, and

### r(x)

in (4) are continuous on

, and

on

### [a, b]

. Then (10) has a unique solution provided that

, where

a≤x≤b

.

### 2.2 – Convergence Analysis

We shall analyze that when

### h

converges to zero, the solution

### u

i of the discrete problem (8) converges to the solution

### y

i of the original continuous problem (5).

### y

i satisfies the following system of equations

i+1

i

i−1

2

2

(4)

i

i

i+1

i−1

2

(3)

i

i

i

i

(11) for

### i = 1, 2, . . . , n − 1

. Subtract (8) from (11) and let

i

i

i, the result is

i+1

i

i−1

2

i

i+1

i−1

i

i

2

i

### , i = 1, 2, . . . , n − 1,

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(37)

## BVP of ODE

19

Since the matrix

### A

is tridiagonal, this system can be solved by a special Gaussian elimination in

2

### )

flops.

Theorem 4 Suppose that

, and

### r(x)

in (4) are continuous on

, and

on

### [a, b]

. Then (10) has a unique solution provided that

, where

a≤x≤b

.

### 2.2 – Convergence Analysis

We shall analyze that when

### h

converges to zero, the solution

### u

i of the discrete problem (8) converges to the solution

### y

i of the original continuous problem (5).

### y

i satisfies the following system of equations

i+1

i

i−1

2

2

(4)

i

i

i+1

i−1

2

(3)

i

i

i

i

(11) for

### i = 1, 2, . . . , n − 1

.

Subtract (8) from (11) and let

i

i

i, the result is

i+1

i

i−1

2

i

i+1

i−1

i

i

2

i

### , i = 1, 2, . . . , n − 1,

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(38)

## BVP of ODE

19

Since the matrix

### A

is tridiagonal, this system can be solved by a special Gaussian elimination in

2

### )

flops.

Theorem 4 Suppose that

, and

### r(x)

in (4) are continuous on

, and

on

### [a, b]

. Then (10) has a unique solution provided that

, where

a≤x≤b

.

### 2.2 – Convergence Analysis

We shall analyze that when

### h

converges to zero, the solution

### u

i of the discrete problem (8) converges to the solution

### y

i of the original continuous problem (5).

### y

i satisfies the following system of equations

i+1

i

i−1

2

2

(4)

i

i

i+1

i−1

2

(3)

i

i

i

i

(11) for

### i = 1, 2, . . . , n − 1

. Subtract (8) from (11) and let

i

i

i, the result is

i+1

i

i−1

2

i

i+1

i−1

i

i

2

i

### , i = 1, 2, . . . , n − 1,

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(39)

## BVP of ODE

20

where

i

(4)

i

i

(3)

i

### ).

After collecting terms and multiplying by

2, we have

i

i−1

2

i

i

i

i+1

4

i

Let

1

2

n−1

T and

k

. Then

2

k

k

k

k−1

k

k+1

4

k

and, hence

2

k

k

k

k−1

k

k+1

4

k

k

k

4

### kgk

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(40)

## BVP of ODE

20

where

i

(4)

i

i

(3)

i

### ).

After collecting terms and multiplying by

2, we have

i

i−1

2

i

i

i

i+1

4

i

Let

1

2

n−1

T and

k

. Then

2

k

k

k

k−1

k

k+1

4

k

and, hence

2

k

k

k

k−1

k

k+1

4

k

k

k

4

### kgk

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

(41)

## BVP of ODE

20

where

i

(4)

i

i

(3)

i

### ).

After collecting terms and multiplying by

2, we have

i

i−1

2

i

i

i

i+1

4

i

Let

1

2

n−1

T and

k

.

Then

2

k

k

k

k−1

k

k+1

4

k

and, hence

2

k

k

k

k−1

k

k+1

4

k

k

k

4

### kgk

Department of Mathematics – NTNU Tsung-Min Hwang December 20, 2003

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